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A Mathematical Analysis of Reaction Diffusion 
Systems in Chemical and Biological Reactors 
with Macro and Micro Structures 
A Thesis presented in partial fulfilment of the requirements for the degree of 
Doctor of Philosophy 
at 
Massey University 
by 
Aroon A. Parshotam, MSc(Hons), MPhil 
1992 

Declaration 
This thesis is submitted to Massey University and has not been submitted for a higher degree to 
any other University or Institution. 
Aroon Parshotam 
March,1992 
Acknowledgements 
I wish to express my sincere gratitude to my supervisors, Prof. Graeme Wake, Dr. Alex McNabb and 
Assoc. Prof. Rao Bhamidimarri. 
Thanks are due also to Dr. Vijay Bhaskar for his help during the numerical part of this thesis and to 
coUegues Dr. Robert Sisson, Dr. Paul Bonnington, Robert Crawford and Mark Byrne for supporting me 
during this study. 
Finally, I wish to thank my parents Kanti and Olga Parshotam for supporting us over all those years 
of study and my wife Kavita for showing such great love and understanding, particularly during the final 
stages of my PhD. 
ii  
Contents 
Declaration 
Acknowledgements 
Contents 
Abstract 
1 
2 
3 
General Introduction 
1.0 Introduction 
1.1 The Physical Problem 
1.2 The Plan of this Thesis 
1 
1.3 New Contributions of this Thesis 
1.4 Other Potential Modelling Areas 
1.5 Notes and Comments 
Model Development 
2.0 Introduction 
9 
2.1 Generalised Particle Reactor Model Development 
2.2 Notes and Comments 
The Unsteady S tate Problem 
3.0 Introduction 
13 
3.1 DefiniLions, NotaLion and some General Results for Linear Parabolic Equations 
3.2 Generalised Comparison Theorems 
3.3 Uniqueness of Solutions to the Unsteady State Problem 
3.4 Imbedding Results 
3.5 Stability of Solutions and Uniqueness of Solutions LO the Steady StaLe Problem 
3.6 Existence of Solutions to the Unsteady State Problem 
3.7 NoLes and Comments 
ii 
ill 
v 
1 
1 
4 
5 
6 
8 
9 
9 
1 1  
13 
13 
23 
46 
47 
5 1  
55 
86 
iii 
4 
5 
CONTENTS i v  
The Steady State Problem 88 
4.0 Introduction 88 
4.1 Definitions, Notation and some General Results for Linear Elliptic Equations 88 
4.2 Imbedding Results 97 
4.3 Existence of Solutions to the Steady State Problem 100 
4.4 Relationships bctwccn Solutions of thc Stcady Statc and Unstcady Statc Problems 127 
4.5 Notes and Comments 138  
The Linear Problem 
5.0 Introduction 
5.1 Matrix Notation 
140 
5.2 Mean Action Times, Time Lag Constants and Mean Residence Times 
5.3 Geometric Factorization 
5.4 Notes and Comments 
140 
141 
142 
145 
152 
6 Examples 1 53 
6.0 Introduction 153 
6.1 A Simple Method for Obtaining Good Bounds for Solutions of a Bioparticle Model 155 
6.2 Analytical Bounds to a Model of a Fluidised Bcd Biofilm Reactor (FBBR) 166 
6.3 Monotone Iteration in the Construction or Upper and Lowcr Solutions 175 
6.4 Uniqueness and Existence Theorems for a Model of an Artificial Kidney 192 
6.5 A Fluidised Bed Biofilm Reactor (FBBR) Model involving Multicomponents 203 
6.6 A Continuous Stirred Basket Reactor (CSTR) Model involving Multicomponents 21 1 
6.7 NOlcS and CommenL<; 224 
Bibliography and References 225 
Abstract 
This thesis is concerned with generalised models for biological and chemical reactors such as the tubular, 
fiuidised, fixed, packed, continuously stirred and trickle bed reactors. 
Suppose n chemical components at concentrations Cj (i = 1 ,2, . . ,n) are "diffusing" and reacting in a 
homogeneous incompressible fluid with a known velocity profile u(z) independent of Cj so that in the reactor 
region A, div u is zero. Immersed in the fluid may be a uniformly distributed population of particles which 
absorb these chemicals and act as local sites for reaction-diffusion phenomena. The particles are sources and 
sinks for the chemicals Cj in the fluid and these fluid concentrations govern the boundary conditions for the 
particle or local behaviour. 
A system of equations is set up as a general model for these complex interactions. The principle 
limitations of this model are firstly that u(t, z), the velocity profile in A is known and not coupled with the 
concentrations Cj in any way, and secondly the particles are assumed to be fixed relative to the coordinate 
system of z in A and sufficiently small so that a representative sample of them can be taken to be in a spatially 
constant concentration environment in A. 
The objectives of this thesis are genemlised comparison theorems for these systems which are used to 
prove uniqueness, existence, stability and other general qualitative features of such models. A number of 
examples from literature arc examined. 
Models conforming to the system described in this thesis have applications in biological wastewater 
treatment, �iochemical manufacture, urea removal by the compact artificial kidney and industrial fermentation 
processes. Other potential modelling areas concern fertiliser or pollutants diffusing in soil moisture and 
reacting with soils, oxidation with product formation in waste deposits and industrial ore reduction 
processes. There are many other industrial and environmental problems with similar interacting macro and 
micro structures. These include the catalytic cracking and synthesis processes in chemical industries ranging 
from the making of synthesis gas from coal to oil refining. 
v 
1 
General Introduction 
1.0 Introduction 
This chapter introduces the physical problem. In section 1 . 1 ,  we give some examples of chemical, 
m icrobiological and biochemical reactors, briefly discussing how and where they may be utilised and giving 
some details of their dynamical biochemical, m icrobiological and chemical aspects. Many of these reactors 
have in common macro and micro structures. This is an important theme of this thesis. Our principal 
motivating example in this study is the fluidised bed biofilm reactor (FBBR). In section l .2, we introduce the 
plan of this thesis and in section 1 .3 ,  discuss some of the new contributions presented. In section lA, we 
discuss some other potential modelling areas and in section 1 .5 ,  survey some relevant literature. 
1.1 The Physical Problem 
This thesis is concerned with generalised models for systems of reaction-diffusion equations based on the 
dynamics and structure of biological and chemical reactors. Typical of these reactors are: (a) the tubular 
reactor, within which homogeneous reactions are often conducted; (b) the packed bed reactor, often used to 
conduct heterogeneous catalyzed reactions; (c) the stirred vessel, operated batchwise, semibaLchwise, or 
continuously (CSTR); (d) the fluidised-bed reactor (FBR), and (e) the trickle bed reactor. These are 
schematized in FIG. 1 . 1  and are described in detail by CARBERRY [45]. 
co-lL-_---::--_--'�C C(z) 
(a) 
C 
Co-'-------' 
Co�C 
C(r.=) 
( b) 
.---�C 
( c) (d) (e) 
FIG. 1.1 Diverse reactor types: (a) tubular (b) packed bed (c) CSTR (d) nuidised bed (e) trickle bcd. 
1.1 THE PHYSICAL PROBLEM 2 
The problem of heat and mass transfer in packed and fixed bed reactors with irregularly shaped 
porous catalyst particles was studied extensively by ARIS [ 19]. In fact, much of this work in chemical reactor 
theory has contributed significantly to the mathematical theory of diffusion, reaction, heat transfer and mass 
transfer. 
An analysis of diffusion in  biological particles using this chemical reactor theory was first proposed 
by ATKINSON and DAOUD [25] .  where immobilised cell particles were treated as catalytic slabs. Other 
authors extended this chemical reactor analogy to microbiological and biochemical reactors such as the 
fluidised bed biofilm reactor (FBBR) (SHIEH el al. [26 1 -266]). Many of the well established procedures for 
heterogeneous chemical catalytic reactor modelling can be adopted to reactors such as the FBBR (RAO 
BHAMIDIMARRI et al. [242]). This  is  because there are analogies between chemical reactions and properties 
of certain biological systems. One such analogy is rather obvious: we may consider living organisms to be a 
dynamic structure built of molecules and ions, many of which react and diffuse. However, unlike most 
chemical reactions, enzymic reactions in microbiological and biochemical reactors are mainly isothermal 
(CHANG [68]). In general however, many concepts such as the effectiveness factors for general reaction rale 
forms have analogous definitions for chemical , microbiological and biochemical systems (SHIEH et al. [26 1 -
266]). The effectiveness factor (defined as the rate of reaction divided by the rate which would occur with no 
resistance to component transfer inside or outside the particles) is an important parameter in reactor 
designing. It has been useful as a rough estimate of inlraparticle diffusion effects in particles such as porous 
catalysts, flocs,  solid spherical supports and adsorbent particles. The effectiveness factor is also used in lieu 
of extensive numerical calculations in these particles. There are many other such analogies and it is now 
acceptable to consider microbiological and biochemical reactors with procedures and concepts developed for 
heterogeneous chemical catalytic reactor modell ing (ATKINSON [26] ,  BAILY and OLLIS [30]). Since this 
thesis is  concerned with generalised models for such reactors, we do not specify in great detail the parameters 
for every special system. However we do go into detail with the nuidised bed biofilm reactor (FBBR) as a 
motivating example in ordcr to get an understanding of the physical problem. 
The Fluidised Bed B iofihn Reactor (FBBR) 
The fluidised bed biofilm reactor (FBBR) is a biochemical processing system with applications to biological 
wastewater treatment and biochemical manufacture (JERIS et. al. [ 1 34] and ATKINSON [26]). It is  a high 
energy, high efficiency system in which the liquid to be treated is passed upwards through a bed of small 
particles such as activated carbon granules at velocities sufficient to fIuidise the bed. Each particle of support 
material (activated carbon or otherwise) provides surface area for biological growth, resulting in a biomass 
for the whole bed, an order of magnitude greater than conventional dispersed growth systems. This growth 
is initiated by seeding the bed with microorganisms to form what is known as a biofilm around the support 
particles which establishes a miniature reaction-diffusion controlled chemical microsystem. The very h igh 
growth support surface afforded by these bioparticles results in denitrification of volatile solid concentrations 
as high as 30,000 mg 1-1 and a bed detention time as low as 6 minutes for 99% nitrate removal. 
It has been demonstrated in numerous pilot studies to be cost effective and has also been investigated 
at least to pilot scale for all of the basic treatment processes, including carbon oxidation, nitrification and 
denitrification for a variety of domestic and industrial wastewuters. A photo or a Iluidised bed biorilm reactor 
(FBBR) is given in FIG. 1 .2. 
Models for all the reactors (except for the tubular reactor) discussed in this section have a system of 
reaction-diffusion equations appropriate to a microsystem coupled via boundary conditions to a system 
modelling a macrosystem composed of convection reaction-dirfusion equations with additional source terms 
1.1 THE PHYSICAL PROBLEM 3 
accounting for outputs from the particles. These convection reaction-diffusion equations provide the 
boundary conditions for the microsystem associated with the particles. 
S uch  models for one system or another have appeared in the literature frequently during the last 
decade (MULCAHY et ai. [ 1 99-202]; LIN [ 170]; LlAPUS and RIPPIN [169]) .  
They generally assume, as we do here that the fluid velocity distribution is divergence free, fixed and 
not coupled with the diffusion and chemical or biochemical processes. 
FIG. 1.2 The Fluidlsed )jed J110film Reactor (FIIIIR) 
In this thesis, an attempt is made to model such reactors both for the interpretat ion o f  performance 
characteristics and for successful process design. Mathematical models for such reactors are needed for the 
description of steiluy-state and dynamic behaviour for process design,  optimisation and control .  While 
insufficient detail can lead to a model incapable of accurately representing the reactor's response to changes in 
operating variables, too much detail can lead to a model that may be computationally impractical. From a 
purely analytical point of v iew, there is no more difficulty in increasing the number of model equations. 
However this may lead to an impractical model in thatu1ere may be too many parameters to identify from the 
1 .2 THE PLAN OF THIS THESIS 4 
operating data. The type of model and its level of complexity in representing the physical system, will depend 
on the use for which the model is being developed. We arc simply presenting a generalised model for such 
reactors. These models turn out to have interesting and useful structural and mathematical properties. 
The mass-heat analogy provides a useful basis for transferring mass transport results to heat flow 
problems. Structurally, the heat transport aspects are mathematically equivalent to another chemical 
component. Since our generalised models consider an arbitrary number of interacting chemical components, 
we may assume that one of these components is temperature. The theory developed in this thesis does not 
therefore exclude heat transfer considerations which are relevant to fluidised bed combustors and to mass 
transfer in nonisothermal reactors (see ARIS [20]) or even the fluidsed bed gasifier (MA [176]) .  Therefore, 
without loss of generality, we will only consider mass transfer models. 
For convenience, we call reactors with the macro and micro structures described in this section, 
"Particle Reactors". 
1.2 The Plan of this Thesis 
In Chapter 2 we set up the coordinate systems and equations representing a generalised model for such a 
particle reactor. A system of equations is presented as a general model for these complex interactions. The 
principle limitations of this model are firstly that U(l, z), the velocity profile in i\ is assumed to be known and 
not coupled with the bulk concentrations Cj in any way, and secondly the particles are fixed relative to the 
coordinate system of z in i\ and sufficiently smalI so that a representative sample of them can be taken to be 
in a spatially constant concentration environment in A 
In Chapter 3, we look at the unsteady state or time dependent problem. Generalised comparison 
theorems are developed and used to study questions of uniqueness, existence and other qualitative features of 
our models. We examine the stability of both the steady state and the unsteady state solutions and study links 
between stability behaviour and uniqueness of steady state solutions. The question of existence of solutions 
is examined by using some recently developed imbedding techniques (McNABB lIH6J). 
In Chapter 4, we look at the steady state or time independent problem. Here we study the question of 
existence of solutions by using the new techniques referred to in Chapter 3. We also study relationships 
between unsteady state or time dependent solutions and steady state or Lime independent solutions. 
In Chapter 5, we show that linear models for panicle reactors treated in this thesis are amenable to 
various algebraic and geometrical factorization transformations which dramatically reduce their 
dimensionality. 
Linear systems of convection reaction-diffusion equations for these reactors have a structure which 
allows a geometric factorization of steady state problems giving a significant reduction in their 
dimensionality. Moreover, convection dominated linear systems with quasisymmetric reaction terms may be 
further simplified by matrix transformations which uncouple the differential equations. The boundary 
conditions are also uncoupled when the diagonal diffusivity matrix D governing diffusion in the particle is a 
scalar multiple of the corresponding matrix H describing the diffusivity characteristic of the fluid boundary 
layers around the particles. The dominant transient behaviour of ule systems may be handled by establishing 
an analogous system of time independent equations for mean action time variables and higher moments. 
These equations have the same amenable structure. Outputs, time lags and various mean residence and first 
passage times associated with establishing steady outputs from a concentration free initial state, can be 
expressed in terms of the steady state solutions and mean action time variables. 
In Chapter 6, a number of reactor models from literature are examined as examples of the theory 
developed in this thesis. Since some of these sections are taken from completed published papers without a 
lot of change, there may be repetition in the model development and problem description. 
1.3 NEW CONTRIBUTIONS OF THIS THESIS 5 
At the end of each chapter is a section on notes and comments where relevant literature is reviewed 
and where future mathematical work and ideas are discussed. At the end of this thesis is a references and 
bibliography section which includes papers which have been referred to in this thesis and papers which are 
not referred to but are still relevant to the subject maller. 
1.3 New Contributions of this Thesis 
The model structure presented in this thesis is novel in the way systems of reaction-diffusion equations 
appropriate to the microsystem are coupled via boundary conditions to systems modelling the macrosystem 
composed of convection reaction-diffusion equations with additional source terms accounting for outputs 
from the particles. Despite the complexity of this coupling, it is shown that many standard and classic 
methods for obtaining stability, uniqueness and existence of unsteady state and steady state problems can be 
extended to this problem. The principal result which makes these extensions possible is a generalised 
comparison theorem similar in type to those for weakly coupled parabolic and elliptic systems. In addition we 
develop some new techniques for our problem which can also be applied to elliptic and parabolic systems 
coupled in other ways. 
Major results of this thesis are uniqueness, existence and stabil ity theorems for solutions to the 
unsteady state and steady state problems. It can be shown that such results may be obtained by imbedding the 
general system which may obey no monotone property into a system of twice the order which does possess a 
monotone properly. The monotone properties of  these imbedded systems suggest the use of monotone 
iterative methods in obtaining existence of solutions and this in tum suggests new methods for numerical 
computation of solutions. It can fortunately be shown that stability, uniqueness and existence may be implied 
in the original system. As a result, our computed solution in the imbedded system also g ives rise to a 
numerical solution of the original system . This imbedding idea is relatively new and is also useful in 
generalising many qualitative features Stich as the rclat.ionships between parabolic and elliptic equations to 
parabolic and elliptic systems of' equations. This idea is even shown to be useful in solving algebraic 
equations by monotone iteration. 
We show that l inear systems of convection reaction-diffusion equations for these reactors have a 
structure which allows a geometric factorization of steady state problems giving a significant reduction in 
their dimensionality. These equations are also amenable to algebraic uncoupling transformations which 
reduce the dimensionality of the problem and which simplify the tasks of obtaining analytic and numerical 
solutions or estimates. 
The recent concept of mean action times (MeN ABB and WAKE [190]) for scalar equations is  
generalised to systems of equations and factorization and uncoupling techniques may be also applied to an 
associated linear system for vectors composed of the mean action time variable for each chemical component. 
These vector functions give the time lags for the various chemical outputs of the system during its  transition 
from one steady output mode to another and the mean first passage times, mean particle residence times and 
higher moments associated with tracer pulse inputs of the chemicals. 
Such results also apply to systems of reaction-diffusion equations where there may be no coupl ing in 
the boundary conditions and also provides us with ways of uncoupling parabolic and elliptic systems. 
Our comparison theory is also shown to be useful in developing some techniques for obtaining upper 
and lower pointwise bounds to solutions. These prove to be favourable compared to results obtained by 
orthogonal collocation and standard finite difference methods. The bounds provide us with some qualitative 
features of solutions of such equations and also prove to be excellent as approximate solutions. 
1.4 OTHER POTENTIAL MODELLING AREAS 
1.4 Other Potential Modelling Areas 
6 
There are a myriad of applications for such macro micro structure models in different disciplines as diverse as 
geology and mining, medicine, agriculture, chemical and biochemical industries and environmental 
technology. We give a brief description and references of some of these that have come to our aLtention. 
Our first example concerns industrial fermentation processes which may be both aerobic and 
anaerobic. Such processes involve a general class of  chemical changes produced in organic compounds 
(substrates) through the activity of micro-organisms and include various alcohol fermentations and the 
production of acetic acid (vinegar), lactic acid, citric acid and gluconic acid, as well as acetone, butanol and 
glycerol .  Microorganisms have the potential to achieve almost any conversion , involving water soluble 
organic compounds by means of complex sequences of enzyme catalysed reactions. 
The basic fermenter types are usually discussed using terminology establised for chemical reactor 
theory, namely: (a) the batch fermenter (BF); (b) the continuous stirred-tank fermenter (CSTF); (c) the 
tubular fermenter (TF) and (d) the fluidised bed fermenter (FBF). 
Many biochemical conversions are not achieveable in the absence of microorganisms and continuous 
fermenters which use suspended organisms suffer from the fact that these organisms can simply be 'washed 
out'. Continuous stirred-tank fermenters (CSTF) are limit.ed in throughput as a result of this phenomena and 
tubular fermenters (TF) with suspended tlocs, become an impossible arrangement without a constant 
resupply of microorganisms at the inlet. An altemativc answer to the last problem is the tluidised bed 
fermenter (FBF) (ANDREWS and PRZEZDZIECKI [17], ATKINSON [261. PARSIIOTAM el at. [2241) which is a 
hybrid hetween the stirred tank fermenter and the tubular fermentcr, in which the microbial particles are 
suspended by the upnowing liquid stream and gravitational forces prevent them from being swept away 
( elutriated). 
The end products of industrial processes involving microorganisms are more microbial mass and 
various biochemicals, one or several of which may be recovered. The main classes of biochemical products 
from industrial fermentation processes are antibiotics, steroids, vitamins, enzymes and organic acids (AlBA 
el at. [2] ) .  The microbial mass produced concurrently with the biochemicals is largely a waste product but 
may be used as an animal feed supplement in view of its protein content (LOEWY and SIEKEVITZ [1 72]) . 
Occasionally, the production of microbial mass is a major objective, as with baker's yeast and the removal of 
organic impurities from polluted water. 
There are other potential modelling areas which are similar in principle to the FBBR and have a 
similar interacting structures as particle reactors. An example of such a system is the Rotating Cylinder (RC). 
This is a biological system where a cylinder is covered by layers of microorganisms held in place by 
extracellular material around a support rotating cylinder. As with the FBBR, ule biological rotating cylinder is 
also an ingeneous method of removing non-settleable organic contents of wastewater. It is however more 
ideal on a small scale. The system involves lowering a rotating cylinder into a stream of wastewater which is 
being pumped through the rotating cylinder bath. The cylinder acts as a medium that provides a large surface 
area for microbial growth and if it is not fully submerged, it gets wel l aerated. Substrate diffuses through this 
biofilm and the organic content is converted into growth of these organisms which can be separated more 
easily. There are also gases, such as C02 being produced by these organisms in aerobic respiration which 
escapes to the aLrnosphere. As with the FBBR, the cylinder may be temporarily removed and cleaned and the 
separated biomass wasted as excess sludge. The Rotating Biological Disk Contactor Plant is similar to the 
rotating cylinder but where slabs are covered by a thin biofilm consisting of layers of microorganisms held in 
place by extracellular material . This provides more surface area for biological film growth (see FIG. 1.3). 
Both these reactors involve reaction diffusion systems within the biofilms with boundary conditions given by 
concentrations in the bulk fluid. 
1 .4 OTHER POTENTIAL MODELLING AREAS 
Drive 
Motor 
'\ 
FIG. 1.3 The Rotating  Diological Contactor Plant 
7 
A medical application concerns modelling urea removal by the compact artificial kidney (LIN (170]) and an 
example of this system will be seen in Chapter 6. There are perfusion models of chemicals and tracers 
through various organs such as the liver and muscle tissue. These sometimes involve a hierachy of structures 
for instance in the muscle tissue model, chemicals in the blood arc transferred from the capillary system by 
diffusion into the interstitial fluid and then into the muscle cells. Each of these systems introduces a new 
dimension in the model. 
For petrochemical industry applications see ARIS [20, 21]. Here the usc of catalytic reactors and the 
consequcnt availability of many inexpensive chemicals, makes the synthesis of a variety of organic acids and 
solvents commercially aLLractive. By  contrast to the petrochemical industry, we find that studies into 
aternative fuel sources from agricultural products such as in the production of the solvents acetone, butanol 
and ethanol (ABE fermentation) in immobilised cell packed bed reactors from whey permeate also involve 
models with similar interacting macro and micro structures (QURESl-lI [236]) . 
Other potential modelling areas concern fertiliser or pollutants diffusing in soil moisture and reacting 
with soils. Interest in the transport of chemicals in soils is motivated by the potential of agricultural and other 
chemicals (fertilizers, pesticides, industrial wastes) to move from the soil surface through the unsaturated 
zone toward the groundwater table. Studies of solute transport in soil adsorption columns (v AN GENUCHTEN 
and PARKER [288]) also result in model equations with interacting macro and micro structures which are 
coupled in the boundary conditions. 
Solute transport studies involving layered media are also important for investigating how restricting 
layers affect rates of solute migration in the soil profile and, more generally, for examining the influence of 
soil heterogeneity on solute transport. The leaching of solutes in the unsaturated zone may be affected greatly 
by the presence of soil layers. Soil stratification is a natural phemenon that is common to many soils. Also, 
artificial barriers (e.g . ,  clay l iners) arc often used to slow down or prevent the movement of certain 
chemicals. There has been considerable attention focused on solute transport through homogeneous soils and 
some work on heterogenous or multiple layered soil profiles which are approximated by a series of 
homogeneous soil layers (SELIM et a/.[259] and LEIJ et a/.[165]) each having its unique physical and 
chemical characteristics. The resulLing mathematical models in literature usually involve one dimensional 
dispersion advection equations involving single chemicals such as a herbicide diffusing in up to three layers 
1 .5 NOTES AND COMMENTS 8 
(LEU et al.[165]). Boundary conditions are necessary in order to maintain the solute concentration continuity 
at the boundary between any two soil layers and there is much discussion about the boundary conditions 
(VAN GENUCHTEN and PARKER [288], LEU et al. [165]) . Usually the boundary condition at the interface of 
one layer is given by the solution to the dispersion advection equation in the adjacent layer. Such equations 
are also found in models of fluid flow through multiple porosity, multiple permeability media that do not 
necessarily have to be layered (LIU [171]). This problem is interesting because it does not involve interacting 
macro and micro structures found in this thesis but it still involves the coupling of reaction diffusion 
equations in the boundary conditions. We can apply our theory without difficulty to systems of reaction 
advection dispersion equations in multilayered soils, multiple porosity and multiple permeability media. 
These may all include solute adsorption and desorption. 
There are many industrial, geological and environmental problems concerned with interacting macro 
and micro structures. These include catalytic cracking and synthesis processes in chemical industries, ore 
reduction (MCNABB [184, 185]), groundwater and geothennal modelling, the making of synthesis gas from 
coal (KATO and WEN [136]), the mining of methane gas from coal beds, the oxidation with product 
formation in waste deposits, combustion theory and various oil refining processes. 
1.S Notes and Comments 
There is considerable literature on the fluidised bed biofilm reactor (FBBR). A history of its development, its 
design and operating conditions is given by RAO BHAMIDIMARRI [241]. Some other examples of fluidised 
bed reactors from li terature are given by ANDREWS [16,17], BOSMAN [42], CHUNG and WEN [71], 
COOPER and ATKINSON [77], DENAC et al. [82], FOSTER [92], HANCHER et al. [113], JERIS and OWENS 
[134], KORNEGAY and ANDREWS [147], MULCAHY et al. [199-202] and SHIEI-1 et al. [261-266]. 
There is considerable literature on the effectiveness factor for particles in chemical and bichemical 
reactors. Some examples are given by ANDREWS [18], ARIS  [21], BISCl Iopr [38], Cl-IANG [67], 
CHURCHILL l72], GoTO l107J, LEE and TSAO l163J, MOO-YOUNG and KOBA YAS I I l  [198], ROBERTS ami 
SATTERFIELD [249], Sl-I lE I I  et al. [261-266], STEWART and V ILLADSEN [275J and WADIAK and 
CARBONELL [300]. 
It should be noted that such mechanistic models described in this section are not the only models 
available and that control models that treat such particle reactors as a black box also prove to be very useful 
(KHANNA and SElNl'ELD [144]). 
Our results on reaction diffusion systems in this thesis may also be applied to population dynamics 
(LADDE et al. [153, p. 181], ZI -IENG [311, 312J, PAO [214]) and bacteriology (PAO [218]). 
2 
Model Development 
2.0 Introduction 
In section 2.1, we shall set up the coordinate system, the spaces and the notation for our generalised particle 
reactor model that will be required for our model development. These shall be used Iluoughoutlhis thesis. 
A system of equations together with their initial and boundary conditions is set up as a generalised 
particle reactor model for the complex interactions in a particle reactor. The various types of equations and 
their various boundary conditions will be examined and Ille assumptions and limitations of this model will 
also be discussed. 
Finally, in section 2.2 we shall discuss some rclevant literature. 
2.1 Generalised Particle Reactor Model Development 
A reactor occupies a fixed region J\ in R3 space and z denotes the coordinates of a point in J\. Fluid flows 
through J\ with a solenoidal fluid velocity disLribuLioll u(z) which in general may vary with time. The 
boundary dJ\ of J\ is subdivided into three surfaces dJ\I' dJ\2 and dJ\3 where fluid flows into J\ through dJ\1 
and out through dJ\3, while dJ\2 is impervious to fluid. If nj denotes the outward normal to dJ\j and vj=lu·njl, 
then for all time 
(2.1.1) 
The reactive particles are fixed relative to Ille z-coordinate system at all time t and !2 is a representative active 
region per unit volume occupied by the particles at a point z in J\. In this sense, the region !2 is multiply 
connected and it represents the shape of this biomass around Ille various particles one would find in a small 
neighbourhood around z, i.e., it represents the region occupied by N particles of various sizes and shapes, 
where N is the number of particles in unit volume of fluid. The particles and this region Q are assumed to be 
small enough so that bulk fluid concentrations are constant spatially about Q. We also assume Q is 
independent of z (the particles are uniformly mixed and distributed) and the particles do not change position 
significantly with time. This is not an unrealistic assumption for fluidised beds with a stratified SLructure and 
is obviously justified for packed bed reactors. This active region is assumed to have an inner boundary d!21 
enclosing inert cores and on which 
dC' 
d� = 0 on (0, T]Xd.a)XJ\, (2.1.2) 
where Cj is the concentration per unit volume in [2, of the ith chemical component. These concentrations are 
9 
2.1 GENERALISED PARTICLE REACTOR MODEL DEVELOPMENT 
assumed to be governed by a system of weakly coupled reaction-diffusion equations 
10  
(2.1.3) 
where x denotes points in Q relative to a suitable coordinate system, V� denotes the Laplacian operator in .Q 
space, Dj is the appropriate d iffusion coefficient for the ith component and fi(t, x, Cj) are Lipschitz 
continuous functions in Cj. On the outer boundary ()il2 of il, the concentrations interact with the fluid 
concentrations Cj(t, z) according to boundary conditions 
(2.1.4) 
where � denotes the gradient of Cj on ()Q2 along the outward normal and the positive constants Hj are mass 
transfer coefficients associated with boundary layer transport. The units for Cj and Cj are assumed to be 
compatible so that when Cj = Cj on ()il2, there is no flux across the boundary layer. We see that the 
concentrations Cj depend not only on x and t, but also on z from the functions Cj(t, z) appearing in (2. 1 .4). 
There may be some chemical components which are immobile in.Q and for these Dj and Hj are zero. Such 
components can only interact with the fluid concentrations Cj in A through the reaction term/; via interacting 
mobile ingredients. The boundary conditions (2. 1 .2) and (2. 1 .4) have no relevance for these immobile 
components. The consideration of the boundary condition (2. 1 .4) includes the Dirichlet type (DJHj == 0), the 
Neumann type (Dj $0, Hj == 0), and the Robin type (Dj $0, H j $ 0). The boundary conditions (2. 1 .2) and 
(2. 1 .4) therefore includes various mixed type of boundary conditions. The initial conditions are of the form 
Cj = Cj,O in .QxA, at t = O. . (2.1.5) 
In the fluid region A, the concentrations Cj are assumed to satisfy the system of weakly coupled convection 
reaction-diffusion equations 
()Cj _ �V2C +u'  Vc +J D· ()Cj = F(t z C) in (0 T]x A 
()t 
I I I a!12 I ()n I "  J ' , (2.1.6) 
where the "diffusion" constant !i)j is assumed to incorporate dispersion effects, V2 is the Laplacian operator 
in A, involving z-coordinates, the fourth term accounts for the flux of chemical i into il representing the N 
particles in unit volume of fluid and the lluid reaction term Fj is the source of Cj due to reactions in the fluid 
itself. The functions Fj like the functions/; above are assumed to be Lipschitz continuous functions in the 
dependent variables Cj. 
The surface integral in equation (2. 1 .6) which is a measure of the total flux of Cj through ()il2 into il 
is only a function of t and z, and can from (2. 1 .4), be expressed in the form I-(S'lC; -I/;J c; , where st i s  an, 
the area of ()il2 (see MIS [2 1 ,  p.24]). 
There is no transport of llu id over ()A2 and fluid concentrations Cj at ()A! and ()A3 are subject to 
boundary conditions as discussed by WEllNER and WILli ELM [306J and DANCKWERTS [79] ,  respectively. 
These boundary condi tions arc of the form 
VICj + !i)/fj = VICj Ion (0, TJXdA1, ani ' 
�j = 0 on (0, TJXdAa, a = 2, 3 ,  
ana 
where Cj,) is the inlet lluid concentration. 
(2.1.7) 
(2.1.8) 
2.2 NOTES AND COMMENTS 1 1  
There may be some chemical components where there is no dispersion in A and for these �j arc zero. 
The boundary condition (2. 1 .8) has no relevance for these components. Furthermore, if u ·  V Cj = 0 for some 
of these chemical components , then the boundary conditions (2. 1 .7) has no relevance as well. 
The consideration of the boundary condition (2. 1 .7) thus includes the Dirichlct type (�/W= 0), the 
Neumann type (!llj 'FO,VI= 0), and the Robin type (!llj 'FO,Vl 'FO). The boundary conditions (2. 1 .7) and 
(2. 1 .8) therefore includes various mixed type of boundary conditions. The initial conditions are of the form: 
Cj = Cj,o in A, at I = O. (2.1.9) 
We now have a system of reaction-diffusion equations (2. 1 .3) coupled with convection reaction-diffusion 
equations (2. 1 .6) via the boundary conditions (2. 1 .4) and the source term in (2. 1 .6) accounting for the flux 
of chemical component j through dQ2. These equations which we label Sn involve the dependent variables 
Cj (l, x, z) defined in [0, T] x Q x A and Cj (l , z) defined in [0, T] x X and the subscript n denotes the 
maximum number of chemical components in either the macro or the m icrosystem. Associated with this 
system are the boundary conditions and initial conditions Bn given by the equations (2. 1 .2) . (2. 1 .5). (2. 1 .7)­
(2. 1 .9). This coupled system Sn. Bn of reaction-diffusion equations is degenerate in (0. l1xf2xA space in the 
sense that the Laplacian operators of equations (2. 1 .3) and (2. 1 .6) involve only the x in Q and z in A 
respectively. The steady slate or time independent system will be denoted as Sn . Bn . 
Our generalised particle reactor equations Sn. Bn form a system of up to 2n weakly coupled equations 
which may be a combination of ordinary differential equations. first order partial differential equations and 
parabolic equations. At steady state. these equations may be a combination of algebraic equations. first order 
partial differential equations and elliptic equations. In many of these cases. the boundary conditions may be 
irrelevant for the same reasons given above. 
Let l be the set of integers corresponding to up to n chemical components in .Q and i be the set of 
integers corresponding to up to n chemical componcnts in 11. If for cxamplc j E / and Dj = /lj = O. thcn we 
assume that j � i. i.e .• there is no corresponding chemical component in A I f. on the other hand j E / and 
Dj. Hj > O. we may assume that there is a corresponding chemical component in 11 and the components Cj are 
coupled to components C j by the boundary condition (2. 1 .4). If j E i.  there may not necessarily  be a 
corresponding component in /. unless Dj. lij > 0 for some j E / and for this. the components Cj are also 
coupled to componenL<; Cj by the boundary condition (2. 1 .4). All the other possibilities are similar. 
We denote by n(l) and n(i) the number of elements in J and i, respectively and the solutions of the 
system Sno Bn are denoted by the ordered pair (Cj, Cj) = (CI .  C2 . . . . . .  Cn(l)' C 10 C2, . . . .  , Cn(J» ' 
The boundaries of dQ and dA are assumed to be of finite curvature so that each point can be found on 
a closed ball of finite radius contained in f2 or X. This is known as the inner sphere property. 
2.2 Notes and Comments 
The physical basis of the boundary condtions (2. 1 .7) and (2. 1 .8) has been discussed at length in the 
l iterature (e.g. DANCKWERTS [791 . WEI-INER and WILHELM [306] , PEARSON [23 1 ] ,  KREFT and ZUBER 
[ 148] and SMITH [270]) .  
The d ispersion coefficient �j i n  (2. 1 .6) reflects two mechanisms for solute spreading. molecular 
diffusion and mechanical dispersion. Much work has been published on this dispersion phenomena. in 
particular to investigate the dispersion tensor. In cyl indrical reactors the dispersion in the direction of flow 
(longitudinal or axial dispersion) is noticeably different from the dispersion perpendicular to the direction of 
flow (transverse or radial dispersion). 
2.2 NOTES AND COMMENTS 12 
Compared to  the work published on the longitudinal dispersion coefficient 9)iL, relatively few resulLs 
have been reported on the transverse dispersion 9)iT. The mechanisms causing transverse or radial dispersion 
are molecular diffusion and "wandering" from the flow path (S IMPSON [267] , LEIJ and DANE [ 1 64]) and in 
some instances i t  may be considered equal to  the coefficient of molecular diffusion (CARBERRY [45]). Values 
for 9)iT are more difficult to obtain than values for 9)iL, because the concentration distribution needs to be 
measured in a direction perpendicular to the flow. Studies of longitudinal or axial dispersion are reported by 
BABCOCK et al. [29], BRITTAN [43J CHUNG and WEN [7 1 ] ,  RASMUSON and NERETNICKS [243] ,  
RASMUSON [244] , MECKLENBURGI I  and HARTLAND [ 1 95 - 197] and in some instances researchers have 
developed criteria based on the reactor length for conditions where axial dispersion can safely be neglected. 
There has been considerable interest in the chemical engineering literature on the transport equation 
(2. 1 .6) neglecting the functional and reaction terms. This equation is commonly known as a convection­
dispersion equation (CDE) or an advection-dispersion equation (ADE). HARLEMAN and RUMER [ 1 1 6] 
presented an analytical solution for the two dimensional ADE for steady state conditions, assuming that 
longitudinal dispersion could be neglected. Analytical solutions for the steady state problem, which 
accounted for longitudinal dispersion were provided by GRANE and GARDNER [ 109] and VERRUIJT [29 1 ] . 
BEAR [34] discusses methods to obtain analytical solutions for some specific problems and LEIJ and DANE 
[ 1 64] provide an analytical solution for the two dimensional transport problem which accounts for both 
longitudinal and transverse dispersion. This solution can also be adapted to three dimensional dispersion. 
Approximate solutions for the equation (2. 1 .6) for a one-dimensional time independent reactor with 
arbitrary kinetics is given by GROTCH [ 1 1 2]. An exact solution to the equation (2. 1 .6) for arbitrary shapes 
and first order kinetics without the additional contribution from the particles is given by VRENTAS and 
VRENTAS [297] with the fluid velocity field zero and by VRENTAS and VRENTAS [298] with a linear source 
term with fluid velocity distribution u as a function of z and t and with div u = O. These solutions are given 
in terms of a Greens's function formulation. 
There is very lillie literature on the coupled system (2. 1 .3) and (2. 1 .6). This set of equations for one 
component without reaction and simpler boundary conditions goes back to the work of DEiSLER and 
WILHELM [8 1 ]. For the case with no dispersion (9) = 0), a classical solution of the coupled system (2. 1 .3) 
and (2. 1 .6) was given by ROSEN [25 1 ]  in terms of an infinite integral. BABCOCK et al. [29] and PELLET!' 
[232] have presented analytical solutions for this case including dispersion. Approximate solutions have also 
been given by RASUMSON and NERETNIEKS l243J and improved by RASMUSON l 244 J .  These are also given 
as an infinite integral which has to detennined numerically. 
This thesis suggests that many of these exact analytical solutions may be used as bounds for our true 
solution with arbitrary reaction kinetics. However, we will not be demonstrating this in this work. 
3 
The Unsteady State Problem 
3.0 Introduction 
This chapter considers the unsteady state or time dependent problem. The objectives are generalised 
comparison theorems. uniqueness. stability and existence theorems for solutions. 
In section 3 .1. we shall collect some notational conventions and basic definitions and give some general 
results and relationships of the spaces that are needed in order to obtain the exact result on the solvability of 
linear parabolic equations. The maximum principle for parabolic equations which will be used throughout this 
thesis will also be defined. 
In section 3 .2 we shall show that although our system is nonstandard. it is governed by generalised 
comparison theorems similar in type to those for weakly coupled parabolic systems. These comparison 
theorems are a useful tool for proving uniqueness. existence and stability of solutions. 
In section 3 .3 .  we use the strong comparison theorem to show that solutions of system Sn. Bn are 
uniquely specified by the functions ci.O. Ci•O and Ci• l . 
In section 3.4. we see that for the purposes of uniqueness. stability and existence theorems. we may 
assume at the outset that the system Sn. Bn is a quasimonotone system. i .e .  Ii and Fi are monotone 
nondecreasing in Cj and Cj respectively for j :¢ i. This is not a restriction on these theorems, since if this 
monotone property is not satisfied. then Sn. Bn with general functions!; and Fi can be imbedded in a system 
S2n. B2n of the same form. It can be shown that solutions of this new system generate solutions of the 
original system and therefore uniqueness. stability and existence can be implied in the original system. 
In section 3 .5. we establish some useful sufficient conditions for the global stability of all solutions of 
the general system Sn. Bn. It is shown that such stability implies the uniqueness of the solutions to the steady 
state or time independent problem. 
In section 3.6. we show that solutions of problem Sn. Bn specified by ci,O. Ci•O and Ci• 1 exist. This is 
done by constructing a sequence of approximate solutions which converges monotonically and uniformly (in 
appropriate function spaces) to a limit function which is shown to be a solution of the system Sn. Bn . 
Finally. in section 3.7. we discuss some relevant li terature and future work. 
3.1 Definitions, Notation and General Results for Linear Parabolic Equations 
It is important in the theory of nonlinear differential equations to obtain theorems on a priori estimates and 
existence of linear differential equations in order to derive estimates for nonlinear differential equations. In 
this section. we shall collect some notational conventions and basic defintions. 
13 
3.1 DEFINITIONS, NOTATION AND GENERAL RESULTS FOR 
LINEAR PARABOLIC EQUATIONS 
14  
We shall also give some general results and relationships of  the spaces that are needed in order to 
obtain the exact result on the solvability of linear parabolic equations. The Maximum Principle for parabolic 
equations which will be used throughout this thesis will also be defined. 
3. 1.1  Definitions and Notation 
The fol lowing notation will be used throughout this section 
x = (XI .  X2 • . . . . •  xm) denotes a point in Rm. 
T >  O. where T denotes a point in R. 
t E  [0. 11 .  
, is a bounded. open. connected domain in Rm. 
iY9 denotes the boundary of ,. 
if denotes the closure of ,. 
Qr = (0. 11x� is regarded as a subset of Rm+ 1 . 
rr = (0. 11xiY9. 
Or= [0. 11xif.  
TT = [0. T]xa�. 
u E R . 
Dxu = (au/ax l • . . . . •  au/axm) .  
Definition 3.1.1. 
A vector field v(t . x) = (VI (I. x) • . . . . •  vm(t. x» is said to be a unit outerward normal (outward normal or 
outemormal) at (I. x) E rr if (t. x-hv) E Qr for small h > O. The outernormal derivalive is then given by 
;"1"(}. vet,){) is � U,, 'jt \/<"do( no(('Il,J +0 IT 
au = lim u(t. x) - u(t. x - hv) . av h-)O h 
3.1.2 General Results and Relationships between Holder, u,q and Sobolev Spaces Spaces 
The following definitions of H61der. u,q and Sobolev Spaces are adapted from LADYZHENSKA YA [ 1 55 .  
ppA-9] .  
HOlder S paces 
For a positive real number. I. say. let [I] denote the greatest integer not exceeding I. For A � QT. we shall say 
thatfE CIl2,/[A. R] iff: A � R is continuous. the partial derivatives off. with respect to x. up to order [I] 
are continuous on A and its [I]th order partial derivatives with respect to x are HOlder continuous on A with 
exponent [ - [I] . and further the partial derivatives off, with respect to t. up to order [//2] are continuous on 
A and its [//2] th order partial derivative with respect to 1 is Holder continuous on A with exponent 1/2-[//2] . 
For 0 < I < 1 and f E CI/2,/[A . R] . we shall use the following notation: 
where 
IIfll� = sup If(t .x)1 • 
(l,x)EA 
3.1 DEFINITIONS, NOT A TION AND GENERAL RESULTS FOR 
LINEAR PARABOLIC EQUATIONS 
L[A (f) _ If(t,x) - f(t,y)1 I� x I - sup I • (I.X).( I.y)eA IIx - yll 
IIz-)4I�p 
HA (f) - If(t,x) - f(t' , x)1 1 1/2 SUp 1/2 • (l.x).(I .x)eA It - 1 ' 1  
11-- I'I�p 
For any I >  0 and f E CIl2•/[A , R), we shall use the folowing notation: 
[/l 
IIfll� = L LIID;D;fll� + (I)� , j=0 2r+s=j 
1 5  
where D: D;f denotes the partial derivative o f  fwith respect to x and t o f  order s and r, respectively, for all r 
and S such that 2r+s <I; 
(I): = IH:I-(I ] (D;D�f) + IH'�/3 (D;D�f), 2r+s=(/) 0</3<1 
for f3 = (l-2r-s)/2. 
Since Qr is closed, it is required that the derivatives of order :5: I can be continuously continued from the 
interior of QT to all of Qr . If a = 1 ,  we also say thatf is Lipschitz continuous. 
Lr,q S pa ces 
L'·q [QT , R) is the Banach space consisting of a l l  equivalent clases of Lebesgue measurable functions u 
defined on Qr into R with a finite norm 
where q � 1 and r � 1 .  
I f  r = q, then Lq.q[Qr ,  R )  i s  denoted by LQ [0 ,  R) and the norm 
by 
Sobolev Spaces 
For nonnegative integer I, W?I [QT' R] is the Banach space consisting of the elements Lq [QT ' R] having 
generalised (weak or distributional) derivatives of the form Dr D� with any r and s satisfying the inequality 
2r+s :5: 2/. The norm in it is defined by 
where the summation L2r+S=j is taken over aU nonnegative integers r and s satisfying the condition 2r+s:5:21. 
3.1 DEFINITIONS, NOTATION AND GENERAL RESULTS FOR 
LINEAR PARABOLIC EQUATIONS 
16 
For nonintegral l, W� [Qr , R) is a Banach space consisting of the elements u of WJIl [Qr ,  R) with fmite nonn 
l Iullw: [�, RJ = l IullwJ/J[�, RJ + l IuIl L�-(/J [�, Rl ' 
where 
and 
L ' denotes summation over all possible derivatives of u of order} satisfying the condition} $; [I); J 
Lj=(I) denotes the summation over all possible derivatives of u of order} = [I] . 
For nonintegral l, the Banach space W�/2. t [r:r ,  Rl is defined analogously (LADYZHENSKAYA [ 1 55 ,  p.8 1 )). 
General Results in  HOlder, U,q and Sobolev Spaces 
We now prove some general results in Holder, U,q ard Sobolev spaces. 
telllllla 3, 1 . 1  
If f, g E CCX/2'CX [A,RJ , then f + g E C(X/2,cx [ A , R] and IIf + gll� $; IIfll� + IIgll� . 
Proof 
I If + gll� = sup If(t,x) + g(t,x)1 + sup l [f(t,x) + g(t,x)] - [f(t,y) + g(t,y)]1 
(l,x)EA (I,X),(I,y)EA I Ix _ ylla 
as required.O 
Lemma 3.1 .2 
IIz-JII$p 
l[f(t,x) + g(t,x)] - [f(t' ,x) + g(t' ,x)]1 + sup a/2 (l,x\,( I ,X)EA I t  - 1' 1 ft-t'l sp 
If(t x) - f(t y)1 Ig(t,x) + g(t,y)1 $; sup If(t,x)1 + sup Ig(t,x)1 + SIlP , ex ' + sup 
{I ,x)EA (l,x)EA (I,X),(I,y)EA I Ix - yll ( I,X),(I ,y)EA I Ix _ ylla 
IIz-JII$p IIz-�!Sp 
If(t,x) - f(t' ,x)1 Ig(t,x ) - g(t' ,x»)1 + sup a /2 + sup a!2 (I,X),{I ,x)EA It - 1' 1  (I,x),{I ,x)EA I t  - 1 '1 
k-I'!$p !I-I'!$p 
$; I l fl l� + l I gl l� , 
If f, g E C a/2,a [A, R] , then fg E Ca /2,a [A, R] and I I fgll� $; I I fll� I I gl l� . 
Proof 
I lfgll� If( ) ( )1 If(t,x)g(t,x) - f(t,y)g(t,y)1 = sup t ,x g t,x + sup a {I x)EA ' (I,x),{I,y)EA I Ix - yll , IIx-JII$p 
If(t,x)g(t,x) - f(t',x)g(t' , x)1 + sup , a/2 (I,X),(I ,X)EA It - t I 
!I-I'!$p 
3.1 DEFINITIONS, NOTATION AND GENERAL RESULTS FOR 
LINEAR PARABOLIC EQUATIONS 
� sup If(I ,x)1 sup Ig(I ,x)1 
(l,x)eA (l ,x)eA 
If(t,x)g(t,x) - f(t ,x)g(t ,y) + f(t,x)g(t,y) - f(t,y)g(t,y)1 + sup 
(I ,X),(I,y)eA I Ix _ ylla 
IIx-)llsp 
If(t ,x)g(t,x) - f(t,x)g(t' ,x) + f(t,x)g(t' ,x) - f(t' ,x)g(t' ,x)1 + sup a!2 ( I ,X),(I ,x)eA 11 - 1' 1 
fl-11Sp 
< If( )1 [ I ( )1 
Ig(t,x) + g(I ,y)1 Ig(t,x )-g(I' ,x)]I] - sup I ,X sup g I ,X + sup a + sup a/2 (l,x)eA (l,x)eA (l,x),(l ly)eA I lx - yll (I ,X),(I ,x)eA 1 1 - 1'1 IIx-yl Sp 1t-11Sp 
I ( )1 [ If(t,x) + f(I ,y)1 If(t,x) - f(t' ,x)]I ] + sup g t,x sup a + sup a!2 (l,x)eA (l ,x),(I,y)eA I Ix - yll ( l ,x),(I ,x)eA I t - (I 
IIx-)lISp �-11Sp 
� I Ifll� I Igll� , 
as required.O 
The next three lemmas and their corollaries that follow are analogous in Holder, Lq and Sobolev spaces. 
Lemma 3.1.3 
Suppose 0 < f3 � a � 1 .  Then CaI2.a(A) C. C/312,/3 (A) . 
Proof 
Suppose f E Ca12,a [A, R]. We are required to show that 
IIfll� = Ilfll� +H:/3(f) + H�/3!2 (f) 
If( )1 
If(I ,x) - f(t,y)1 If(t,x) - f(t' ,x)1 = sup t ,x + sup /3 + sup /3/2 (I,x)eA (l,x),(I,y)eA I Ix -yll (l,x),(I ,x)eA I t - 1'1 Ix-)IISp �-11Sp 
is finite, given that 
IIfll� = IIfll� +H:a(f) + HI�a/2 (f) 
If( )1 
If(t,x) - f(t,y)1 I f(t ,x )- f(I' ,x)1 = sup I ,X + sup a + sup ' /2 (l,x)eA (l,x),(I,y)eA I Ix -yll (I,X),(I ,x)eA 1 1 - t' la 
Ix-)lISp 11-11Sp 
is finite. 
Suppose that (I, x) , (I , y) E A and 0 < I Ix- yll < 1 .  Then 
If(t,x) - f(t,y)1 = If(I ,x) -f�t,Y)1 I Ix - ylla-/3 � If(t,x) -f(t,y)1 
IIx - yliP I Ix -yll I Ix -ylla 
since a - f3 � O. On the other hand, if (t, x), (I, y) E A and I Ix - yll � I ,  
where 
If(t,x) - f(I ,y)1 
I Ix - yll/3 
= If(t ,x) - f(t,y)1 IIx _ ylla-/3 � ka-/3 
If(t,x) - f�I ,y)I , 
I Ix -ylia IIx -yl l 
k = sup ( lix -yl l : x, y E �) . 
A similar argument holds if we have to show that 
17  
3.1 DEFINITIONS, NOTATION AND GENERAL RESULTS FOR 
LINEAR PARABOLIC EQUATIONS 
If(t ,x) - f(t',x)1 sup 
(l,x),(I ,X)EA I t - ( 1 13/2 
11-11Sp 
is finite and the result follows by combining these arguments.O 
A trivial consequent of this result is the following corollary: 
Corollary 3.1.1 
Suppose 0 < a, {3 � 1. Then Ca/2,a (A) 11 C{3f2,{3 (A) = cyf2, Y (A) where r=  min ( a, {3l . 
The following lemma is stated without proof in BURKILL r44] 
Lemma 3.1.4 
Suppose 1 � P I � p2. Then LP2 fA, R] c:; LP' [A,  R).  
Corollary 3.1 .2 
Suppose 1 � P I ,  P2. Then U2 [A ,  R] 11 UI r A, R ]  = UtA, R] where P = mirl (P I ,  P2 l .  
The following lemma and its corollary follows from Lemma 3 . 1 .4 and the defini tion of Sobolev spaces. 
Lemma 3.1.5 
Suppose 1 $ PI $ P2 ' Then W;�21 [A , R] � W;;21 [A, R] . 
Corollary 3.1.3 
Suppose 1 � P I , P2 . Then W;;21 [A, R] 11 W;;21 [A, R] = W?' [A , R] where p = min (PI , P2 l .  
We define the following operator which takes a space into itself 
Definition 3.1.2 
The Nemytskii operator .A(u) is defined by 
.A(u)(t, x) = h(t, x, u), (t, x) E Qr 
18  
We now present a result concerning the Nemytskii operator .A(u). The proof is found in LADDE e t  al. ( 1 53 ,  
p.22 1 ]  and TEMME (282, p.65] .  
Lemma 3.1 . 6  
Let h E  Ca/2,a [ [O, T) x if x R, R) , and let .A(u) be the Nemytskii operator .A(u). Then 
(i) JYE C[c(l+a)12. 1+a[QT '  RJ , ca12 ,a[QT , R)) ;  
(ii) JY takes bounded sets in C(1+a)12, I+a[QT ' R) into bounded sets in ca12,a[Qr ,  R) . 
Remark 3.1 .1  
From the fact  that the space c(I+a)/2,l +CX[Qr . R] is  compactly imbedded in CO, I [Qr ,  RJ , the Nemytski i  
operator belongs to C[ CO, I [QT ' R] ,  C[QT, R] ] . 
3.1 DEFINITIONS, NOTATION AND GENERAL RESULTS FOR 
LINEAR PARABOLIC EQUATIONS 
We present another result concerning the Nemytskii opcnltor J1(u). 
Lemma 3.1 .7  
Let h E  Cal2,a rrO. T] x iff x R. R] . and let J1(u) be the Nemytskii operator J1(u). Then 
(i) JfE C[U,q [Qr .  R ) .  U,q [Or .  R]] ;  
(ii) Jf takes bounded sets in u,q [Or .  R] into bounded sets in u,q [Or .  R]. 
Proof 
19  
Jf maps a l l  of u,q [Qr .  R] into u,q [Qr .  R )  since Qr is bounded and h E  CaI2,a [[O. T) x iff x R. R) .  
Therefore Jf is a continllolls and bounded operator.O 
The Relationships between these Spaces 
The spaces defined in this section are needed to obtain the exact resull on the solvability of boundary value 
problems for linear parabolic equations in spaces Wd ,2 [Qr , R]. This is related with the fact that the 
differential properties of the boundary values of functions from the classes Wd,2 [Qr . R] and of their 
derivatives can be exactly described in terms of the spaces W;I2,/ [rr . R] with nonintegral /: 1 =  k-l/q. where 
k is an integer. This relationship is g iven in the following result. For details. see LADYZHENSKA YA [ 1 55 .  
pp .  79-82] or  LADDE e t  al. [ 1 53 .  p.2 1 8] . 
Lemma 3,1 .8 
If u E W� ,2 [Q; . • R] , then for all nonnegative integers r and s. 
and 
2r + S < 2 - 2/q. 
Dr DSul E W2-2r-s-2Iq [r'ii Rj 1 X 1=0 q (1' "  
Moreover,jor 2r + s < 2 - l/q . 
Dr DS 1_ E WI-r-sl2-1/2q,2-2r-s-I/q [[:,. R] 1 xU rT q 7 .  • 
I lul lwl-r- SI2- 1/2Q.Z-Zr- S-l/Q [ 'F,. Rj�  CliuIlW 1,Z [Q-: Rj '  Q 1 , Q r , 
We say how smooth functions in a Soholev space are, by imbedding Soholev spaces continuousl y into 
HOlder spaces. This is called the Sobolev Lemma or the Imbedding Theorem. We shall firstly require the 
following defintion: 
Definition 3.1 .3 
Let X and Y be normed spaces. We say that the normed space X is imbedded in the normed space Y i ff 
(i) X is a vector subspace of Y; 
(ii) the identity operator e defined on X into Y by ex = x for all x E X is continuous. 
Also required is the following definition which g ives the smoothness properties of the boundary df9 that is 
required in the Sobolev Lemma. 
3.1 DEFINITIONS, NOTATION AND GENERAL RESULTS FOR 
LINEAR PARABOLIC EQUATIONS 
Definition 3.1.4 
20 
Let rI be an m-dimensional domain with boundary (JrI. We say that (JrI belongs to class C2+a, if for every 
x E drJ, there exists a neighbourhood U of x such that drJ n U can be represented in the form 
for some i, 1 $; i $; m, where h E c2+a[(JrI, R] .  
We now state the following imbedding theorem. For proof, see LADYZl-ffiNSKA YA [ 1 55 ,  p.60] . 
Theorem 3. 1.1 (Imbedding Theorem or Sobolev Lemma) 
Let rI � Rm and let drJ be of class C2+a. Suppose that q = (m+2)/(I-a) for 0 < a < 1 .  Then WJ·2[Qr , R] 
is imbedded in C(1+a)/2. !+a [Qr , R] . 
3.1 .3 Solvability of Linear Parabol ic Equations 
We will discuss in this section a specific example of a parabol ic equation that occurs in this thesis. For the 
definitions of more general linear and quasilinear second order equations of parabolic type, as well as their 
uniformly parabolic conditions, see LADYZHENSKAYA [ 1 55, p. 1 1 ] . 
Let aij, bi and c belong to Ca/2.a [Or , R] and let c $; O  in Or . Let !£ be a second order differential operator 
defined by 
(J !£ = - - L, (3.1 .1)  (Jt 
where 
m a2 m a L = Laij(t ,  x)� + Lb; (t ,  x)-;- +c(t, x) . i.j=! OXjOX j j=! OXj 
Definition 3.1 .5 
(3. 1 .2) 
The differential operator !£ defined by (3. 1 . 1 )  is said to be parabolic at a point (t, x) E QT, if the coefficient 
matrix ai/t, x) is positive definite, that is there exist two functions 1 and I such that 
for all � E Rm - { O} . If 
1(t, x) > 0 in QT, 
then!1J is called parabolic in QT. If 
1(t, x)�A.o>O, 
for a positive number Ao then !1J is called almost strictly parabolic in QT. If 
I(t, x) $; K, 
1(t, x) 
i 
(3.1 .3) 
for some positive number K, then !£ is called strictly uniformly parabolic in QT, that is (3. 1 .3) can be 
written as 
3.1 DEFINITIONS, NOTATION AND GENERAL RESULTS FOR 
LINEAR PARABOLIC EQUATIONS 
for all � E Rm, (t, X)E QT. 
21 
Let rJ be a bounded domain in Rm ,  QT  = (0, T]xrJ for T > ° and rT = (0, T]X drJ. Let p ,  q E 
C(1+a)/2, l+a[1";T , R] be nonnegative functions and let Vet, x) be the unit outward normal vector field on drJ 
(which belongs to the class C2+a )  for t E [0, TJ . 
Consider the nonl inear second order parabolic initial boundary value problem (lRV? for short): 
where 
and 
flu = h ( t ,  x ,  u )  i n  QT'} 
!i1u = ¢J( t ,  x) on TT, 
u (O ,  x) = ¢Jo(x) in iff , 
¢J E Cel +a)/2, 1 +ar 7;. , R] ,  
¢Jo E C2+a[� , R] , 
h E CaJ2,a[[o, T]xij xR, R] ,  
du fiJu = pet, x)u + q(t, x)- . 
d v  
Definitioll 3.1 .6 
(3.1 .4) 
(3. 1 .5) 
(3. 1.6) 
(3.1 .7) 
(3.1 .8) 
We shall say that the compatibility conditions of order k � 0 are ful filled for the IRV? (3 . 1 .4)-(3 . 1 .8) i f  
�(j) ( . ') ( ') ( '  ') du(J) (x) ( ') L..J . (p }-I (x)u ) (x) + q }-I (x) ) = ¢J ) (x) 
;=0 l dv 
on d' for j = 0, 1 ,  . . . .  , k , where, 
and 
( j) ( ') _ (j) (0 ) _ dju(t , x) 1 u ) - u  , x  - , , dt} 1=0 
±(�')(p(j-;) (O, x)u(j) (O, x) + q(j-i) (0, x) du
(
j
) (O, x» = �jj (p(t , x)u(t , x) + q(t ,x) duCt ,  x» 1 . ;=0 l dv at dv 1=0 
We now consider the linear second order parabolic initial boundary value problem (lRV? for short) 
flu = h (t ,  x) in QT,  } 
fiJu = !P(t ,  x) on r,/" 
u(O ,  x) = !Po(x) i n  � .  
(3.1 .9) 
3.1 DEFINITIONS, NOTATION AND GENERAL RESULTS FOR 
LINEAR PARABOLIC EQUATIONS 
22 
Let liS now s ta le the classical ex islence and uniq ueness theorem whose proof can be found in 
LADYZHENSKAYA [ 1 55, p.320] and FR IEDMAN [94, p. l44 J .  
Theorem 3.1 .2. Assume that 
(i) aij, bi, c E Cal2,arITr ,  RJ, c(t, x) � 0 and!J! is strictly uniformly parabolic in QT; 
(ii) p, q E C( l Hx)/2, 1 HXI7;. ,  R"l for p and q nonnegative funct ions and there exists )1 1 >  0 such that 
p � )1 1 ; 
(iii) d<9 belongs to class c2+a; 
(iv) h E ccc/2,a[Qr ,  R] ; 
(v) I/J E c(1 +a)/2, l+a[G" R] and cfJo E C2+a[�, R] ; 
(vi) the IBV? (3 . 1 .9) satisfies the compatibility cOlic/it ion of order 1 ( 1 +a)/2 1 .  
Then the linear parabolic IBV? (3 . 1 .9) has a unique solution u such that u E Cl+a/2,2+a[Q;. , R] . 
The following result provides the global a priori Schauder-type estimates for classical solutions of 
IBVP (3 . 1 .9) . 
Theorem 3.1.3. 
Assume that the hypotheses of Theorem 3 . 12  hold. Then for any u E C1 +cx/2 ,2+cx[Qr , R] ,  there exists a 
positive constant C which is independent of u such that 
" " V/' < C'(" (fJ " V/' " un  " f/' " ,h ,,'9 U 2+u - JA� U + .:;vU I+u + 'l'0 2+u ) '  
Moreover, if u is the classical solution of the IBV? (3 . 1 .9), then (3. 1 . 1  0) reduces to 
I Iull�:;t $ C(",,"S'" + I I<P IIl:� + I IlPo II?"a ) ' 
Remark 3.1.2 
(3. 1 . 1 0) 
(3. 1 . 1  I )  
Analogous theorems to Theorem 3 . 1 .2 and Theorem 3 . 1 .3 hold for the IBVP (3 . 1 .9) with Dirichlet 
boundary conditions. In this case it is required that I/J E Cl+a/2,2+a[.Fr , R] with the compatibility conditions 
of order 1 +a/2 (see LADYZlIENSKAYA [ 1 55, p.320]). 
We shall state some results for solutions in the Sobolev spaces W?[Qr ,  R] , q > 1 analogous to the 
Schauder results in the HOlder spaces Cl+cx/2,2+cx[Qr ,  R) .  Let us state the following uniqueness and 
existence theorem that provides us with generalised (weak) solutions of (3 . 1 .9). Its proof can be found in 
LADYZHENSKAYA [ 1 55 ,  p.34 1 ] .  
Theorem 3.1.4 Assume that 
(i) aij, bi, c E Cal2,a[Qr ,  R], c(t, x) � 0 and fl! is strictly uniformly parabolic in Qr; 
(ii) p, q E C(I+U)/2, I+a rI;· ,  R] ,for p and q nonnegative functions and there exists )11> 0 such that 
p � )1 1 ;  
(iii) d<9 belongs to class C2+a; 
(iv) h E Lq[�, R] for q > 1 ;  
(v) I/J E W�I2-1/2q,I-l!q [.Fr ,  R] and I/Jo E Wi-2Iq [�, R] ;  
(vi) the IBVP (3 . 1 .9) satisfies the compatibility condition of order [(q-3)/2q] .  
Then the linear parabolic IBVP (3 . 1 .9) has a unique solution u such that u E W� ,2 [Qr , R) . 
3.2 GENERALISED COMPARISON THEOREMS 23 
The following theorem provides global a priori Agmon-Douglis-Nirenberg type of estimates for generalised 
(weak) solutions of the IBVP (3 . 1 .9) . Its proof can be found in LADYZI IENSKA YA [ 1 55,  p.342] . 
Theorem 3.1.5 
Assume that the hypotheses of Theorem 3 .1 .4 hold. Then for any u E wd·2 [Qr ,  RJ ,  there is a constant C 
which is independent of u such that 
lIuI lW1.2 [-Q Rj � C(II9?uIlLq [-Q Rj + lI3Jull - - [- j
+ l I<Po l lw2-2Iq [- Rj ) '  q T .  T .  Wqlf2 lf2q.1 I/q rT . R q , .  
Moreover, if u is a generalized solution belonging to Wd·2 [Qr , RJ , then 
l IuIIWJ.2 [Qr. Rj
� C(I IFIIU [QT '  Rj
+ 1 I<PlIw�f2-1!2q.H,q [rT ' Rj+ l I<PoIIWq2-2Iq [� . Rj ) '  
Remark 3.1.3 
(3.1 .12) 
(3.1 .13) 
Analogous theorems to Theorem 3 . 1 .2 and Theorem 3 . 1 .3 hold for the IBVP (3 . 1 .9) with Dirichlet 
boundary conditions (LADYZI IENSKAY A [ 1 55 ,  p.341 )). 
3.1 .5 The Maximum Principle for Parabolic Equations 
Throughout th is thesis, we will use various forms of the maximum principle for the parabolic operator to 
obtain information about t1le solutions of our equations. A simple form of the maximum principle that we will 
find useful is the following 
Lemma 3.1.8 (Maximum Principle) 
Let lJI E  CI.2 [QT ' R] be such that 9?lJI � 0 in QT, 3JlJI� 0 on rT and 1f1(O, x) � 0 in iff . Then lJI�  0 on Or.  
Other forms of the maximum principle for the parabol ic operator are given by PROTTER and WEINBERGER 
[234] and SPERB [27 1 ] .  In section 3 .2 we shall develop some Generalised Comparison Theorems which are 
also useful in obtaining information about solutions of our equations. 
3.2 General ised Comparison Theorems 
There are important techn iques in the theory of differential equations which are concerned with estimating a 
function satisfying a differential equation by extremal solutions of an associated differential inequality. One 
technique is available when the solution of the differential equation satisfy certain comparison theorems. A 
tremendous advantage offered by theorems of this type arises from the relative ease with which one can find 
solutions of inequalities as opposed to equations and the useful feature that each solution of the inequality is a 
constraint on a whole class of perhaps unknown solutions. These comparison theorems are a useful tool for 
proving uniqueness, existence and stabil ity theorems for solutions. The maximum principle for parabolic and 
elliptic equations is an example of such a theorem. 
Our generalised particle reactor equations Sn ' Bn form a system of up to 2n weakly coupled, 
degenerate equations which may be a combination of ordinary differential equations, first order partial 
differential equations and parabolic equations. The degeneracy is is not only concerned with the possibility of 
D j  or 9>j being zero for some i but is also associated with the fact that in the case that Dj is nonzero, the 
Laplacian for the Cj equations involves only the x dependent variables whereas for the Cj equations it involves 
only the z dependent variables. The weak coupling is also non standard in that through equation (2. 1 .6) the 
Cj equations have a functional connection to me Cj variables via J Dj dCj • Despite these features we are aQ2 dn 
3.2 GENERALISED COMPARISON THEOREMS 24 
able to show in this section that this system is governed by generalised comparison theorems similar in type 
to those for weakly coupled parabolic systems in McNABB  [ 1 82, 1 86] .  However, additional smoothness 
properties such as the inner sphere property is used for comparison theorems and this smoothness hypothesis 
together with some Lipschitz continuity conditions on Ii and Fj are required for our generalised strong 
comparison theorem. We are also able to show by a counterexample that only in some cases are there 
analogous theorems for the corresponding time independent or steady state system. 
3.2.1 Some Basic Comparison Theorems 
We first require some weak and strong comparison theorems for ordinary differential equations, first order 
partial differential equations and parabolic equations. These weak comparison theorems are similar in that 
they assume at the outset that a solution is bounded by comparison functions and rules out the existence of a 
contact point for these functions for all time. There are no restrictions on the nonlinear functions. The strong 
comparison theorems are similar in that they provide stronger results for a solution bounded by comparison 
functions and spells out the consequences of the existence of contact points of these comparison functions. 
There are however restrictions on the nonlinear functions for such strong comparison theorems. 
Weak and Strong Comparison Theorems for Ordinary Differential Equations 
We shall firstly look at weak and strong comparison theorems for ordinary di fferential equations. The first 
two theorems are from McNABB [1 86] and are only included here for the sake of completeness. 
Theorem 3.2.1 (Weak Comparison Theorem) Suppose that 
(i) The functions CI and C2 are defined and are continuous in [0, TJ, their first order t derivatives 
exist and are uniformly bounded and continuous in the region (0, T] ; 
(ii) 
(iii) 
Then 
Proof 
dCI dC2 --h(t, cl ) <--h(t, c2 ) on (0, T] . dt dt 
C I (t) < C2(t) on [0, T ] .  
Let us  suppose that cI � c2 somewhere in [0, T] . Then, since cl - c2 is continuous in [0, T] ,  and C I (O)­
q(O) < 0, there is  a point t* in (0, T] such that CI (t* ) = C2(t* ) and cI < c2 on [0, to ) .  But then 
aCI (t* ) �  aC2 (t* ) ,  and h(t: cl ) = h(t: C2) '  Since this violates (ii i), at t, no such t* exists in [0, T] and at at 
therefore CI (t) < C2(t) on [0, T] . 0 
I f  h(t, c) satisfies a Lipschitz condition then a stronger result can be stated. 
Theorem 3.2.2 (Strong Comparison Theorem) Suppose that 
(i) The functions CI and C2 are defined and are continuous in [0, TJ . their first order t derivatives 
exist and are uniformly bounded and continuous in the region (0, T] ; 
(ii) CI (0) < C2(0); 
3.2 GENERALISED COMPARISON THEOREMS 
(iii) 
(iv) The function h is Lipschitz continuous in c so that there is a finite constant K > 0 for which 
Ih(t, cI ) - h(I, c2 )1 � Klcl - C2 1 in rO, T]. 
Then 
CJ (t) < C2(t) on [0, T] .  
Proof 
Let us suppose that C2(0) - CI(O) = A >  0, and let 
Then 
w = CI + 
A e-2K1 
2 
dw dCI -2KI -
dt 
- h(t, w) = --KAe - h(t, cI ) - [h(t, w) - h(t, CI )] dt . 
�1 �� � [-- h(t, cI )] + Klw - cl l-KAe dt 
dC2 KA -2K -2K � [Tt- h(t, c2)] +Te I - KAe 
I 
dC2 < --h(t , c2 )' 
dt 
25 
Now w(O) < C2(0) and so by Theorem 3.2.1 (Weak Comparison Theorem), W < C2 on [0, T] and cI < W < c2 
on rO, '1'] .0 
The following theorem is  also a strong comparison theorem 
Theorem 3.2.3 (Strong Comparison Theorem) Suppose that 
(i) The functions CI and C2 are defined and are continuous in [0, T] , their first order t derivatives 
exist and are uniformly bounded and continuous in the region (0, T] ; 
(ii) 
(iii) 
(iv) 
Then 
Proof 
Let 
c) (0) � C2(0); 
The function h is Lipschitz continuous in c so that there is a finite constant K > 0 for which 
Ih(t, cI ) - hCt, c2)1 � Klcl - C2 1 in [0, T] . 
W = c2 + ;l.e2K1 . 
Then 
3.2 GENERALISED COMPARISON THEOREMS 
�� - h(t, w) = d;
t
2 - h(t , w) + 2KAe2K1 + h(t, c2 ) - h(t, C2 ) 
� �] - h(t, cI ) + 2KAe2K1 - [h(t, w) - h(t, c2 )] 
dCI 2KI � -- h(t, cI ) + 2KAe - Klw - c2 1 dt 
� dc] _ h(t, cI ) + 2K Ae2K1 - KAe2K1 
dt 
� dC] _ h(t, c) ) + KAe2K1 
dt 
dCI < --h(t, cI ) '  
dt 
26 
Now w(o) < c, (0) and so by Theorem 3 .2. 1 (Weak Comparison Theorem), w < c] on [0, T] for all A which 
implies that c] � C2.0 
The following theorem provides a stronger result and spells out the consequences of the existence of contact 
points of these comparison functions. 
Theorem 3.2.4 (Strollg Comparison (COil tact) Theorem) Suppose that 
(i) The functions CI and C2 are defined and are continuous in [0, T] , their first order t derivatives 
exist and are uniformly bounded and continuous in the region (0, T] ; 
(i i) 
(iii) 
(iv) 
c] � C2 in [0, '1'1; 
dCI dC2 . 
-;;;- - h(t, cI) � dt- h(t, C2 ) on (0, 1 ] ;  
The function h is Lipschitz continuous in c so that there is a finite constant K > 0 for which 
Then either 
C I (t) < C2(t) on [0, T J .  
or there is a constant To in (0, T] such that 
C I (I) < C2(t) in (10 ' T] and CI (t) = C2(t) for all t in [0, To ] . 
Proof 
Either c ) < C2 in [0, '1'1 or there is some To in [0, TJ s lIch that C I = C2. Let To be the greatest such time. 
Suppose that CI < C2 at 1; < To, Then Theorem 3 .2,2 (Strong Comparison Theorem) i mplies that Cl  < C2 in 
(T; ,  T] which is a contradiction and hence the theorem must follow .O 
Weilk and Strong Comparison Theorems for First Order Partial Ditl'erentilll Equations 
We now look at weak and strong comparison theorems for first. order partial d i fferential equations. The 
proofs of these theorems arc extensions of comparison theorems for ord inary d i fferential equations that we 
3.2 GENERALISED COMPARISON THEOREMS 27 
have examined. We shall g ive such a weak and slrong comparison theorem for first order partial differential 
equations of the form 
dC f. dC . Ic = -:;- - £.Jai (t, x)-= h(t, x, c) m (0, T] x D, at i=] dXi (3.2.1) 
where D is a fin ite domain in Rm, T < 00 and 'has bounded coefficients ai(t, x) 
Theorem 3.2.5 (Weak Comparison Theorem) Suppose that 
(i) The functions c] and C2 are defined and are continuous in [0, T] x D,  their first order Xi 
derivatives exist and are continuous in (0, T] x D and their t derivatives exist and are uniformly 
hounded and continuous in the region (0, nxD; 
(iii) CI < C2 on (0, T] x aD ; 
Then 
c] < C2 in [0, Tl x D .  
Proof 
Let us suppose that CI � c2 somewhere in [0, T] x D .  Then, since cI - c2 is continuous in [0, T] x 15,  and (ii) 
and (i i i) hold at t = 0 and (0, T] x (JD , respectively,  there is a poi nt ( t� x· ) in (0, T] x D such that 
• •  • •  • - dCI • • dC2 . .  dCI dC2 CI (t, X ) = C2 (t , x )  and cI < c2 on [O, t ) x D. But then -;-(t, x ) �:l(t , x ) , :}=-;- and at at �i uXi 
h(t: x: cI )= h(t: x: c2) '  Since this violates (iv), at (t� x· ) ,  no such (t� x· ) exists in [0, T] x D . D  
I f  h satisfies a Lipschitz condition then a slronger result can be stated. 
Theorem 3.2.6 (Strong Comparison Theorem) Suppose that 
(i) The functions C I  and C2 are defined and are continuous in [0, T] x 15, their first order Xi 
derivatives exist and are continuous in (0, T] x D ,  and their t derivatives exist and are 
uniformly bounded and continuous in the region (0, T] x D ;  
(iii) C] $; C2 on (0, T] x aD ; 
Then 
CI $; C2 in [0, T] x D . 
Proof 
Let 
Then 
3.2 GENERALISED COMPARISON THEOREMS 
w = C2 + Ae2K1 . 
tCI -h(t, x, w)= leI - h(t, x, w) + 2KAe2K1 + h(t, x, C2) - h(t, x, C2 ) 
� leI - h(t, x, c\ ) + 2KAe2K1 - [h(t, X, w) - h(t, x, C2 )] 
� leI - h(t, x, cI ) + 2KAe2K1 - Klw - C2 1 
� leI - h(t, X, CI ) + 2KAe2K1 - KAe2K1 
� leI - h(t, x, CI ) + KAe2K1 
< tCI - h(t, x, cI ) '  
28 
Now w(O, x) < C l (O, x) and w < c I on (0, 1'] x dD , so by Theorem 3 .2.5 (Weak Comparison Theorem), for 
ordinary differential equations, w < C l an [0, T] x D for all A which implies that CI � C2 on [0, T] x D.D 
The following theorem provides a slronger result and spells out the consequences o f  the existence of  contact 
points of these comparison fllnctions. 
Theorem 3.2. 7 (Strong Comparisoll (Contact) Theorem) Suppose Ihal 
(i) The functions Cl and C2 are defined and are continuous in [0, T] x D, their first order Xj 
derivatives exist and are continuous in (0, T] x D and their I derivatives exist and are uniformly 
bounded and continuous in the region (0, 1'] x D; 
(ii) CI (t, x) � C2 (t, x) in [0, T] x D ; 
(iii) leI - h(t, x, cI ) � tC2 - h(t, x, c2 ) in (0, T] x D ;  
(iv) The function h is Lipschitz continuous in c so that there is a finite constant K > ° for which 
I h(t, x, cl ) - h(t, x, c2 )1� Klcl - C2 1 in [0, 1'] x D x  R .  
Then either 
Cl (t, x) < ez(t, x) in [0, T] x D 
or there is a constant To in (0, T] and a point x· in D such that 
C I  (t, x) < C2(t, x) in (10,  T] x D and C I  == C2 at those points which lie in [0, 10 ] x D and also lie 
along the characteristic curve given by dx = a(t,x) , x = x· at To for t in [0, To] . 
dt 
Proof 
Either C l  < C2 in [0, 1'] x D or there is a constant To in (0, 1'] and a point x· i n  D such that C l  = C2 at 
(To , x·) in (0, T] x D . Let tcl = h(t, x, cl ) + A1 (t, x, Cl ) and tC2 = h(t, x, c2) + A2 (t, x, C2) where A I  � A2 
3.2 GENERALISED COMPARISON THEOREMS 29 
in (0, T] x D .  Then along the characteristic curve dx = a(t, x), x = x· at T o ,  we have dCI = 
dC2 dt . dCI dt h(t, x, cI ) + A1 (t, x, CI ) and -=h(t, x, c2 ) + A1 (t, x, C2 ) ho ld ing ,  so tha t  --h(t, x, CI ) �  
dC2 - h(t, x, c2 ) in (0, T]X D , an� hence from (ii) and (iv) and Theorem 3 .2.4, i f  CI (To ,
d�· ) = C2 (To ,  x· ) dt 
then cI (t, x) = c2 (t, x) for t in [0, To] , x on the characteristic curve and (t, X)E [0, T] x D . O  
Weak and Strong Comparison Theorems for Parabol ic Differential Equations 
We now give weak and strong comparison theorems for quasilinear parabolic equations in one dependent 
variable of the form 
aC m a2c m ac . !JX = T - L aj (t, x)-.a-. - Lbj (t, x)-:;-= h(t, x, c) In (0, T]xD, t j.j=1 ax, Xl j=1 aXj 
(3.2.2) 
where D is a finite domain in Rm, T < 00 and 9! is a uniformly parabolic operator with bounded coefficients 
aij (t, x) and bj(t, x) . The boundary aD of D must satisfy the inner sphere property which requires every 
point on aD to lie on the surface of an open sphere contained in D. 
The following two theorems are standard. The proofs follow along the lines for comparison theorems already 
discussed (see FRIEDMAN [93, 94]). 
Theorem 3.2.8 (Weak Comparisoll Theorem) Suppose that 
(i) The functions CI and C2 are defined and are continuous in [0, T] x D ,  their first order Xj 
derivatives exist and are continuous in (0, T] x D ,  their second order xi) derivatives exist and 
are uniformly bounded and continuous in the region (0, 1'1 x D and their t derivatives exist and 
are uniformly bounded and continuous in the region (0, TJ x D ;  
(ii) 
(iii) 
(iv) (Jel aC2 a(t, x)cI + {J(t , X) Tn < a(t, x)c2 + (J(t, x) an on (0, T] x aD, 
where aCt, x) � 0, (J(t, x) � 0 on (0, T]xJD and a+{J > 0 at each point. 
Then 
CI < c2 in [0, 1'] x D .  
If, h(t, x, c) satisfies a Lipschitz condition, then a stronger result can be stated . 
Theorem 3.2.9 (Strong Compariso1l Theorem) Suppose that 
(i) The functions Cl and C2 are defined and are continuous in [0, T] x D ,  their first order Xj 
derivatives exist and are continuous in (0, T] x D ,  their second order xi) derivatives exist and 
are uniformly bounded and continuous in the region (0, 1'] x D and their t derivatives exist and 
are uniformly bounded and continuous in the region (0, T] x D ;  
(ii) Cl(O, x) $; C2 (0 , x) ; 
3.2 GENERALISED COMPARISON THEOREMS 30 
(iii) !l!cl - h(t, X, CI ) ::; !l!c2 - h(t, x, c2 ) in (0, T] x D ;  
dCI dC2 , aCt, X)CI +{3(t, x)-;- ::; aCt , X)C2 +{3(t, x)-on (O, 1 ] x  dD, on dn 
(iv) 
where a(t, x) � 0, {3(t, x) � 0 on (0, l1xdD and a+{3 > ° at each point; 
(v) The function II is Lipschitz continuous in c so that there is a fInite constant K > 0 for which 
Then 
CI ::; c2 in [0, '1'] x D. 
The following theorem provides a stronger result and spells out the consequences of  the existence of  contact 
points of these comparison functions. 
Theorem 3.2.1 0  (Strong Comparison (Contact) Theorem) Suppose that 
(i) The functions C I  and C2 are defined and are continuous in [0, T] x D, their first order Xj 
derivatives exist and are continuous in (0, T] x 75,  their second order xij derivatives exist and 
are uniformly bounded and continuous in the region (0, T] x D and their t derivatives exist and 
are uniformly bounded and continuous in the region (0, T] x D ;  
(ii) CI ::; c2 in [0, 1'] x 75;  
(iii) !l!cl - h(t, x, CI ) ::; !l!c2 - h(t, x, c2 ) in (0, T] x D ;  
(iv) aCt, x)cI + {3(t , x) ��::;  a(t, x)c2 +{3(t, X)�� on (0, T] X dD,  
where aU, x) � 0, /3U, x) � ° on  (0, llxdD and a+/3 > 0 a t  each point; 
(v) The function h is Lipschitz continuous in c so that there is a finite constant K > 0 for which 
Then either 
CI < c2 in [0, T] x D, (3.2.3) 
or there is a constant To in (0, 1'] such that 
(3.2.4) 
If {3 > 0 on (0, l1xaD, (3 .2.4) can be replaced by the stronger result 
CI < c2 in (To ,  T] x D  and ci = c2 in [0, To] x D .  (3.2.5) 
Proof 
Define functions VI and V2 by 
(3.2.6) 
where K is the Lipschitz constant for h, so that 
3.2 GENERALISED COMPARISON THEOREMS 31 
(3.2.7) 
and 
!lvl � !lv2 in (0, T] x D. (3.2.8) 
Either VI < V2 and hence CI < C2 in (0, 1'] x D or there is a To in (0, 1' ] such that VI < V2 in (10 , 1'] x D and 
V I  = V2 at a point P in 15 at time To. But if P is in D,  this implies that V I  == V2 and hence C }  == C2 in  
[0 ,  To ] x D by the strong comparison theorem of Nirenberg for parabolic equations (NIRENBERG [204]). On 
the other hand, if V I  = V2 at  P on aD at  time To, and f3 > ° on (0, TJxaD,  then from (v), �VI � ;2 at P 
h ' l  h . f d " ( O O .) 
aVI aV2 h h 
aVI dV2 P I th· 
n 
h 
n 
w 1 e at t e same tIme, Tom con Itlon I I I , - � - t ere, so t at -= - at . n IS case t e strong an dn an dn 
comparison theorem of Friedman implies V I  == V2 in [0, To ] x D (FRIEDMAN [93]), so that our conclusions 
(3 .2.3) and (3 .2.5) are established. If f3 can vanish on (0, T] x aD ,  there is a To such that V I  < V2 in 
(7o , '1'1 x D and VI = V2 at a point P in D at time To. The Nirenberg theorem then establ ishes (3.2.4).0 
3.2.2 General ised Weak and Strong Comparison Theorems 
We have developed a number of comparison theorems for scalar equations. Extensions of these results to 
systems of equations where CI , C2 and h are taken as n-vectors in previous theorems will not do. A simple 
counterexample for ordinary differential equations is given by McNABB [ 1 86] and a counterexample for 
multicomponent diffusion systems is given by WAKE [302]. However, an extension can be obtained by the 
concurrent use of upper and lower bounds in the formulation of the comparison theorems by redefining h. 
We therefore look for comparison theorems for the system S,. , 8,. by defining the following functions 
for 
where, 
I(t, X, fk ' c/ ) = inffi(t, x, OJ), 
/j(1, x, fk '  c/) = supfi(t, x, OJ), 
L(t, z, �k '  C/) = inf Fj(t, z, 8j), 
F;(t, z,  �k' C/) = sup Fj(t, z, 8), 
OJ = fj in L, OJ = Cj in /j ' 
8j = �j in Ij ' 8j = Cj in F; , 
(3.2.9) 
(3.2.10) 
(3.2. 11)  
(3.2.12) 
(3.2.13) 
(3.2.14) 
(3.2.15) 
(3.2.16) 
i .e., OJ lies in the closed interval bounded by fj and Cj for all } :t- i and 8j lies in the closed interval bounded 
by r;;,j and Cj for all } :t- i. 
3.2 GENERALISED COMPARISON THEOREMS 32 
Theorem 3.2 . 1 1  (Generalised Weak Comparison Theorem) 
Suppose (ci ' Ci ) is a solution of the system Sn ' Bn and the functions £" ci ' L and Ci are defined and 
satisfy the following continuity properties and inequalities: 
(i) For components i E I ,  where Di > 0, £i ' ci and ci are continuous in [0, T] x n x A ,  their first­
order xj-derivatives exist in (0, T] x !2 x A , their second order xjXk-derivatives and first order 
t-derivatives exist and are continuous and uniformly bounded in (0, T] x n x A ; 
(ii) For components i E I ,  where Di = Hi = 0, fi ' ci and ci are continuous in [0, T] x n x A and 
their first order t-derivatives exist and are continuous and uniformly bounded in (0, T] x n x A ; 
(iii) For components i E ], where !lJi > 0, �i , Ci and Ci are continuous in [0, 1'] x A , their first order 
Z derivatives exist in (0, T] x A,  their second order zjZk-derivatives and first order t-derivatives 
exist and are continuous and uniformly bounded in (0, T] x A ; 
(iv) For components i E ], where 9>i = 0, U ·  VCi $0, fi ' C i and Cj are continuous in [0, T] x A ,  
their first order Zj derivatives exist in (0, T] x A and their first order t-derivatives are 
continuous and uniformly bounded in (0, 1'] x A ; 
(v) For components i E .I ,  where �j = 0, It · VCi == 0, �j , Cj and Cj are continuous in r O, 'f ] x A and 
their first order t-derivatives exist and are continuous and uniformly bounded in (0, T] x A ; 
(vi) !2j < Cj < Cj in .Q x A and �j < Ci < G in A at t = 0; 
(vii) 
dc · dC' de (viii) -=!.. < -' < -' on (0 T]xdn xA' dn dn dn ' I ,
(ix) 
(x) 
de · dc dc· -D· -=!.. - fJ. (C - c · ) < D -' - H · (C - c· ) < D -' - H · (C; -c )  on (0 71xdn2xi\' , an ' -, -, , an " , , an ' " , 
d�j - 9J.,V
2 C,. + u . VC . + H,. f (C,. - c, ) -F, (t, z, Ck '  C,) at - -, Jil2 - - - -
< aCj - 9J.V2C + u . VC + I-I J (C - c ) - F (t z C) dt " " JIl2 ' "  , , J 
(xi) �j. !  < C,.! < G.! , 
VI= -u 'n l is uniformly bounded and continuous in (0, 71xdA, 
(3.2.17) 
(3.2.18) 
(3.2.19) 
(3.2.20) 
(3.2.21) 
(3.2.22) 
(xii) 
3.2 GENERALISED COMPARISON THEOREMS 33 
_ .. _------- --_._-_ .. _----_ .... _. -_._---------- -_._------ --_._--
so that 
ac. ae. - aCi 
V&:i + � �-I < VICi + �-I < VI Ci + �� on (O, T] x aAI ; anI anI anI 
ac .  ae. ae -=!..< __ I <� on (O, T] X dAa ,  a = 2, 3 .  dna ana ana 
(3.2.24) 
(3.2.25) 
Then fi < ci < ci in [0, T] x Q x A and (;.i < Ci < Ci in [0, T] x A . 
Proof 
We w ill only look at the case when Di, 9Ji > 0 for all i. The proofs for the other cases follow along similar 
lines of Theorem 3 .2 . 1  (Weak Comparison Theorem) for ordinary differential equations and Theorem 3 .2.5 
(Weak Comparison Theorem) for first order partial differential equat.ions. 
We let ui = ci - fi and vi = ci - ci in [0, T] x Q x A and Vi = Ci -L and Vi = Ci - C; in [0, T] x A . 
Then if the conclusion is not true then either there exists a point (t� x� z" ) in [0, T] x Q x A and an index 
i E I  such that Uj (t, x, z )� O � v/t, x, z) on [0, t" ] x Q  x A ,  j '# i and ui (t, x, z) < O < vi (t, x, z) on 
[0, t" ] x Q x A ,  or there exists a point (I: x" ) in [0, T] x A  and an index i E J  such that 
Vj (t, z) � 0 � Vj (t , z) on [0, t" ] x A, j '# i and Vi (t, z) < 0 < Vi(t ,  z) on rO, t o )  x A .  
Suppose firstly that there exists such a point (t � x� z" ) in [0, T] x Q x A .  B y  continuity, we have 
either ui (t� x: z" ) = O or vi (t: x: z" ) = O . 
If (I: x: z") E (0, T] x ()QJ x A , then either ()Ui � 0 or aVi � 0, which is impossible by (viii) 
()n an 
If (I: x� z" )  E (0 ,T] x Q x A ,  then (I� x: z" )  is either a point of minimum of Ui or a point of maximum 
of Vi and these minimum and maximum values at these points are equal to zero. Hence if  0: x: z* ) E 
(0, T] x Q x A , we have either 
Ui (t: • x, 
or 
Vi (I: " x ,  
" ) = 0 aUi ( • z , a t ,  Xj 
' ) = 0  aVi ( ' z , a t ,  X ·  ) 
• x, 
" x ,  
n a2 " ) - 0 LA?�(t' z - , 1 a 2 ' j=1 Xj 
n a2 
" ) - 0 LAL�(t" z - ,  1 a 2 ' j=1 Xj 
• Z·) � O ,  x ,  
" z· ) � O .  x ,  
Suppose that the first alternative holds, then it implies that at (I: X: z" ) ,  
ac .  -=.L(t" • � , x ,  
" ) aCi ( ' " ' ) Z = - t x ,  z , aXj , dXj 
n �2 
"" , 2 (!'....S.. (t' " L.J/I., a 2 ' x ,  j=1 Xj 
" )
' a2 fi ( "  " z --a 2 t ,  x ,  Xj z" » � O .  
X �  z" ) .  Since aUi = �(Ci - c )  � 0 also, it follows that dt dt -I 
3.2 GENERALISED COMPARISON THEOREMS 
dc· dc · 2 2 -' -----=l. - D V c· + D· V  c ·  < 0 dt dt ' x , ' x  -'" - . 
Hence, we have 
at the point (t: x: z * )  which is a contradiction. 
34 
* * * au· avo If (t , x ,  z ) E (0, T] x af22 x A , then either Di -' + Hiui � 0 or Di -' + Hivi � 0 and so there must exist a 
- � � -
point (t: z + ) in [0, T] x A  and an index i E  J such that Vj (t, z) � O � Vj(t , z) on [0, t* ] x A , J :T. i and 
Vi (t , z) < O < v; U, z) on [0, t * ) x A . 
Suppose there exists such a point in [0, T] x A . By continu ity , we have either Vi (t: z * ) = O  or 
v; U: z* ) = O .  
If (t: z * )  E (0, T] x A , then (r: z * )  is either a point of minimum of Vj or a point of maximum of Vi and these 
minimum and maximum values at these points are equal to zero. Hence if Ct: z * )  E (0, T] x A ,  we have 
either 
or 
. + + _ aVi + + _ � , 2 a2Vi + + > V, (t , z ) - 0 ,  "l (t , Z ) - 0 , LJ /I., 2 (t , z ) _ 0 , aZj j=1 aZj 
* + aV; * + � , 2 a
2Vj * + \ti (t , Z ) = 0 , --;-:-Ct , Z ) = 0 , LJ /l.j -2-. (t , Z ) � O. az; j=l az;
Suppose that the first alternative holds, then it implies that at (t: Z * ) ,  
ac 2 J ae 2 J -='- - 9J.V C + u · VC + fI . (C - c ) � -' - �V C· + u · VC + lI  (C· -c · ) .  dl , -, -, ' an-z -, -, dl " " an2 " 
Hence, we have 
d C .  2 f 
-;;' - 9lV �j + u · V�j + /-/, ail2 (�j - fj ) -E, (r , z, �k '  C, ) 3C 2 f > -' - 9! V C + u · VC + fI . (C - c · ) - F (I Z C - ) - dt ' J  J J ail2 J J J "  J ' 
at the point (t: z+ ) which is a contradiction. The second alternative is treated similarly. 
3.2 GENERALISED COMPARISON THEOREMS 
If (t: z' ) E (0, T] x dA I , then vlVj + � dVj � 0 or VI Vi + 91 dVi � 0, which is impossible by (ix). dnl dnl 
If (I: z' ) E (0, T] x dAa ,  a = 2, 3 ,  then either dVj � 0 or �Vi � 0, which is impossible by (xii). 
dna ana 
35 
Thus, we get Uj (t ,  x, z)< O<  Vj (t, x, z) on [0, 1') x Q x A and Vj (t , z) < 0 < Vi(t, z) on [0, 1') x A for all i. 
We would arrive at the same conclusions if we suppose firstly that there exists such a point (t: z' )  in 
[0, T) x X and this proves the claim of the theorem.O 
Ifli and Fj satisfy a Lipschitz condition of the following form, a stronger result can be stated which allow 
general inequalities in (3.2. 17)-(3.2.25). We shall need the following assumption onli and Fj. 
(HI) Ii and Fj satisfy a uniform Lipschitz condition in Cj and Cj respectively on any fini te interval, so that 
there are positive constants kj and Kj such that 
(3.2.26) 
It can be shown that our assumptions (HI)  of Lipschitz continuity properties for the functions .ti, Fj with 
respect Lo the variables Cj and Cj imply similar properties for L, j; ,  Ei and F,. in the variables f.k ' cl ' (k and 
� .  We will first require the following lemma: 
Lemma 3.2.1 
Suppose that we choose the points XI , X2, Y I  and Y2 where X I  < Y I  and X2 < Y2 and suppose that there 
exists () E [X I ,  yd .  Then we can always find ¢I E [X2, Y2] such that 
Proof 
Either 8 E [X2 , Y2] or 8 e [X2, Y2] .  I f  8 E [X2 , Y2] ,  we may lake ¢I = () so  thal min I ()  - ¢II = O. If 
¢ 
() e [X2 , Y2] ,  then ei ther () < X2 or e > Y2 . If () < X2, then XI <()  < X2 and we may take ¢I = X2 so that 
min Ie - ¢II = e - X2 < IXI-X21 or if e > Y2, then YI > Y2 and we may take ¢I = Yl so that min Ie - ¢II = e -Y2 
¢ ¢ 
< IYI-Y21 .  Hence, we can always find ¢I E [X2, Y2] such that min I e  - ¢II � max(lxI - x2 1, IYI - Y2 1} .0 
¢ 
We are now able to prove the following 
Lemma 3.2.2 
Our assumptions (/-I I) of Lipschitz continuity properties for the functions .ti, Fj with respect to the 
variables Cj and Cj imply similar properties for L, j; .  E and F,. in the variables f-k ' Cl ' kk and Et and 
so there are positive constants kj and Kj given by (/-I I) such that 
I L (t , X, fk ' c,)-L (t, X, f: ' ct)1 � kj skUf(lfk -f: I , I CI -ct l ) ,} 
I A (t, X, fk ' cl )-A (t, x, f:, ct)1 � kj Sf,f(lfk -ft l ,  I C1 - Cj* I ) ,  
(3.2.27) 
and 
Proof 
3.2 GENERALISED COMPARISON THEOREMS 
I£j (t ,  Z, �k ' 0 )-£, (t , z, �Z , 0* )1 � Kj sup(l�k -�Z I ,  1 0 - 0* 1 ) ,} k,1 
I�(t ,  z, �k ' 0 )-�(t , z, �Z , 0* )1 � Kj sup(l�k -�Z I ,  1 0 - 0* 1 ) .  k,1 
We will only prove the first inequality. The other inequalities follow similarly. 
36 
(3.2.28) 
Assume wi thout loss of generality that f j < Cj and f j < Cj* so that 1.£ (t , x, fk ' c/ ) - .£ (t, x, fZ , "0* )1 = 
Ih(t, x, 0j)-h (t ,  x, OJ )l , where by definition, OJ E [fj , cj l and OJ E [fj , cn for all j. Suppose that 
I h (t ,  x, OJ) "? h(t ,  x, OJ )1 . Then Ih (t ,  x, 8j ) - h(t ,  x, OJ)1 � h(t, x, 0j ) - h(t ,  x, CPj ) '  where CPj is any 
point in [fj , en . By Lemma 3.2. 1 ,  we can choose CPj so that min 18j - cpj l � max {Ifj - fj l, lej - e!l} for 
tPj 
all j. Then for this choice of CPj, 
If;(t, x, 8j)-f;(t, x, OJ )1 � kj I Oj - cpj l 
= kj sup IOj - cpj l  
J 
� kj suP(lfj - fj l ,  I Cj - c! I) , 
J 
for all j and the theorem follows .0 
This theorem is the basis for the following comparison theorems for solutions of the system Sn ' Bn . 
Theorem 3.2.12 (Generalised Strong Comparison Theorem) 
Suppose (Cj , Cj ) is a solution of the system Sn' Bn and the functions fj ' Cj , C and Cj are defined and 
satisfy the following continuity properties and inequalities 
(i) For components i E I, where Dj > 0, fj , Cj and Cj are continuous in [0, T) x .f2  x A ,  their first­
order xrderivatives exist in (0, T) x .f2  x A ,  their second order xjXk-derivatives and first order 
t-derivatives exist and are continuous and uniformly bounded in (0, T) x Q x A ; 
(ii) For components i E I .  where Dj = H j = 0, fj ' Cj and Cj are continuous in [0 , T) x Q x A and 
their first order t-derivatives exist and are continuous and umformly bounded in (0, T) x .f2  x A ;  
(iii) For components i E J, where 9)j > 0, �j , Cj and Cj are continuous in [0, T) x if, their first order 
Z derivatives exist in (0, T) x A , their second order zjZk-derivatives and first order t-derivatives 
exist and are continuous and uniformly bounded in (0, T) x A ; 
(iv) For components i E J. where q)j = 0, U ·  VCj $0, �j ,  C j and Cj are continu.ous in [0, T) x if , 
their first order Zj derivatives exist in (0, T] x A and their first order t-derivatives are 
continuous and uniformly bounded in (0, T) x A ; 
(v) For components i E J, where q)j = 0, U ·  VCj == 0, �j , Cj and Cj are continuous in [0, T) x .lf  and 
their first order t-derivatives exist and are continuous and uniformly bounded in (0, T) x A ; 
(vi) f.j � Cj � ej in QxA  and �j � c.. � Ci in A at t = 0; (3.2.29) 
(vii) 
--------------------------- - -- - -
3.2 GENERALISED COMPARISON THEOREMS 
as 2 -Tt-DjVx fj -L (t , X, fk '  C, ) 
ac· 2 < -' - D·V c· - F (t x c · ) - at ' x , Jj , , J 
< aCj D n 2- -I ( - ) . (0 11 /0., • -Tt- j V xCj - j t, X, fk '  Cl In , . XH"';\, 
(viii) afj < aCj < aCj (0 71 aQ ;\ . an - an - an on , x 1 X ,
(ix) 
(xi) 
(xii) 
ac . ac· ac· - _ D-=1..- /1 (C - c · ) < D -' - J1 . (C- - c ) < D -' - II-(C,' - c·)  on (0 Tl xaf22xA , an , -, -'" - ' an " " - ' an ' , ' . ,
vI=-u 'n l is uniformly bounded and COnlinuous in (0, T]xa;\ \ ,  
so lhat 
(xiii) Ij and Fj satisfy the uniform Lipschitz condition (J-J I ) '  
Proof 
Construct the functions : 
I.. , 1(/ -I.. - , 1(/ CI.. C ' 1(/ C-I.. C- , 1(/ , 0 0 fj = fj - Iloe • Cj = Cj + Iloe • _j = _j - Iloe , j = j + Iloe • Ilo > , K > . 
37 
(3.2.30) 
(3.2.31) 
(3.2.32) 
(3.2.33) 
(3.2.34) 
(3.2.36) 
(3.2.37) 
(3.2.38) 
The conditions of Theorem 3 .2. 1 1  (Generalised Weak Comparison Theorem) can be shown to be satisfied by 
these functions. We give the details for the only difficult aspect; to show that assumptions (vii) and (x) can 
hold for all A. > ° if K is large enough. Our assumptions of Lipschitz continuity properties for the functions/; 
and Fj with respect to the variables Cj and Cj imply similar properties for 1.-, j; ,  E, and F; in the variables 
fk ' Cl ' Ck and 'Et by Lemma 3 .2.2. The inequalities (3 .2.30) and (3 .2.33) in assumptions (vii) and (x) 
----------- - - - -
3.2 GENERALISED COMPARISON THEOREMS 38 
follow for our functions �t ' c/o. , �t ancl � A. if /( is chosen bigger than !<:.j , is ,  !S.j ancl 1( .  We give the 
argument for the first inequality of (3.2.30): 
[ aCj D v2 ; ( ) aft V2 A. f ( A. -A. Ii- j xCj - Jj I , x, Cj ] - [Tt - Dj x fj - _j I , X, fk ' C, )] 
= e + AxeKl + i./I , X, fk - A.e� c, + A.eKl ) -i/I , x, d,  c/ )  
� e + A.( K"-kj )e 1(/ 
> 0, 
aC' 2 ac . 2 where e = [--:f - DjVxcj - hCt , x, cJ )] - [� - DjVx cj - f(t, x, Ck , c,)l > ° from (3.2.30). al al - -I -
(3.2.39) 
The other inequalities follow in a similar fashion. Hence for all A. > 0, we have from Theorem 3.2. 1 1 ,  
d < ct < CjA. in [0, T] x Q x A and r.t < CjA. < E/ in [0, T] x A , (3.2.40) 
ancl lllerefore in the limit as A. tencls \.0 zero, 
(3.2.41) 
and the result follows. 0 
We have seen some generalised weak and strong comparison theorems. While these theorems are useful for 
solutions of Sn, Bn where initial values at t = ° are given, they provide no information about the steady state 
solutions for which only boundary value data is given. The following theorem provides a stronger resul t  and 
spells out the consequences of the existence of contact points of these comparison functions. Its corollary 
provides useful information about the steady state solutions. 
Theorem 3.2.1 3  (Generalised Strong Comparison (Contact) Theorem) 
Suppose (Cj ' Cj ) is a solution of the syslem Sn ' Bn and Ihe funclions f; ,  Cj ' �j and Cj are defined and 
salisfy the following continuity properties and inequalities: 
(i) For components i E I ,  where Dj > 0, f.j ' Cj and Cj are continuous in [0, T] x Q x A ,  their first­
order xj-derivatives exist in (0, T] x Q x A , their second order xjXk-derivatives and first order 
I-derivatives exisl and are conlinuous and uniformly bounded in (0, T] x Q x A ;  
(ii) For components i E I ,  where Dj = I-I j = 0, f.j ' Cj and Cj are continuous in [0, T] x Q x A and 
Iheir first order t-derivatives exist and are continuous and uniformly bounded in (0, T] x Q x A ;  
(iii) For components i E J, where q)j > 0, �j , Cj and Cj are continuous in [0, T] x A ,  their first order 
Z derivatives exist in (0, T] x A ,  their second order zjzk-derivatives and first order t-derivatives 
exist and are continuous and uniformly bounded in (0, T] x A ; 
(iy) For components i E J, where 9)j = O, u , VCj �o,  �j , C j and Cj are continuous in [0, T] x A , 
their first order Zj derivalives exist in (0, T] x A and their first order t-derivatives are 
conlinuous and uniformly bounded in (0, T] x A ; 
3.2 GENERALISED COMPARISON THEOREMS 39 
------------
(v) For componeflls i E J, where 9)j = 0, U ·  VCj == 0, �j ,  Cj and Cj (lrc cOnlinuous in rO, T] x A and 
their first order t-derivatives exist and are continuous and uniformly bounded in (0, 1'] x A ;  
(vi) 
(vii) 
(2j � Cj � Cj in [0, T] x Q x A and �j � Cj � � in [0, T] x A ; 
afj 2 -
Tt-DjVxfj -1/t, x, fk ' CI ) 
ac· 2 < -' - D·V C· - r et x c · ) - at • x .  Jj , , J 
< aCj _ D n2-. _ -f ( - ) ' (0 1'] t>., A .  - at , v xc, j t, X, fk ' C, In , X�"A.Il, 
( . . .  ) afj < aCj < aCj (0 T] an A .  Vlll an - an - an on , _ x �"l x.ll, 
(ix) 
(x) 
D· afj - II (C · -c) < D aCj - I/ . (C- - c )  < D rJc. - H (C,' - c· ) on (0 TJxrJ!22XA' • an • -, -, - ' an " " - ' an ' '" • 
ac .  2 f ---=i.- g)V c · + u · vC · + /-1 (C · - c . ) -P, (t, z, Ck ' C,) at , -, -" iJ!l2 -, -, -, -
ac- 2 f � -' - !l)V c- + u ' vc- + /-I (C- - c) - F (t z C - )  at " " iJ!l2 " "  , , J 
aCj 2- - f - - - - . � -- �v Ci + U ·  VCj + Hj (Cj - cJ- F; (t, z, Ck ' C, ) In (0, T]xA; � � -
(xi) C.I :os; Cj•1 :os; �.1 ' 
(xii) 
V1=-u 'n 1  is uniformly bounded and continuous in (0, T]xaA I ,  
so that 
ae < aCj < aCj (0 T) x a A 2 3 
a - a - a 
on , "a' a = • ; na na na 
(xiii) Ii and Fj satisfy the uniform Lipschitz condition (/-I I ) '  
Then 
(3.2.42) 
(3.2.43) 
(3.2.44) 
(3.2.45) 
(3.2.46) 
(3.2.47) 
(3.2.49) 
(3.2.50) 
(I) For components i E I, where Dj • Hj > 0, 9)j>O, either fj < Cj <Cj in (0, T]xQ  x A and f:2j < Ci < Cj 
in (0. T) x if or there are constants Tj in (0, T] such that fj < Cj < Cj in (1; , T] x .Q  x A ,  C j == Cj 
(or Cj == fj ) in [0, 1i ) x Q x A ,  �j < Ci < Cj in (7; , T) x X  and Cj == C; (or C, == �j ) in [O. 7; ) x X. 
(II) For components i E I, where Dj , Hj > 0. Si)j = 0. u ·  vej $ 0, e ither fj < Cj < Cj in (0. T) x Q x A 
and �j < Cj < Cj i n  (0, T] x if or there are constants Tj in (0. 1'] such that fj < Cj < Cj j n 
(7; . T] X .Q  x A . �j < Cj < Cj in (7; , T) x X and there is at least one point z· in if such that 
3.2 GENERALISED COMPARISON THEOREMS 40 
C j = Cj (or Cj = �) i n  rO, 7; ] x .Q x A  a n d  Cj =� (or Cj = �j ) in 1 0, 1; ] x A  along the 
characteristic curve given by dz = u(t, z) , z = z · a t  T dor t in [0, 1] or until z reaches the 
dt boundary dA\ .  
(III) For components i E I, where Dj , Hj > 0, g)j = 0, U ·  VCj = 0, either fj < Cj < Cj in (0, T] x ..Q  x A 
and �j < Cj < Cj i n  (0, TJ x A or there are constants Tj in (0, T] such that fj < Cj < Cj i n  
(1'; , T] x ..Q  x A and �j < Ci < Cj in (1'; , T] x A and there is at least one point z· in A such that 
Cj = Cj (or Cj = f;) in [0, 1';] x ..Q  x A and Cj = C; (or Cj = �j) in [0, 7;. ] x A for t in [0, 1'; ]  at z·. 
(IV) For components i E /, where Dj = Hj = O, e i t h e r  �j < Cj < Cj i n  (0, T] x ..Q x A  or there are 
constants Tj in (0, T] such that �j < Cj < Cj in (1i , T] x .Q x A and there is at least one point 
(x� z* ) in ..Q* x A such that Cj = Cj (or Cj = fj) in [0, 1'; J x  ..Q* x A for t in [0, 1'; ]  at (x� z*) ,  
where .Q * is some simply connected part of..Q containing x* . 
(V) For components i E i, where Dj = Hj = 0, g)j > 0, either (;2j < Ci < Cj in (0, T] x A or there are 
constants Tj in (0, 1'] such that �j < Ci < Cj in (1'; , T] x A and Cj = C; (or Cj = (;2j ) in [0, 1'; ]  x A. 
(VI) For components i E .1 ,  where Dj = Hj = 0, g)j = 0, U ·  VCj "$ 0 ,  either �j < Ci < Cj in (0, T] x A, 
or there are constants 1i in (0, T] such that �j < Ci < Cj in (1'; , T] x A and there is at least one 
point z· in A such that Cj =�. (or Cj = (;2j ) in [0, 7; ] x A along the characteristic curve given by 
dz = u(t, z) , z = z· at T;/or t in [0, 1';] or until z reaches the boundary dA! .  
dt 
(VII) For components i E i, where Dj = Hj = 0, g)j = 0, U · VCj =0 ,  either (;2j < Ci < Cj in (0, T] x A or 
there are constants Tj in (0, T] such that (;2j < Cj < Cj in (1'; , TJ x A and at least one point z· in 
if such that Cj = C;  (or Cj = �j ) in [0, 7; J x  if for t in [0, 7; ] at z* . 
Proof 
Let us define 
(Mt, x, Uj ) = /; (t ,  x, cj ) where ci = Uj and cj = Cj for j 1= i ,  
and 
Then from (vi) and (ix), we have, 
D dCj LJ D dCj '·1 <D dc, LI ­·�+ £ · c < . - + r: ·c · . _ + 1,  ·c · 1 dn 1 -I - 1 dn 1 1 - 1 dn I "  
from (vi), (vii) and the definition of j . ,  1; and CPj, we have 
-I 
and from (vii), (x) and the definition of f.j ' F,. and c1>j, we have, 
(3.2.51 )  
(3.2.52) 
(3.2.53) 
(3.2.54) 
(3.2.55) 
3.2 GENERALISED COMPARISON THEOREMS 41 
The inequalities governing the functions fj ' Cj , Cj , �j ' Cj and C; are now all uncoupled and our conclusion 
is a consequence of Theorem 3 .2.4 (Strong Comparison (Contact) Theorem) for ordinary differential 
equations, Theorem 3.2.7 (Strong Comparison (Contact) Theorem) for first order partial differential 
equations and Theorem 3.2. 1 0  (Strong Comparison (Contact) Theorem) for parabolic equations. 
(I) For components i E I, where Dj , Hj > 0, 9)j>O, either fj < Cj < Cj in (0, T] x £2 x A and {2j < Cj < Cj 
in (0, T] x A ,  or there are constants Tj in (0, T] such that fj < Cj < Cj in (� , T] x £2 x A ,  �j < Cj < Cj in -
* * * *  - -(�, T] x A ,  cj = cj (= f;) at a point (� , x ,  Z ) ,  where (x , Z ) E £2  x A and Cj = Cj (= C) at a point 
* -(�, Z ) where Z· E A . 
Suppose that cj = Cj at a point (� , x *, Z * ) .  Theorem 3.2. 1 0  then implies cj == cj in (0, � ]  x £2* x A , 
where n· is some simply connected part of £2 at (1; ,  Z * ) containing x· . Condition (ix) shows Cj = C; at 
(7; , Z * ) and hence Cj == C; in [0, 1; ] x A.  This same resul t  is obtained from the alternative assumption that 
Cj =C; at a point (1; , z*) .  Condition (x) now impl ies cj = e;  on an x A  at l=1'j and hence cj == cj in 
[0, �]  x n x A and Cj == C; in [0, T; ] x X.  
(II) For components i E I, where Dj, I-Ij > 0 ,  9)j = 0 ,  u · VCj $0, either fj < Cj < Cj in (0, T ]  x £2 x A and 
{2j < Ci < Cj in (0, 1'] x A ,  or there are constants 1'j in (0, 1'] such that fj < Cj < Cj in (T; , T] x £2 x A ,  
!'2j < Cj < Cj in (1; , T] x A,  Cj = Cj (= fj ) at a point 0; , x*, z* ) ,  where (x; z* ) E £2 x A and Cj = C; (= !'2;) 
* -at a point (T; , z ) where z· E A . 
Suppose that cj = Cj at a point (� , x *, z * ) .  Theorem 3.2. 1 0  then implies cj == cj in (0, 1; ] x £2* x A , 
where £2. is some simply connected part of £2 at CT; , z* ) containing x·. Condition (ix) shows Cj = C; at 
(T; , z *) and hence Cj == 1:; (or Cj == !'2j ) in [0, 7; ] x A along the characteristic curve given by dz = U(I, z) , 
z = z ·  at Tj for 1 in [0 , T; ] or until z reaches the boundary (JA t .  This same result is obtaing� from the 
alternative assumption that Cj = C; at a point (1i , z *) . Condition (x) now implies cj = cj on a£2 x A at I=Tj 
and hence cj == cj in [0, � ]  x n x A and Cj == C; in [0, � ]  x A along the characteristic curve given by 
dz 
= U(I , z) , Z = z· at 1'j for I in [0, 1; ] or until z reaches the boundary (JAt .  
dl 
(III) For components i E I, where Dj , Hj > 0, 9)j = 0, U ·  VCj == 0 ,  either fj < Cj < Cj in (0, T] x £2 x A and 
!'2j < Ci < Cj in (0, T] x A ,  or there are constants Tj in (0, T] such that fj < Cj < C, in (T; , T] x £2 x A ,  
!'2j < C; < C j in (T; ,  T] x If,  Cj = cj (= fj)  at a point (T; , x *, z * ) ,  where (x·, z· )  E £2 x A and Cj = 1:; (=!'2j) 
* -at a point (T;, z ) where z· E A . 
Suppose that cj = cj at a point (T; ,  x *, z * ) .  Theorem 3.2. 1 0  then impl ies cj == cj in (0, 1i ] x £2* x A , 
where £2. is some simply connected part of £2 at (�, z * ) containing x· . Condition (ix) shows Cj = C; at 
(T;, z *) and hence Cj == C; in [0, � ]  x A at z*. This same result is obtained from the alternative assumption 
that Cj = 1:; at a point (T;, z * ) .  Condition (x) now impl ies cj = C. on an x A at I=Tj and hence cj == cj in 
[0, T; ] x £2 x A and Cj == C; in [0, T; ] x A for 1 in [0, 1i ]  at z* in A . 
(IV) For components i E I, where Dj = I-/j = 0, either fj < Cj < Cj in (0, T] x £2 x A ,  or there are constants 
Tj in (0, T] such that fj < Cj < Cj in (�, T] x n x A and cj = e; (= fJ at a point (�,  x *, Z* ) ,  where (x� Z*)E 
£2 x A .  
Suppose that cj = cj at a point 0; , x *, z* ) .  Theorem 3.2.4 then impl ies cj == cj in (0, T; ] x £2* x A , at 
(x; z* ) ,  where n* is some simply connected part of £2 at (1; , z*) containing x*. 
3.2 GENERALISED COMPARISON THEOREMS 42 
(V) For components i E J, where Di = Hi = 0, !1Ji > 0, ei ther �j < Ci < Cj in (0, T] x A , or there are 
constants Ti in (0, T] such that �j < C. < Ci in [0, 1; ] x A and Cj = C, (= �J at a point (1'; , z*)  where 
z· E i\ . 
Suppose that Cj = C, (= �J at a point (1';, z* ) .  Theorem 3.2 . 1 0  then implies C; = Ci in [0, 1';]  x A .  
(VI) For components i E J, where Di = Hi = O, !1Ji = O, u ,VCi $O, either �i < Ci < Ci in (0, T] x A ,  or 
there are constants Ti in (0, T] such that �j < C; < Cj in (1';, 1'] x A and Ci = C, (= �) at a point (1'; , z *) 
where z·  E i\ . 
Suppose that Ci = c, (= C )  at a point (1'; , z*) . Theorem 3 .2.7 then impl ies Cj = c,  (or Cj = �j) in 
ro, 'Ii 1 x A along Ihe dlaraClcristic curve given by dz = u(t , z) , Z = z· at 'J'i for ( in r o, 'Ii 1 or until z reaches 
dt 
the boundary oi\) .  
(VII) For components i E J ,  where Dj = Hj = 0, !1Jj = 0, U ·  VCj=O, either �i < Ci < Cj i n  (0, T ]  x A,  or 
there are constants Tj in (0, T] such that �i < C; < Ci in (1';, T] x i\ and Ci = C, (= �j) at a point (1';, z*) 
where z· E if .  
Suppose that Ci = C. (= �j ) at a point (1'; , z*) .  Theorem 3 .2 .4 then implies Cj = Ci (or C i = �i) in 
[0, 1';] x A for I in [0, 'Ii ] at z* .O 
An immediate corallary that i s  independent of time I i s  the fol lowing where we  assume that the functionsJi 
and Fj are independent of I, so that L,  .� E... and F; are independent of I. 
Corollary 3.2.1 
Suppose (ci ' Cj )  is a sleady slale SolUlion of Ihe system S, I' Bn and the functions fi ' Gi ' �j and Cj are 
defined and satisfy the following continuity properties and inequalilies: 
(i) For components i E I, where Di > 0, fj ' Cj and C; are continuous in Q x i\ ,  (heir first-order Xj­
derivatives exisl in Q x i\ and their second order xjXlcderivatives exist and are continuous and 
uniformly bounded in Q x i\ ; 
(ii) For components i E I, where Di = Hj = 0, fj '  Cj and Gi are continuous in Q x i\ ;  
(iii) For components i E J, where !jj >  0, �j , C i and Cj are continuous in i\ , their first order z 
derivatives exist in if and their second order zjZk-derivatives are continuous and are uniformly 
bounded in i\; 
(iv) For components i E J, where 9Jj = 0, u ·  VCj $ 0 ,  �j , Cj and Cj are continuous in A and their 
first order Zj derivatives exist in i\ and are uniformly bounded in A; 
(v) For components i E J, where 9Jj = 0 ,  u , VCj = 0 ,  �j , Cj and Cj are continuous in if; 
(vi) 
(vii) 
(viii) 
-DjV;fj -.L(x, fk ' G, ) � -DiV;Ci -!;(x, cj) �-D,V;Gi -!j (x, fk ' G, ) in Qxi\; 
Of.j < OCj < dCj ::)n A .  
dn - dn -
Tn on O�"lXJl, 
(3.2.56) 
(3.2.57) 
(3.2.58) 
(ix) 
(xii) 
3.2 GENERALISED COMPARISON THEOREMS 
D (}fj - H(C . - c . ) < D dCj - l/. (C- -c· ) < D· dCj -H(C '  - c )  on {}Q2xA-1 dn 1 -I -I - 1 dn 1 1 1 - 1 dn 1 1 1 , 
vI=-u 'nl is uniformly bounded and continuous in dAI ,  
so that 
43 
(3.2.59) 
(3.2.60) 
(3.2.61 )  
(3.2.62) 
(3.2.63) 
(3.2.64) 
(xiii) fj and Fj satisfy the uniform Lipschitz condition (H I ) .  
Then 
(I) For components i E I, where Dj ,  Ilj > 0, g)j > 0, either fj < Cj < Cj in {J x A and Qj < C < Cj in if 
or c j == Cj (or Cj == fj) in n x A and Cj == � (or Cj == Qj ) in if. 
(II) For components i E I, where Dj , Hj > 0, g)j = 0, U ·  VCj $ 0, e ither fj < Cj < Cj in n x A and 
�j < Cj < Cj in if,  or Cj == Cj (or Cj == f) in Q x A and there is at least one point z .. in if such ' 
that Cj == � (or Cj == C)  in if along the characteristic curve given by � = u(z) , z = z* at Z l  for 
_ dZI u�(z) Z l  in Al until z reaches the boundary JAI (Here U I  is chosen without loss OJ generality to be the 
first component of u(z) that is nonzero, Z l  corresponds to this component and A == A I U An-I 
where Al is the interval [a, b] with a, b E R) . 
(III) For components i E /, where Dj, H j > 0, g)j = 0, u · VCj==  0, either fj < Cj < Cj in n x A and 
�j < Cj < Cj in If or there is at least one point z* in If such that Cj == Cj (or C1 == f.) in Q x A and 
Cj == � (or Cj == �j )  in if at z*. 
(IV) For components i E I, where Dj = Hj = 0, either fj < Cj < Cj in Q x A or there is at least one point 
z .. in If such that Cj == Cj (or Cj == f) in n '  x A  at the point (x', z' ) ,  where Q' is some simply 
connected part of n at z' containing x· . 
3.2 GENERALISED COMPARISON THEOREMS 44 
(V) For components i E J, where Dj = IIj = 0, !lJj > 0, either �j < Cj < Cj in .If or Cj == C; (or Cj == �j )  
in .If.  
(VI) For components i E J, where Dj = Hj = 0, !lJj = 0, U · VCj $0,  e ither �j < C; < Cj in A or there is 
al leasl one point z· in .If such that Cj == C; (or Cj == �j )  along the characteristic curve given by 
dz u(z) . . - . -=--, z = z at z l for Z I In AI unlll Z reaches Ihe boundary cHI . 
dz] UI (z) 
(VIn For components i E J, where Dj = Hj = O, !lJj = 0, u ·VCj==O , e ither �j < Cj < Cj in A or there is 
at least one point z· in .lf such that Cj ==C; (or Cj == �j ) in .If at z* . 
Remark 3.2.1 
Consider the system where Dj ,  !lJj > 0 for all i. From conditions (vii) and (x) of Theorem 3.2. 1 3  
(General ised S trong Comparison (Contact) Theorem) , we must also have /;(t, x, c) =J; (t ,  X ,  fk ' Ct ) 
(= 1/t, x, fk ' cJ )) in [0, l; J x .Q x A  and F;(t, z, Cj ) = �(t , z, 0. ,  CJ ) (= f.; (t, z, 0. ,  Ct» in [0, 1; ) x A  
if.f; and Fj are strictly monotone increasing in Cj and Cj, respectively. These imply Cj coincides with one of 
the bounds Cj or fj and hence Tj =Tj or else ]; is independent of Cj or fj and in this range the inequalities 
for Cj (f) become uncoupled from the Cj , fj system and Tj and Tj may be unrelated . There are similar 
consequences for F; and the C;, Q system. From conditions (vii) and (x) of Corollary 3.2. 1 ,  we see that 
similar consequences also hold for the time independent case. 
Our next comparison theorem for solutions of Sn, Bn shows the consequences of Theorem 3 .2. 1 3  if some of 
the inequalities are strict. Let us consider the case where strict inequalities are required of the initial conditions 
and the boundary condition at JA 1 and therefore, the severe constraints that (vi) in Theorem 3 .2. 1 3  hold at 
the outset and (xi) hold at the boundary JA I are relinquished. Similar theorems hold i f  strict inequalities are 
required of the differential inequalities. 
Theorem 3.2.14 
Assume that our assumptions in Theorem 3.2 . 13  all hold with the exception of (vi) which is replaced by 
the following 
(vi ') fj < Cj < Cj in .QxA  and �j < Cj < C;  in A at t = 0, (3.2.65) 
and for components iE J, where !lJj = 0, U · VCj $0, (xi) is replaced by the following 
(xi ') (3.2.66) 
Then the inequalities in (3 .2.65) holdfor all t in [0, T) . 
Proof 
If the inequalities (3.2.65) are violated in [0, T] ,  then there is a first time I· when the strict inequality is 
violated. At such a t* there are points in A where, at a point in .Q x A , one, some or all of the fol lowing 
happen: 
(3.2.67) 
for some value or values of i. 
3.2 GENERALISED COMPARISON THEOREMS 45 
Suppose this happens at (I ; x; z* ) .  Then from Theorem 3.2. 1 3  this impl ies that fi or ci == ci in 
[0, t* ] x Q x J\ at (x; z* ) and hence this same equality is satisfied at (0, x; z* ) .  If we suppose that this 
happens at (t; z* ) then Theorem 3 .2. 1 3  implies that �j or C; == Cj in [0, t * ] x if at z* and hence this same 
equality is  satisfied at (0, z* )  or in the case oF components i E J, where q,j = 0, u ,  '\lCj $ 0, this equality may 
be satisfied at dJ\\ .  This is contrary to assumptions, hence the strict inequality (3.2.65) must hold for all t in 
[0, T] .O  
It  i s  interesting to note that theorems analagous to theorems 3 .2. 1 1 (General ised Weak Comparison 
Theorem) and 3.2. 1 2  (General ised Strong Comparison Theorem) do not hold in general in the case of the 
corresponding steady slate or time independent problem Sn ' En ' For example, consider the problem 
,:Pc --2 = 8c+ 1 for ° < x < 1 ,  0 < z < 1 ,  ax 
dC 
ax = 0 at x = 0, 0 < z < 1 ,  
ac j-;" + c = C at x = 1 ,  0 < z < 1 ,  
d2C dC ac l . - -2- + - + - = O m O < z < l , dz dz ax x=l 
ac C + - = 1 at z = 1 ,  
dz 
ac a; = 0 at z = O. 
The functions f. = (4x4+3)z, e = 8x2z, � = z2 and E = z2 + z satisfy the following inequalities: 
d2C d2C [--::cT-(8e + 1) ]-[---:cf-(8f+ 1 )] = [- 1 6z-(8·8x2z+ 1 )]-[ -48x2z-(8( 4x4+ 3)z+ 1 )] dX dx 
= [-1 6z-64x2z- 1 ]-[-48x2z-32x4z-24z- 1 )] 
= (32xL1 6x2+8)z 
= [2(4xL1 )2+6]z 
> 0 in 0 < x < 1 , 0 < z < 1 ,  
dC dc :l = � = 0 � 0 at x = 0, 0 < z < 1 ,  oX dx 
de - _ de - -[ dx -(C -c)]-[ d
�-(� -f}) = [ 1 6z-(C -8z)]-[ 1 6z-(� -7z)] = � -C +z = 0 � 0 at x = 1 , 0 < z < 1 ,  
[_ d
2E + it + dc l ] _ [_ d2e;. + de;. + af l ] = [-2 + (2z + 1) + 1 6z] - [-2+ 2z + 16z] dz2 dz dx x=1 dz2 dz ax x=1 
= 1 � 0 in 0 < z < 1 ,  
- aE dC [C + -] - [C + -=] = [(z2 + z) + (2z + 1)] - [z2 + 2z]  = z + l  � O at z = l , az - az 
dE dC 
- = 1 ,  -= = 0 at z = O.  dz dz 
Thus ,  
3.3 UNIQUENESS OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 
d2C d2C [--a 2 - (8c + I) ) > [-2 - (8f + 1 )] for 0 < x < 1 , 0 < Z < I ,  x dX 
de dC 
- > -= at x = 0 0 < Z < 1 dX - dX ' , 
de - _ dC [ dX - (C - c )] � [ d�- (� - f)] at x = I , O < z <  1 ,  
[_ 
d2C + dC + de l ] � [_ d2f + d� + df l ] for 0 < z < I ,  
dz2 dz dX x=l dz dz dX x=1 
dC dC 
- � --= at z = 0, dZ dZ 
[C + dC] � [� + d,� ] at z = l .  (Jz (JZ 
46 
If  theorems analogous to Theorems 3 .2. 1 2  (Generalised S trong Comparison Theorem) held, we would also 
expect that at least that f � c for all 0 � x � 1 ,  0 < Z < 1 and � � E for 0 < Z < 1 .  Clearly ,  from the 
definition of � and E, we have � � E for all z. However, when x = t, £(t) = (4(t)4 +3)z = 3tz ,  
c(t) = 8(t)2 Z = 2z and 3tz 4. 2z , so theorems analogous to Theorems 3.2. 1 2  do not hold in this case. 
Remark 3.2.2 
Equations Sn ' Bn have been chosen with bioreactor applications in mind , but the theory can be readily 
generalised in a number of ways. Our proofs are still valid for D/V;cj replaced by V x . (Dj (x, Cj )V xCj )  and 
q)jV2e; replaced by V · (� (z, Cj )VCj ) ,  provided that we have uniform elliptici ty conditions for these more 
general equations. Furthermore, the mass transfer coefficients I/j could be functions of x and I, provided that 
these functions are still positive and satisfy appropriate continuity properties, and a wider class of coupling 
functions is pennissible, since.fi and Fj may be permiLLed to depend on VCj and VCj , respectively. 
In section 3.3 we discuss the uniqueness of solutions of the system Sn ' Bn . 
3.3 Uniqueness of Solutions to the Unsteady State Problem 
The numerical analyst needs a knowledge of classical theory in order to decide whether a problem has a 
solution and whether it is a unique solution or not. The experimentalists who try to val idate mathematical 
models also need to know which solution they are comparing their experimental data against if there is more 
than one solution. 
We use Theorem 3 .2. 1 2  (Generalised Strong Comparison Theorem) to show that solutions of system 
Sn ' Bn are uniquely specified by the functions Cj,Q, Cj,Q and Cj, l ' 
Theorem 3.3.1 (Generalised Uniqueness Theorem) 
Suppose (Cj, Cj) is a solUlion of the system Sn ' Bn which satisfies the following continuity properties 
(i) For components i E I, where Dj, Hj > 0, Cj are continuous in [0, T] x Q x J\ , their first-order Xj­
derivatives exist in (0, T] x Q x A ,  their second order xjxk-derivalives and firs 1 order 1-
derivatives exist and are continuous and uniformly bounded in (0, T] x Q x J\ ; 
------------------------------------------ ----- - -
3.4 IMBEDDING RESULTS 47 
(ii) For components i E I , where Dj = IIj = 0, Cj are continuous in 1 0 ,  '1'1 x !2  x A and their first order 
t-derivatives exist and are continuous and uniformly bounded in (0, T] x !2 x A ; 
(iii) For components i E J, where !lJj > 0, C I are continuous in [0, T] x A ,  their first order Z 
derivatives exist in (0, T] x A , their second order zjZk-derivatives and first order t-derivatives 
exist and are continuous and uniformly bounded in (0, T] x A ; 
(iv) For components i E J, where !lJj = 0, U ·  VCj 'F 0, Cj are continuous in [0, T] x A , their first order 
Zj derivatives exist in (0, T] x A and their first order t-derivatives are continuous and uniformly 
bounded in (0, T] x A ; 
(v) For components i E J, where !lJj =0, U ·  VCj == 0, Cj are continuous in [0, T] x A and their first 
order t-derivatives exist and are continuous and uniformly bounded in (0, T] x A ; 
(vi) fj and Fj satisfy the uniform Lipschitz condition (/1 1 ) .  
Then there can be  a t  most one solution to  the system S, I' Bn . 
Proof 
Theorem 3 .2. 1 2  (Generalised Strong Comparison Theorem) implies uniqueness of the initial value problem 
Sn' Bn , since if (ciI ' Cj l) and (Cj2' Cjz) are two solutions coinciding at t = 0, and at JAJ ,  then 
(3.3. 1 )  
and therefore cil coincides with Cj2 and Cil with Cj2 for I > O. 0 
Remark 3.3.1 
It should be poimed out that although the uniqueness conclusion for the system Sn' 8n for given initial 
conditions holds for all finite T, it has l i llie relevance to questions concerning the uniqueness of the steady­
state problem Sn ' Bn . There may be many steady state solutions of the system Sn ' Bn satisfying all but the 
initial conditions of section 3 .2 but we see from Theorem 3 .2. 1 2, that each must arise from d ifferem initial 
conditions. 
In section 3 .4 we shall present some imbedding results for the system Sn, Bn . 
3.4 Imbedding Results 
In this section, we see that for the purposes of uniqueness, stability and existence theorems, we may assume 
at the outset that the system Sn, Bn is a quasimonotone system, i .e. Ii and Fj are monotone nondecreasing in 
Cj and Cj respectively for j � i. This is not a restriction on these theorems of this chapter since if this 
monotone property is not satisfied, then the system Sn, Bn with general functions Ii and Fj can be imbedded 
in a system S2n , 82n of the same form where h(t, x, Cj) is replaced by ]; (t , x, fl< ' cl ) for the first n(1) 
dependent variables Cj and by fj (I , x, f" ,  cl ) for the next n(1) dependent variables (;'j. Also, Fi(t ,  z, Cj) is 
replaced by F;(t, z ,  h:/c , Ct) for the first n(J) dependent variables � and by E.jU, z, h:/c , Ct) for the next 
n(l) dependent variables C; . I t  can be shown that solutions of this new system generate solutions of the 
original system and thererore uniqueness, sl4lbility and ex istence can be impl ied in the original system. 
3.4 IMBEDDING RESULTS 48 
We consider the new system S2n, B2n of up to twice the order satisfied by fj ' ej ' �j and C; in Ule 
following equations: 
dc · de· 
d
� =0 ,  
d
� =0  on (0, TjxdDlxA, 
dc ·  de· -DI d� = Hj (�j - f.;) , Dj d� = IIj (C; -e;) on (0, TJxdD2xA, 
d�j _ 9>,V2�j + U · vC· + IIifao2 (�i - fi ) =[i (l, z, �k ' (;, ) in (0, TlxA, 
L(O, x, z) = Cj,O , [j (O, x, z) = Cj,O in iliA, 
�j (O, z) = Cj,o , C;(O, z) = Cj,o in A, 
where L. l; E.i and F;are defined in (3.2.7)-(3 .2. 1 4) .  
Note that the functions [ J ,  -f) and ( F; , -E.d obey a mixed quasimonotone properly (mqmp) in 
ilie sense of LADDE et al. [ 153 , p. 16 1 ] .  i .e, the functions j; and -f are monotone nondccreasing in  el and -I 
monotone nonincreasing in fk for all i � k. I and the functions F; and -f.; are monotone nondecreasing in CI 
and monotone nonincreasing in �k for all i � k. I . 
If we therefore slightly modify the system S2n, B2n as suggested by McNABB [ 1 86] , by introducing 
new variables Vj =Cj ,  v n(l)+ j = -fj for i= 1 ,  . .  ,n(/) and Vi = C; ,  V n(J)+j = -C for i= 1 , . .  ,n(J) and if we set 
f;* =7j •  fn*(l)+j = -L for i= l ,  .. ,n(f) and 1';* = F; ,  Fn(J)+j = -f.; for i= l • . . •  n(.l) then we obtain a new 
system S�n ' B�n for which t* and F;* are nondecreasing functions of Vj and Vj• respectively , for all j ;t: i. It 
can be shown by Lemma 3 .2.2, that these new functions have the Lipschitz properties that were imposed on 
the original functionsfi and Fj• 
Every solution (Ci ,  C;) of Sn, Bn generates a solution Vi = Cj, vn(l)+j = -Cj and Vj = Cj, Vn(J)+j = -Cj 
of Ihe new system S�n ' Bin with the special property that for i � n(f) , Vj + Vn(l )+j == 0 in [0, T] x D x A and 
for i � n(J), Vi + Vn(J)+j == 0 in [0, T] x A .  Conversely ,  any solution (Vj, V;) of the new system Sin '  Bin 
with the special property that for i � n(f), Vj + vn( l)+j == 0 in [0. 1'1 x .f2 x A at t = 0 and for i � n (J) .  
Vi + Vn(J)+i == 0 in X at 1 = 0 and Vi,I + Vn(J)+j, 1 == 0 on (0, 1'lxdA I .  is shown in the fol lowing theorem to 
give rise to a solution of ilie system Sn, Bn. 
Theorem 3.4.1. 
The general system Sn, Bnfor which fi and Fj are Lipschitz continuous in Cj and Cj respectively. may be 
imbedded in a system Sin ' Bin of twice the order which is coupled by monotone functions fi' and 1';* of 
3.4 IMBEDDING RESULTS 49 
the new dependent variables Vj and Vj. Moreover, all the solutions (Cj, C j) of the system Sn, Bn are 
solutions of the new system, where 
Vj = Cj, Vn( l )+j= -Cj for i= I , .. ,n(1) 
and 
Vj = Cj, Vn(J)+j= -Cj for i= l , . .  ,n(.l) 
and all the solutions (Vj, Vj) of Sin '  Bin for which 
Wj = Vj + Vn(l)+j= 0 in DxA at t = 0, 
� = v.. + Vn(J)+j= ° in A at t = 0, 
dW -' = ° on (0, T] x dDI X A ,  dn 
D dWj - f! (W - w) = O on (O T] X dD2 x A  
l an 1 1 1 , . 
, 
dWj - �v�w. +u·VWj = F/'(t , z, V) } +  Fn*(J )+ j (t , z, Vk ) - lIjf ( Wi - Wj )  in (0, TjxA, � � 
VIWj + �j �Wj = Wj I = Vi I + Vn(J)+j I = 0 on (0, T] x (JAI , anI . .  . 
(3.4.1)  
(3.4.2) 
(3.4.3) 
(3.4.4) 
(3.4.5) 
(3.4.6) 
(3.4.7) 
(3.4.8) 
(3.4.9) 
(3.4.10) 
with Wj = Vi + vn(l )+Jor all i = 1 ,  .. ,nU) and Wj = Vi + Vn(J)+i , for all i = 1 ,  . . ,n(J) generate solutions 
(Cj , C) of the system Sn, Bn. 
Proof 
We first note that if we set fj == Cj == Cj and [j == c.. == Cj for all t, where (Cj, Cj) is a solution of Sn, Bn, then 
we have a solution of the new system S2n, B2n, so that the solution set of this new system contains all of the 
solutions of the original system Sn, Bn. In this system, we make the variable change 
(3.4.1 1) 
Vi =c.. , Vn(J)+j = -c;..j , for i= I , . .  ,n(J) , (3.4.12) 
so that the coupl ing functions f/ and Fj* are nondecreasing runctions of all the new dependent variables Vj 
and Vj' respectively, for all j ,to i. Denote this system by Sin '  Bin .  
The solutions o f  Sn, Bn generate solutions in Sin '  Bin for which Wj = Vj + vn{l )+, =  ° for i= 1 , . .  ,nU) 
in [0, T] x D x A and Wj = Vi + Vn(J)+i= 0 for i= l , . .  ,n(.I) in [0, T] x A . 
Suppose we have a solution (Vi, Vj) of Sin ' Bin for which (Wj, W;) defined above satisfy Wi = Wi = 0 
everywhere at t = ° and conditions (3 .4.5)-(3 .4. 10) are satisfied. We then obtain the following system of 
equations, S� for (Wi, Wj) : 
3.4 IMBEDDING RESULTS 
= �(t, z, �j ' Cd -Ej (t ,  z, �j ' Ck ) - lIjL.a
2 
(HI; - Wj ) 
50 
(3.4.13) 
(3.4. 14) 
=F;(t, z, Cj-Wj, Ck)-E;(t, z, Ci-Wj, Ck)-HjIa.a2 ny.·-w.} in (0, T]xJ1 , 
(3.4.15) 
In addition , (Wj, Wj) satisfies the boundary condi tions B� given by Bn with zero in itial condi tions and 
Wj, I =Vi,1 + Vn(J)+j.l = O. 
But 1 ;(1 , x, Cj - Wj '  ck ) -L(t , x, Cj '- Wj '  ck ) and F;(I ,  z, Cj _. Wj ' Ck )-Ej(t , z, Cj - Wi ' Ck ) 
-HjI (HI; -Wj ) vanish when Wj == Wj == 0 for all j, and since (Wj, Wj) == 0 is a solution of this  initial value d.a2 
problem and by Theorem 3 . 3 . 1  (Generalised Uniqueness theorem), is the only solution, we conclude that 
W j == 0 in [0, T] x .Q x A and Wj == 0 in rO, T] x A. The conclusion of our tlleorem must follow.O 
An immediate consequence of  these imbedding results is that existence, uniqueness and stability 
results for the system S;" ,B;" implies existence, uniqueness and stability for the corresponding solution of 
S", B". Of course, solutions of S", B" may be stable in S", B" , but unstable in the larger setting Sill ' Bi" . A 
direct implication of these results is that nonexistence results of the system Sill ,Bi" implies nonexistence for 
the system S", B". There are other impl ications of these imbedding results that are d iscussed in section 3.7. 
Remark 3.4.1 
Note that the functions 
and 
in the right hand sides of (3.4. 1 4) and (3.4. 1 5) are monotone nondecreasing in Wj and Wi, respectively. 
In section 3.5 we shall study the stabil ity of solutions of the system S", B" and uniqueness of soutions to the 
steady state system '�'" 8" . 
3.5 STABILITY OF SOLUTIONS AND UNIQUENESS OF SOLUTIONS 51 
TO THE STEADY STATE PROBLEM 
3.5 Stabil ity of Solutions and Uniqueness of Solutions to the Steady State Problem 
In this section, we establish some useful sufficient conditions for the global stability of all solutions of the 
general system Sn, Bn. By global stability of a solution (Cj, Cj) of Sn, Bn, we mean that arbitrary changes in 
Cj,O and Cj,O decay to zero with increasing t. Such stability implies the uniqueness of the solutions to the 
steady state problem,  since if (Cj l , CiI) and (Cj2, Cj2) are two steady state solutions of Sn, Bn satisfying the 
same boundary conditions, the disturbances (Cj2 - cil '  Cj2 - Cil )  in the initial conditions of (Cjl , Cjl )  must 
decay to zero, implying that only one of these could be a time independent solution. 
We assume at the outset that the system Sn, Bn is a quasimonotone system, i .e. /; and Fj are 
monotone nondecreasing in Cj and Cj respectively for j :;C i. This is not a restriction on the theorems of this 
section since if this monotone property is not satisfied, then the system Sn, Bn with general functions/; and 
Fj can be imbedded in a system S2n ' B2n of the same form where I;(t, x, Cj) is replaced by 1; (t , X, fk ' c,) for 
the first n(T) dependent variables Cj and by jj (t, X, fk ' c, ) for the next n(T) dependent variables f.j. Also, 
Fj(t, z, Cj) is replaced by F;(t, z, �k ' �) for the first n(J) dependent variables E; and by E.j (t, z, �k ' �) 
for the next n(J) dependent variables Cj• The stability and uniqueness results obtained for this new system 
then apply to the original system by the imbedding results of section 3 .4. 
Suppose (Cj, Cj ) is a solution of a quasimonotone system Sn, Bn for which the velocity distribution U 
is independent of time. We set OUl to find positive functions (Uj, Vj) and conditions on /; and Fj so that 
(Cj + AUj , Cj + AUj ) are upper functions and (Cj - AUj , C, - A Vj ) are lower functions for all A >  0. When 
such functions (Uj, Vj) exist, they give us a family of comparison functions associated with (Cj, Cj ) for all 
A. > ° which have a bearing on the stability of the solutions of the system Sn, Bn, including the steady state 
ones and provide a means of establ ishing conditions under which these are unique. We first note that the 
functions (Uj, Vj ) need to satisfy the following conditions if they are to generate upper and lower functions 
for all A >  0: 
(i) For components i E I, where Dj, IIj > 0, Uj are continuous in [0, T] x {2 x A ,  their first-order xr 
derivatives exist in (0, T] x {2 x A ,  their second order xjxk-derivatives and first order t­
derivatives exist and are continuous and uniformly bounded in (0, 71x{2xA; 
(ii) For components i E I, where Dj = Hj = 0, Uj are continuous in [0, T] x {2 x A and their first order 
t-derivatives exist and are continuous and un(formly bounded in (0, 71x{2xA; 
(iii) For components iE J, where 9)j > 0, V j are continuous in [0, T] x A ,  their first order z 
derivatives exist in (0, T] x A ,  their second order zjZk-derivatives and first order t-derivatives 
exist and are continuous and uniformly bounded in (0, T] xA; 
(iv) For components i E J, where 9)j = 0, U · VUj $.0, Vj are continuous in [0, T] x A , their first order 
Zj derivatives exist in (0, T] x A and their first order t-derivatives are continuous and uniformly 
bounded in (0, T]xA; 
(v) For components i E J, where Ellj = 0, U · VVj == 0, Vj are continuous in [0, T] x A and their first 
order t-derivatives exist and are continuous and uniformly bounded in (0, T]xA; 
(vi) 
(vii) 
Uj > ° in {2 x A and Uj > ° in if at t = 0; 
au · 
-' � ° on (0, T] x a{2l xA; an 
(3.5.1) 
(3.5.2) 
3.5 STABILITY OF SOLUTIONS AND UNIQUENESS OF SOLUTIONS 52 
TO THE STEADY STATE PROBLEM 
(viii) D· 
OUj 
-H ·(U - u·) > 0 on (0 T] x oD2xA l dn " , - , ,
(ix) 
(x) 
(xi) 
(xii) 
(3.5.3) 
(3.5.4) 
(3.5.5) 
(3.5.6) 
(3.5.7) 
If Ii and Fj are assumed to be differentiable, then it follows from the mean value theorem that the right hand 
sides of (3 .5 .6) and (3 .5.7) are equal t�ah (t, x, Cj + Aiuj )u, an�aF; (t, z, Cj + Ai*Vj )Vi. respectively, 
dC ' 0 ac u 
" , * , ** . (0 ' ) . "" \ ) . _ ) lor some I\.j , I\.j In , I\. .  -.I ..) - I 
The following conditions on the derivatives of Ii ,  Fj and on the solutions (Uj, Vj) of a l inear system 
of equations Pee) derived from Sn, En are sufficient to provide a family of upper and lower functions of the 
type we seek. 
Assume there exists COlIstUllts aj} such thai. (!�; ( I ,  x, Ok ) � aU for ull (I, x ,  z) ill 1 0, '1' 1  x !2 x It ulld 
dej dF ' for all 8k and assume also that there exists constants Ai} such that ac�(I ,  z ,  ed � Ai}  for a l l  (t, z) in  
_ ) [0, T] x It and for all Bk. Note that ail � 0 and Ai} � 0 for j "# j since we have already assumed thatli and Fj 
were quasimonotone non decreasing. We look for posil.ive functions, (Uj, Vj) of separated variables form 
and 
V. = e-elw. 1 I '  
(3.5.8) 
(3.5.9) 
for some e > 0, satisfying the l inear inequalities (3.5 . 1 )-(3.5 .5), together with the following l inearised 
versions of (3.5 .6)-(3 .5 .7): 
and 
dUj -D.V2u . -� a .u . > O in (0 T]xDxIt 
at 1 x 1 LJ I) ) - , , 
) 
(3.5.10) 
aVj 
- 9).V2V · +u ·VU +H ·sflj · -� A-V >I-I J u· in (0 T]xlt (3.5.1 1) at 1 1 I I I LJ I) ) - 1 a� I ' • J 
Such functions exist and generate upper and lower functions for any solution of Sn, En and any A >  0, i f  we 
can find Wj and Wj positive in D x lt  and It respectively, satisfying the inequalities (3 .5. 1 2)-(3.5. 17) 
below. The functions Uj and Vj defined by (3 .5 .8)-(3 .5.9) then satisfy the inequalities (3 .5. 1)-(3.5.5) and 
(3.5. 10)-(3.5. 1 1) .  Matrix notation is convenient for the linearised system which follow. 
Let W denote the n(l)-vector with components Wj and let D, H, In(l) denote the diagonal n(l)th order 
matrices with diagonal elements Dj, Hj and 1 where In(l) is the n(l)xn(1) unit matrix. Let W, WI denote the 
n(J)-vector with components Wj, Wj, ! and let 9), In(J) denote diagonal n(J)th order matrices with diagonal 
elements 9)j and 1 where In(J) is the n(J)xn(J) unit matrix. Also, let a and A be the matrices with elements ajj 
and Ai} respectively. The vectors (w, W) are required to be positive solutions of the matrix system pee): 
3.5 STABILITY OF SOLUTIONS AND UNIQUENESS OF SOLUTIONS 53 
TO THE STEADY STATE PROBLEM 
(3.5.12) 
(3.5. 13) 
(3.5.14) 
(3.5.15) 
(3.5.16) 
(3.5.17) 
Theorem 3.5.2 
If (w, W) have positive components in n x A and A which are solutions of Pee) for some e > 0, then all 
solutions of Sn, Bn (including the steady state ones) are globally stable. 
Proof 
Let (ei l ,  CI) be a solution of problem Sn, Bn satisfying I.hc continuity requirement or the Theorems 3 .2. 1 2  
(Generalised Strong Comparison theorem). Consider the functions 
- 1 -£1 1 -151 C- C 1 -elW C C 1 -elW Cjl = Cjl +l\.e Wj ,  fjJ = Cjl - l\.e Wj , jl = jJ +l\.e j , _jl = il - I\.e j , 
where w, and Wj are positive solutions of problem Pee) for some e >  O. 
(3.5.18) 
It is a tedious but trivial exercise to show that when A > 0 and Ii(t, x, Cjl )  and F;(t ,  z, Cjl) are 
monotone nondecreasing in Cjl and Cjl for j :F- i, then (fjJ ' {2jl ) and (Gil ' Cjl ) defined by equations (3.5. 1 8) 
satisfy all the requirements of Theorem 3.2. 1 4. I f  (Cj2, Cj2) is any other solution of problem Sn, Bn satisfying 
the same boundary conditions (2. 1 .  7) (except when 9)j = 0, U· V Uj ;{= 0), but different initial conditions, we 
may take A large enough so that (Cj2, Cj2) is bounded by (fil ' {2jl ) and (ejl ' C;I ) at t = O. Theorem 3 .2. 1 4  
then implies these bounds are sustained for al l  finite time, and since these bounds decay exponentially (with 
exponent -et) to (Cj l ,Cjl ) ,  we see that all solutions are exponentially stable .  Moreover, if  the steady state 
problem has a solution, then it is unique and exponentially stable.O 
If n = n(I) = n(J), and all w are coupled to W by (3 .5 . 1 4) ,  then the system Pee) can be uncoupled in the x 
and z variable by means of the matrix transformation 
W(x,z) = O(x)SW(z),  
where 8(x) satisfies the boundary value problem: 
DV�O+(eI +a)8 =O  in n, 
ao 
an = 0 on aDI , 
D a8 =11(S-I - 8) on anz, an 
(3.5.19) 
(3.5.20) 
(3.5.21) 
(3.5.22) 
3.5 STABILITY OF SOLUTIONS AND UNIQUENESS OF SOLUTIONS 54 
TO THE STEADY STATE PROBLEM 
and vector W(z) satisfies the differential inequal ities: 
9>V2W - U . VW + (eI - ..9IH + A)W + ( Hf e)SW � O  in A,  
dil2 (3.5.23) 
(3.5.24) 
(3.5.25) 
Suppose we impose further conditions on Sn, Bn, by assuming the matrix a is the product of a positive 
diagonal matrix d and a symmetric matrix b (i .e. , a = db), a not uncommon feature of kinetic equations, 
while D and H are proportional so that 11 = fD for a positive number y. In these circumstances, the equations 
for e may, by a suitable choice of S (in fact S = T- I ) be expressed in the form e(x)=Tcp(x) whence, by a 
choice of T shown below, qi,.x) can be made diagonal. The matrix cp satisfies the equation 
(3.5.26) 
where T' denotes the transpose of T. The matrix T can be chosen so that T(Erl +b)T=p, a diagonal matrix 
and T d-IDT = J, where J is diagonal with the ith diagonal element equal to one if Dj > 0 and equal to zero 
if Di is zero. We may choose E so that p is nonsingular. If S-I = T, then in Q we have 
and on the boundaries of (JQ, 
J �� = 0 on r)QI , 
(Jcp J-=  rJ(J - cp) on (JQ2. (In 
We see that all the components of cp are uncoupled, and w = TcpT- I W. 
(3.5.27) 
(3.5.28) 
(3.5.29) 
Problem P*(E), where the inequalities are replaced by equalities may have non zero solutions when 
WI = 0 for certain values of £ (eigenvalues). If £1 is the smallest eigenvalue of £ and £1 is positive, then in 
general, we can find a positive E*<EI for which P(E*) has positive solutions (w, W). The argument functions 
of E on any closed interval not containing an eigenvalue. When E is very large and negative, w = W = K, a 
positive (in every component) constant vector is an upper solution of P · (E), while w = W = 0 is a lower 
solution, and so a positive solution exists. As E is increased, the solution (w, W) must remain positive since 
it depends continuously on £ and Theorem 3 .2. 1 3  (Generalised Strong Comparison (Contact) Theorem) does 
not allow for the existence of an E for which w � 0 in Q x A and W � 0 in A while one of these vanish at a 
point in Q x A or A . This situation prevails for increasing £ unless w or W have components which tend to 
infinity as £ increases towards a finite £1 . If M is the greatest value of all the components of (w , W), then 
w· = �, W· = � is the solution of ? * (E) for W( = � and as E lends to EI and M tends to infinity, 
( w· , W*) tends to a non zero solution of P*(£I) with WI = 0, and so EI is an eigenvalue. 
In section 3.6 we shall study the existence of solutions to the system Sm Bn. 
3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 55 
3.6 Existence of Solutions to the Unsteady State Problem 
There are many methods of proving existence of solutions. Often a fixed-point algorithm in an abstract 
function space is used. A number of existence-comparison theorems are available for weakly coupled 
parabolic systems. These can be established by various methods. Some examples are the work by KUIPER 
[ 149] using a functional analytic approach and by AMANN [ 1 0] and BEBERNES et al. [35-37] in connection 
with invariant sets. The technique we concentrate on in this section that is intimately connected with the 
maximum principles is the so called method of sub- and supersolutions or the method of upper and lower 
solutions. 
The theory of monotone operators together with the method of upper and lower solutions can be used 
to construct convergent monotone sequences and thereby establ ishes an existence-comparison theorem, 
provided that a suitable initial iteration can be chosen. It turns out that this initial iteration can be taken as 
either an upper solution or a lower solution satisfying certain inequalities associated with the original system. 
It is shown that the existence of a lower solution �i and an upper solution ui (these are also known as 
subsolutions and supersolutions or under and oversolutions, respectively) with �i � ui guarantees the 
existence of at least one solution Ui in the interval [�i '  ild . More precisely, there exists a minimal solution Hi 
and a maximal solution iii in the interval [�i '  ui ] such that the solution Ui satisfies Hi � ui � il;. The minimal 
and maximal solutions will, respectively, be obtained as the supremum of all possible lower solutions and the 
infimum of all possible upper solutions. 
The theory of monotone operators together with the method of upper and lower solutions in proving 
existence theorems for scalar parabolic equations is usually attributed to SA lTINGER [257]. Similar results 
are also available for scalar elliptic equations (HUDJAEV [ 133],  KELLER [ 14 1 ] ,  AMANN [5]). In these cases, 
the nonlinear term is assumed to be quasimonotone nondecreasing in the sense that the nonlinear term could 
be made monotone nondecreasing by the addition of a suitably large enough linear term. This method of 
proving existence theorems can also be extended to systems of equations and had been used earlier for 
multicomponent parabolic systems where some diffusion coefficients were allowed to vanish (MCNABB 
[ 1 82]). It has also been used for infinite systems of ordinary differential equations (CHANDRA, et al. [66]). 
All these techniques employ the maximum principle or an appropriate comparison theorem and in particular 
for systems, all these results require the nonlinearities h(t ,  x, c) to be quasimonotone nondecreasing 
(non increasing) in the sense that Ii is monotone nondecreasing (nonincreasing) in Cj for all j '# i and there 
exists a constant Mi (the L ipschitz constant if Ii is Lipschitz continuous in Cj) so thatJi+Mci is monotone 
nondecreasing (nonincreasing) . The method of upper and lower solutions can also be applied to systems 
where the restriction thatJi is  quasimonotone nondecreasing (non increasing) is not satisfied but instead the 
nonlinearities may satisfy other monotone properties. NORMAN [205] and CHANDRA, et al. [65] gave an 
example of a system which obeyed a mixed quasimonotone property (mqmp) in the sense that for all i, the 
nonlinearities h(t, x, Cj ) are quasimonol.One nondecreasing in Ci, monotone nondecreasing in the variables 
cj, i  '# i and monotone nonincreasing in the the variables Ck, k '# i, i. PAO [2 16] categorised systems of 
parabolic equations into three basic types according to their relative quasimonotone property (a) type I where 
the nonlinearities h(t, x, Cj ) were quasimonotone nondecreasing in the variables Cj, (b) type II where the 
nonl inearities were quasimonoLone nonincreasing in the variables Cj, and (c) type III  where for some i, the 
nonlinearities were quasimonotone nondecreasing in Cj, and for all remainding i, the nonlinearities were 
quasimonotone nonincreasing in Cj. For mixed quasimonotone systems, type I ,  type II and type III  systems, 
an existence-comparison theorem in terms of upper and lower solutions could still be established along the 
monotone and regularity arguments of SATfINGER [257] for scalar equations. These techniques could also be 
applied to other systems of di fferential equations provided that the nonlinearities obey some sort of a 
monotone property . LADDE, et al. [ 1 53] defined monotone iterative techniques for systems of first order 
--------------------------------------- -- -
3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 56 
differential equations, second order differential equations, elliptic equations, parabolic equations and first 
order partial differential equations where the nonl inearity possesses a mixed quasimonotone property 
(mqrnp). These solutions are defined in terms of coupled quasisolutions. Coupled upper and lower 
quasisolutions are defined by using the mixed quasimonotone property (mqmp) and quasi solutions may be 
constructed by monotone iteration. It is then required to show that these quasisolutions are actual solutions. 
The stability for solutions of reaction d iffusion systems may also be examined by the method of 
quasisolutions (LAKSHMIKANTHAM and V ATSALA [ 1 58]) as well as other qualitative properties of such 
solutions. 
I t  is not always necessary that in order to establish such an existence-comparison theorem in terms of 
upper and lower solutions that the nonlinearities must obey a monotone property . A monotone i teration 
technique was developed by KHA VANIN and LAKSHMIKANTHAM [ 145] for weakly coupled first and second 
order boundary value problems. This technique, known as the method of mixed monotony made it possible 
to construct monotone sequences even when the nonlinearities did not possess any monotone properties. 
CARL [46] , also proposed a monotone scheme for the solutions of weakly coupled systems of reaction­
diffusion equations without any monotonicity property required of the nonlinear reaction terms. The 
considerations including the definitions of upper and lower solutions are all taken within the framework of 
weak solutions. CARL and GROSSMAN [47] proposed a method of imbedding a system of n equations into a 
system of 2n equations and obtaining existence and enclosing theorems for elliptic and parabolic systems 
with nonmonotone nonlinearities in terms of upper and lower solutions. These theorems implied existence of 
solutions of the original system and the method was allowed to work under very low regularity conditions. 
Their definition of coupled upper and lower solutions coincides with the notion of coupled upper and lower 
quasisolutions defined by LAD DE et al. [ 1 53].  This method of imbedding a system of n reaction diffusion 
equations where reaction functions had no monotonicity requirement into a system of order 2n with reaction 
functions redefined had also been proposed by McNABB [ 1 86] for systems of ordinary differential equations 
and parabolic systems. However, it was furthermore shown that this system of twice the order may be 
modified to give a quasimonotone nondecreasing system of the type given earlier by McNABB [ 1 82] . All the 
earlier notions of monotone iteration were then applicable and coupled upper and lower solutions could be 
uncoupled into upper and lower solutions. These results also show that stability, uniqueness and existence 
results of this new system implies stability, uniqueness and existence results of the original system. 
Most of the discussions in the current literature are devoted to systems which are coupled through the 
reaction functions in the diffusion equations. However, in many situations, the nonlinearities may occur in 
the boundary conditions as well as in the equations. KELLER [ 14 1 ]  and AMANN [5 , 6, 8]  considered a class 
of mildly nonlinear scalar elliptic boundary value problems that are suggested by several steady state 
diffusion processes of interest where the nonl inearities are in the boundary conditions and ARONSON and 
PELETIER [23] considered an interesting problem of an initial boundary value problem that is in the presence 
of several distinct steady states that is coupled in the boundary conditions. PAO [21 1 ]  studied a scalar 
parabolic  equation with nonlinear boundary conditions in an unbounded or bounded domain. The 
nonlinearities in the reaction functions as well as in the boundary condi tions are assumed to be 
quasimonotone. The question of stability of such a parabolic equation in bounded domains was also 
examined by PAO [21 2] by using this monotone property in the boundary conditions. PAO [21 5] considers a 
system of two reaction diffusion equations which are coupled through the boundary conditions and not in the 
reaction functions. The nonlinearities in the boundary conditions are defined as earlier for reaction functions 
in terms of type I, type II and type III  functions depending on their relative quasimonotone property and a 
montone iteration scheme is proposed for each type. IL is this monotone property together with suitable 
maximum principles which guarantee the use of monotone iterations in proving existence theorems. There 
3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 57 
has been no literature on systems of coupled reaction diffusion equations which are also coupled through the 
boundary conditions where there are no monotone properties in both the reaction functions and the boundary 
conditons. 
Our generalised particle reactor equations Sn ' Bn fonn a system of up to 2n weakly  coupled, 
degenerate equations which may be a combination of ordinary differential equations, first order partial 
differential equations and parabolic equations. The degenemcy is is not only concerned with the possibility of 
Dj or 9Jj being zero for some i ,j but is also associated with the fact that in the case that Dj is nonzero, the 
Laplacian for the Cj equations involves only the x dependent variables whereas for the Cj equations it involves 
only the z dependent variables. The weak coupling is also non standard in that through equation (2. 1 .6) the 
Cj equations have a functional connection to the Cj variables via J Dj aa
Cj 
• Our reaction diffusion system 
aQ2 n 
within a particle is therefore not only coupled through the boundary conditions but the nonlinearity in the 
boundary conditions are more general than the examples discussed by PAO [21 5] and are given as coupled 
functionals (see PARSHOTAM et al. [224] and WAKE [304] for examples of functional boundary value 
problems), i .e. in the sense that the boundary data is given by solutions to coupled reaction convection 
equations. These reaction convection equations may also have reaction functions which have no monotonicity 
requirement. 
Despite all these features we are able to show in this section that this system is governed by 
generalised existence theorems similar in type to those for weakly coupled parabolic systems in McNABB 
[ 182]. However, additional smoothness properties such as HlHder continuity conditions on !; and Fj are 
required for our generalised ex istence theorem. 
In general, the methods of proving existence of nonlinear differential equations consists of three steps: 
(i) Construction of a scqucnce of approximate solutions; 
(ii) Showing that the constructed sequence or approximate solutions converges to a limit function; 
(iii) Proving that this limit function is actually a solution of the given problem. 
Steps (i) and (iii) are standard and straighforward. It is, however step (ii) that needs attention. In this section, 
we shall construct a sequence of approx imate solutions by setting up an i teration scheme of linear equations. 
The additional continuity properties are required for the existence of these linear equations. We show that this 
constructed sequence of approximate solutions converges monotonically and uniformly (in appropriate 
spaces) to a limit function which is actually a solution of the system Sn ' Bn ' 
We demonstrate in this section that solutions of problem Sn ' Bn specified by Cj,O, Cj,o and Cj, l exist. By a 
solution, we shall understand a classical solution (Cj, Cj) of Sn ' Bn where 
(i) For components i E I where Dj, Hj > 0, Cj are continuous in [0, T] x.Q  x A ,  have continuous first 
order Xj derivatives in (0, T] x.Q  x A ,  continuous second order Xj derivatives in (0, T] X.Q  x A and 
continuous first order t derivatives in (0, TJx.QxA. In this case, we shall look for classical solutions 
of the fonn Cj (t, x, Z) E C1,2,O [(0, T] x.Q x A , Rn(/) ] .  
(ii) For components i E I where Dj = Hj = 0, Cj are continuous in [0, 'J'] x.Q  x A and have continuous 
first order t derivatives in (0, TJx.QxA. In this case, we shall look for classical solutions of the fonn 
Cj (t, x, Z) E CI,o,o [(O, T] x.Q xA ,  Rn(l) ] .  
3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 58 
(iii) For components i E J where gJi > 0, Ci are continuous in [0, T1 x X, have continuous first order Zj 
derivatives in  (0, T] x X, continuous second order Zj derivatives in (0, T) x A and continuous first 
order 1 derivatives in (0, T) x A .  In this case we shall look for classical SolUlions of the form 
Ci (l, Z) E Ci,2 [(0, T) x A , Rn(J») .  
(iv) For components i E J where gJi = 0. U ·  '\lCi =f- 0, Ci are continuous in [0, T) x A ,  have continuous 
first order Zj derivatives in (0, T) x A and continuous first order 1 derivatives in (0, T) x A .  In this 
case we shall look for classical solutions of the form Ci(t, z) E Ci.i [(O, T) x A ,  Rn(J» ) .  
(v) For components i E J where gJj = 0, U ·  '\lCj == O. Cj are continuolls in rO, T] x A and have 
continuolls first order 1 derivatives in (0, T) x A. In this case we shall look for classical solulions of 
the form Ci (l, Z) E Ci.O [(O, T) x A ,  Rn(J» ) .  
Comparison theorems are used i n  this section i n  conjunction with theorems on a priori estimates and 
existence of linear parabolic equations to derive estimates of the system Sn' Bn and to prove the existence of 
solutions to this system. There may be some cases where Di or gJi may be zero and these cases are treated 
by using standard results. 
We shall need some additional continuity properties to establish existence of the corresponding linear system 
and make the following assumptions on h (l ,  x, Cj )  and I� (t, z, Cj) .  
(H2) (i) h(l ,  X, Cj ) E Ca/2.a [[0, T] x .Q x R
n(l) , Rn(l ) ] , i .e . ,  h(l, x, Cj )  is H()\der continuous in 1 and x 
with exponent aJ2 and a ,  respectively, for each fixed value of Cj. 
(ii) � (I ,  Z, Cj) E  Ca/2•lX I [O, T] x A  x Rn(.l) , R"(.I » ), i .e . ,  � (I, z, Cj ) is H()\der continuous in I and Z 
with exponent a/2 and a respectively, for each fixed value of Cj. 
From Lemma 3 . 1 .3 ,  we see that exponents a in both cases may be assumed to be identical. 
3.6. 1 The Monotone System Sm Bn 
We may assume at  the outset that the system Sn, Bn is a monotone system in the sense that h (I, x, c) i s  
monotone nondecreasing in  Ci and F;(t, z,  C) is monotone nondecreasing in Ci. This is not a restriction on 
the theorems of this section since if this monotone property is not satisfied we may make the fol lowing 
substitution to obtain a system of the same type but with new functions that are monotone nondecreasing in Ci 
and Ci . We first observe that if (Ci, Ci) is a solution of Sn, Bn, then the functions (Wi, Wi) defined by 
and 
satisfies the following system of equations: 
aWi D n2 K KI F ( -KI ) . (0 1'J �'A -- i V xWj = Wj + e  Jj t , x, e Wj In , X �A , at 
aw· -' = ° on (0, 7lxa.aixA, 
an 
(3.6.1) 
(3.6.2) 
3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 59 
- -------.-.---- - - -_._- - _._----------- --- - -- -- --------
Wj (O, x, z) = Cj ,O in .QxA, 
Wj(O, z) = Cj,o in J\. 
This system is similar to the original system Sn' Bn with the nonlinear coupling function h (t, x, Cj) for Cj 
components replaced by 
K Kl f( ,-KI ) wj + e  )j t , x, e  wi ' 
and nonlinear coupling function F;(t, z, Cj ) for the Cj components replaced by 
KW; + eKt F;(t ,  z, e-KtWj ) .  
These functions satisfy a monotone property given by the following lemma: 
Lemma 3.6.1 
(3.6.3) 
(3.6.4) 
Our assumptions (N \) of Lipschitz continuity properties for the functions h and Fj with respect to Cj and 
Cj, imply that the functions KWj + eKtl;(t, x, e-Ktwj) and KW; + eK'F;(t , z, e-K1Wj ) with Cj = e-K1wj and 
Cj = e-K1W; , are monotone tlondecreasing in Wj and Wj, respectively. 
Proof 
Assume that Wj � wj so that Cj � cj . 
From (H I) ,  we see that 
and therefore, 
[KWj + eK'I; (t , x, e-K'w)J- fKwt +eK'I;(t ,  x, e-K'wj )l 
= KeK1 (cj - c; ) + eK' [I;(t , x, c)-I;(t, x, cj )] 
� -eK1 [K(c; - Cj ) - Kj (cj - c)] 
� 0, 
i f  K is chosen to be large enough. This shows that 
K Kl f ( -KI ) < K * Kl f ( -KI * ) wj + e  )j t , x, e Wj _ Wj + e  )j t , x, e Wj , 
so that this new coupling function KWj +eK'I; (t, x, e-K1wj ) is monotone nondecreasing in Wj. A similar 
argument holds if we have to show that KW; + eK1 F; (t, z, e-K1Wj) is monotone nondecreasing in Wj.O 
3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 60 
Remark 3.6.1 
The substitution Cj = e-K1wj and Cj = e-K1W; which gives us a simi lar system with new functions 
KWj + eK1/;(t , x, e-K1wj )  and KW,' + eK1fj(t , z, e-K1W) from fj and Fj and which are monotone 
nondecreasing in Cj and Cj, respectively is discussed by FRIEDMAN [94, p.202] and PAD [208] .  Note that if 
K is chosen large enough our new functions are strictly monotone increasing in Cj and Cj . 
Remark 3.6.2 
From the Lipschitz continuity properties of the functions j; and Fj, it can be shown thatj;+kjcj and Fj+KjCj 
are monotone nondecreasing in Cj and Cj, respectively, where kj and Kj are Lipschitz constants off,' and Fj, 
respectively (PAD [2 1 1  D. I f  furthermore, the functions j; and Fj are monotone nodecreasing in Cj and Cj, 
respectively, for j ':I- i, thenj; and Fj will be quasimonotone nondecreasing. 
It can also be shown that these new functions also satisfy the same Lipschitz and H61der continuity 
properties as our original functions. 
Lemma 3.6.2 
Our assumptions (HI ) of Lipschitz continuity properties for the functions /; and Fj with respect to the 
variables Cj and Cj imply similar Lipschitz continuity properties for the functions KWj + 
eK1/;(t, x, e-K1wj) and KW; + eK'F;(t ,  z, e-K1Wj )  with respect to the variables Wj and Wj, w h ere  
c - = e-K1w- and C. = e-K1W l J J J '  
Proof 
We need to only show that 
where, 
I fKWj +eK1/;(t , x, e-K'wj)l - [Kwi + eK1/;(t, x, e-K'wj )l l  
� Klwj - wi l+eK' I/;(t , x, e-K1wj ) -/;(t, x, e-K1wj )1 
< Klw- - w" l+eK1k· le-K1(w . - w�)1 - I I I J J 
< Klw- - w� l+k· lw · - w" 1 - I I I J J 
� k sup IWj - wj l ,  
J 
k = max (K, kj ) .  
J 
The first part of the proof is complete and the rest of the proof follows simiiarly.O 
Lemma 3.6.3 
Our assumptions (H 1) of Lipschitz continuity properties for the functions Ii, with respect to the 
variables Cj and assumptions (H2) of Holder continuity properties for the functions /;, with respect to 
the variables t and x with Cj fixed imply similar Holder continuity properties for the functions 
KWj + eK1/;(t, x, e-K1wj ) with respect to the variables t and x with wj fixed, where cj = e-K1wj . 
Similarly, our assumptions (111) of Lipschitz continuity properties for the functions Fj, with respect to 
the variables Cj and assumptions (H2) of Holder continuity properties for the functions Fj, with respect 
to the variables t and z with Cj fixed imply similar Holder continuity properties for the functions 
KW; + eKI F;(t, z, e-K1Wj )  with respect to the variables t and z with Wjfixed, where Cj = e-K1W; .  
Proof 
-- - . .  _ - - -- - ------- - - - -
3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 61 
We shall only show that h(t ,  x, e-Ktwj ) is Holder continuous in t. The rest of the proof is similar and 
follows from Lemma 3 . 1 . 1  and Lemma 3. 1 .2. 
1 1i(1, x, e-Ktwj )_ Ii( t: x, e-Kt• wj)1 
where 
Remark 3.6.3 
- I j: ( -Kt ) j: ( * -Kt ) j: ( * -Kt ) j:( * -Kt· )1 Ji I , x, e  Wj - Ji I ,  x, e Wj + Ji I ,  x, e Wj - Ji I ,  x, e Wj 
< I j: ( -Kt ) j: ( o -Kt )1 I j: ( o -Kt ) j: ( o -Kt· )1 - Ji t , x, e  Wj - Ji t ,  x, e Wj + Ji t ,  x, e Wj - Ji t ,  x, e Wj 
< k ( f )l t  - t* lal2 + k ( f )le-Ktw · _ e-Kt• w · 1  - t Ji 1 Ji J J 
� kt (Ii)I I - lo laI2 + ki (Ii)l wj l le-Kt - e-Kt · 1 
� kt (Ii )I I - l o laI2 +ki (Ii )l wj IKTI-aI2 I t  _ t* IIXI2 
� k i t  - t* la/2 , 
Note also that we may just as well have chosen the substitution ci = e -Kxl wi and Ci = e -Kzq.� , (where XI 
and Z l  are chosen without loss of general ity to be the first components of X and z) instead of (3.6. 1 )-(3 .6.2) 
and arrived at the same conclusions in Lemma 3 .6. 1 ,  Lemma 3 .6.2 and Lemma 3 .6.3. 
We may assume that the substitution (3.6. 1 )-(3 .6.2) has been made and that Ii (t, x, c) is  monotone 
nondecreasing in Ci and Fi(t ,  z, Cj )  is monotone nondecreasing in Ci. If, on the other hand the monotone 
property is not satisfied by al l  the other variables in these two functions, then the system Sn '  Bn with general 
functions/.' and Fi can be imbedded in a system S2n ' B2n of the same form where /'(1, x, Cj ) is replaced by 
lCt , x, f,k ' c/ ) for the first n(l) dependent variables ci and by jj (I, X, f,k ' c/) for the next n(!) dependent 
variables fi ' Also, FiU, z, Cj ) is replaced by �U, z, �k ' E;) for rhe first n(.!) dependent variables C; and by 
E.i (t , z, �k ' 'Et) for the next n(.!) dependent variables kj . The existence results obtained for this new system 
of twice the order satisfied by (f2i ' L )  and (ci ' C;) then implies a solution of our original system Sn ' Bn by 
the imbedding results of section 3 .4. 
It has been shown in Lemma 3.2.2 that the functions L,  l , E.j and F; satisfy the same Lipschitz 
continuity properties as our original functions .li and Fi in the system Sn ' Bn . It can also be shown in the 
following lemma that these new functions satisfy the same Holder continuity properties as our original 
functions. 
Lemma 3.6.4 
Our assumptions (H2) of Holder continuity properties for the functions h' with respect to the variables 
t and x with Cj fixed, imply similar I-Iolder continuity properties for L and l with respect to the 
variables t and X with f,k and CI fixed and so there are constants kt and kx such that 
If · (I , x, fb C/ ) -f ·U: x, fk ' Cj)1 � kt (f · )lt - t* I(X/2 , -, -, -, 
- - - * - - * a/2 Ih(t, x, fk ' C/ )-h Ct ,  x, 12k ' c/ )1 � kt (h)l t - t  I , 
If · (t, X, fk ' c[ )-f · (t, x: fk ' c[)1 � kx(/ · )lIx -x* l Ia , 
-I -I � 
(3.6.5) 
3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 62 
Similarly, our assumptions (H2) of Holder continuity properties for the functions Fj with respect LO the 
variables t and z with Cjfixed, imply similar Holder continuity properties for Ej and � with respect to 
the variables t and z with (;.k and C; fixed and so there are constants K, and Kz, such that 
Proof 
- - - .  - - .  a/2 1F; (t, z, (;.k ' C[ )- F;(t , z, �k ' C[ )I � K, (F; )lt - t  I , 
- • - • a IEj (t, z, �k ' C, ) -Ej (t, z , �k ' C,)I � Kz(E;)lIz - z II , 
I�(t, Z, �k ' C; ) - �(t,  z: �k ' C;)I � Kz (�)lIz - z· l Ia . 
(3.6.6) 
Since f.k ' C" �k and C; are fixed, lhis lemma follows direclly from the definitions of L, ];, Ej and F; in 
(3.2.9)-(3 .2. 1 2) and assumplion (H2).O 
For the purposes of our existence proof, we may henceforth assume our coupling functions fj and Fj are 
monotone nondecreasing in the variables Cj and Cj, respectively, for all). 
Remark 3.6.4 
CARL and GROSSMAN [47 1 have an analogous defin it ion for the funct ions L and l;. It is shown that i f  
h(t ,  x ,  u) are Caratheodory type functions , thul i s  for almosl all x E [2, the funclions ii arc conlinuous on 
Rm, and for all u E Rm, the functions Ii are measurable on n, then so are the functions L and l;. This is 
proved using a measurable seleclion theorem which is usually known from optimisation theory. 
3.6.2 Upper and Lower Solutions and Monotone Iteration 
We shall now introduce the concepts of upper and lower solutions relative to the monotone system Sn' En . 
Definition 3.6.1 . 
Assume that 
(i) For components i E I, where Dj > 0, fj and Cj are continuous functions in [0, T] x Q x A with 
continuous firsl order Xj derivatives in (0, T] x n x A ,  continuous second order Xj derivatives in 
(0, T] x n x A and conlinuous first order t derivati ves in (0, T] x n x A ; 
(ii) For componenls i E I, where Dj = H, = 0, fj and Cj are continuous functions in [0, T] x n x A 
with continuous first order t derivatives in (0, T] x n x A ; 
(iii) For components i E J, where 9Jj > 0, C and Cj are continuous functions in [0, T] x A ,  with 
continuous first order Zj derivatives in (0, T] x if,  continuous second order Zj derivatives in 
(0, Tl x A and continuous first order t derivatives in ; 
(iv) For components i E J, where 9Jj = 0, U ·  V' C ,  U ·  V' Cj $0, �j and Cj are continuous functions in 
[0, T] x A ,  with continuous first order Zj derivatives in (0, T) x A and continuous first order t 
derivatives in (0, T] x A ; 
(v) For components i E J where 9Jj = 0, u ' V'�j , u ,V'Cj == 0, C and Cj are continuous functions in 
[0, T] x A , with continuous first order I derivatives in (0, T] x A ; 
3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 63 
The ordered pair of functions (fi ' qi) and (ci ' Ei ) with fi � ci on [0, T] x.Q  x A and qi � Ei on [0, T] x if 
are said to be lower and upper solutions of Sn' Bn respectively, if they satisfy: 
tlnd 
afi 2 . Tt-D/Vx fi � h Ct , x, f) In (0, l1x.QxA, 
aCi 
a� � ° on (0, T]xaQ]xA, 
aco Di a� � Hi (qi- fi )  on (0, l1xaQ2xA, 
aCi 2 
f �- g)V C+u ·VC+H (C - c ) < F (t z C -) in (0 11xA at 1 _I _I 1 Bil2 _I 1 - 1 ' , -J ' , 
ac 
VI (;i + !lli -:-,_' � v]Ci.] on (0, l1xaA] , on] 
aCi 0 
-:-, - � ° on (0, 71xaAa, a = 2, 3 ,  ona 
fi CO, x, z) � ci,O in QxA, 
��i -DjV� Ci "? h(t, X, Cj ) in (0, T]x.QxA, 
�CI "? 0 on (0, T]xrJQ]xA, on 
D aCj > Ho (E - c )  on (0 TJx Y22xA , an - l " , a , 
aE 2 - - f -
-
-' - 91\1 C + u o VC+ l- I  (C- c » F(t z C - ) in (0  l1xA at 1 1 1 1 Bil2 1 1 - 1 ' ' J ' , 
- aE VI Ci + !lli �"? v]Cj ] on (0, l1xaAI , on] , 
�Ei "? 0 on (0, T]xaAa, a = 2, 3 ,  ona 
respectively. 
The strong comparison theorem shows that if (fj , q; ) and (Cj ,  Ei) are lower and upper solutions of Sn, Bn 
and (Cj, Cj) is a solution of Sn' Bn , then fi � Cj � Cj and (;j � Ci � Ei . 
The existence of monotone sequences depend therefore on a suitable pair of lower and upper 
solutions. This is by no means ensured without addition restrictions on the nonlinear reaction functions. PAO 
[222] gives sufficient conditions for the existence of lower and upper solutions for parabolic equations. 
. , at; . aF . These will reqUIre thatli or _0_' to be umformly bounded and Fi or _, to be umformly bounded. CARL ac ae J J 
3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY ST ATE PROBLEM 64 
[46] gives a method to show how these lower and upper solutions can be constructed . TAM [278-28 1 ]  used 
comparison theorems to construct upper and lower solutions for parabol ic equations and compared these 
solutions with the exact numerical solutions. It is important to note that these lower and upper solutions 
provide lower and upper bounds for solutions and that these bounds can be improved by monotone i terative 
techniques. We shall for the purposes of an existence theorem. assume tJ1at upper and lower solutions exist. 
In order to establ ish an existence llieorem for Sn . Bn in terms of upper and lower solutions. we define 
a transformation fi, by 
(dk) dk» = ff(c<.k-I) c<.k-I» J ' I  I J ' J ' 
and consider the sequences ( (c(k). C(k» } where c(k) is obtained from the linear system " , 
ac(k) 
_' _ _  D·V2C<k) = r (1 x c<.k-I» in (0 l1xQxA 
at J x J Jj , , ) , , 
ac(k) 
a
� = ° on (0 . 'J'jxoQ1xA. 
aC<k) 
D· -' -+ Hc(k) = I-f .C(k-l) on (0 l1xaf22xA I an I I J , ' , 
C(k)(O. x. z) = C, 0 in Q xA. , . 
and C<k) is obtained from the l inear system , 
C<k\O. z) = Cj 0 in A. , . 
'th < (k-l) < - [0 T] n A d C < C(k-ll < C- [0 T] A f k - 1 WI £j _ Cj _ Cj on • x �.: X an _j _ j _ j on • x or . . . . . 
(3.6.7) 
(3.6.8) 
(3.6.9) 
(3.6.10) 
(3.6. 1 1 )  
(3.6. 1 2) 
(3.6.13) 
(3.6.14) 
(3.6.15) 
For each k. the system (3 .6.8) consists of n(J) l inear. completely uncoupled initial value problems with 
boundary and initial conditions given by (3 .6.9)-(3 .6. 1 1 ) and this system is uncoupled from the system 
(3.6. l 2) which consists of n(J) linear. completely uncoupled initial value problems willi initial and boundary 
conditions given by (3 .6. 1 3)-(3.6. 1 5). 
Since cfk) (I, x, z) is not d i fferentiated with respect to z in (3.6.8), A may be considered to be a 
parameter space in (3 .6.8)-(3.6. 1 1 ) .  For functions dk) (t , x, z) where D j  = H j  = 0, f2 may also be 
considered to be a parameter space and for functions C?)(t. z) .  where 9)j = 0, U ·  vcfk-l) = O. we may 
simi larly treat A as a parameter space. The ex istence and uniqueness of sequences ( Cfk) , Cj(k» } may 
therefore follow from solving standard scalar systems of linear parabol ic equations (LADYSHENKAYA [ \55] 
or FRIEDMAN [94]) which may or may not depend on parameters, systems of ordinary differential equations 
which depend on parameters (HARTMAN [ 1 19 ,  p. 93]) and systems of first order partial differential equations 
(LAKSHMIKANTHAM el al. [ 1 60]). 
3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 65 
All these theorems will require H6Ider continuity properties on the functions h (t ,  x, C)k- I » and 
F (t z C<.k- I » + Hf cl k- I )  which are satisfied i f  either c�k- 1 )  E C( I + cx)I2,I + a,cx [ (0 T] x Q x A Rn( l ) ]  , , ' J ' aQ2 " , J " , 
c(k-I) E Ccx/2,cx,a [ (0 T] x .Q  x A Rn( l ) ] C<.k-I ) E C(1 + CX)/2,I + CX [(0 T] x A Rn( J ) ]  or C<.k-l )  E Cal2,a ) " - , J " j  
[(0, T ]  x A ,  Rn( J ) ] . 
These theorems will also require Holder continuity properties in the boundary and initial conditions 
and so we make the following assumptions on ci,O , Ci,o and Ci, l ' We will assume that iJQ and iJA belong to 
class C2+cx. 
(H3) 
(i) For components j E I, where D i, Hi > 0, 
C- E C2+lX, CX [Q X A Rn( l» ) . I ,D " 
(ii) For components i E /, where D i = Hi = 0, 
(iii) For components i E J, where 9)i > 0, 
(iv) For components i E J, where 9)i = 0 and u , VCi $. 0, 
c · E C t +a r A  Rn(J» ) and C ·  E Ca12,a r(O T) x iJA Rn( J» ) "  & ,0 ' 1, 1 , 1 t , 
(v) For component.s i E .I, where 9) .. = 0 and u ·VC .. == 0 ,  
The velocity distribution vector function U(I ,  z )  is  also required to  satisfy the following Holder continuity 
property: 
For components i E J, where 9Ji = 0, U· VCi $ 0, we shall also need the following additional assumptions 
(Hs) 
(i) For each (10, zo) E [ O,T] x A , there exists a unique solution z(t, 10, zo) of 
dz - = U(I, z ) ,  z(to) = Zo, on [0, T] ;  dt 
(ii) z(t, 10, zo) is  continuously differentiable with respect to (to, zo); 
(iii) The relationship 
holds. 
(3.6.16) 
(3.6.17) 
(H6) 
3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 66 
--_. --_. __ ._--_._ .. _--_._-----
(i) For each Zo E A and YjO E Rn(J) , there exists a unique solution Yj(t , 0, YjO; ZO) of 
at; - F (  ( ) C(k-I» Ii �JV I_ J f (k-I) ( ( . ° - ( :l j I, z I, 10 ' zo , j - j vHj + r. j cj I ,  x, z 1 , /0 , zo» , Y,( ) - YjO, 3.6.18) al Jil2 
on [0, 11,  where Z(/ , 10, ZO) is the unique solution of (3.6. 1 6); 
(ii) YlI, 0, YjO; ZO) is continuously differentiable with respect to (YjO, Zo) . 
Note that assumptions (H3)-(Hs) will hold in either our original system Sn ' Bn or the monotone system 
S2n ' B2n and (H6) can be shown to hold in the monotone system S2n ' B2n if it holds in our original system 
Sn , Bn · 
I.emma 3.6.5 
Consider Ihe IBVP (3 .6.8)-(3 .6 . 15) and suppose Ihat the assumptions (// 1 )-(114) hold. Let there exiSI 
(fj , C) and (Cj ' Cj ) which are lower and upper Solulions of Sn ' Bn wilh 9.j S; cjk-I) S; Cj on [0, T]xnxA 
d C < C(k- I ) < C- [0 T] A an -i - i - j on , x .11 • 
Assume Ihal 
(i) For componenls j E I, where Dj, I-Ij > 0, e)k- I ) E c(\ + a)/2,1 + 1l.t1 [(0, T] x .Q  x A ,  Rn( l ) ] ;  
(ii) For components j E I, where Dj = IIj = 0, C)k-I ) E C a/2 , a, a [(0, T ] x .Q x A ,  Rn( l ) ] ;  
(iii) For componenls j E i, where 9Jj > 0, C)k-I) E c( l +a)/2, I +a [(0, T] x A ,  Rn( J ) ] ;  
(iv) For componenls j E i, where 9Jj = 0, U · VCY-I) $0, CY-I) E C(\+a)/2,a reO, T ) x A ,  Rn( J » ) ,  
and assump/ions (/-/s)-(H6) hold; 
(v) For compOnenlS )' E i where 9J = ° U · VC(k-l) == ° C(k-I) E C(\+a)/2.a [(0 T) x A R n( J» )  , J ' J ' J " .  
Then the IBVP (3.6.8)-(3 .6.15) possesses a unique solution (c;k) , C[*» , where 
(I) 
(II) 
(III) 
(IV) 
(V) 
For componenls i E I, where D" J-J j > 0, c1k) E C I +a/2,2+ a,a [(0, T ]  x .Q  x A ,  Rn( l ) ] ;  
For componenls i E I ,  where D j = II j = 0, cfk) E C I+a/2,a,a [(0, T ]  x n x A ,  Rn( l ) ] ; 
For components i E i, where 9Jj > 0, C?) E CI +a/2,2+ a [(0, T ] x A ,  Rn( J ) j ;  
For componenls i E i ,  where 9Jj = 0 ,  U ·  VCfk) $ 0, Cfk) E C I+a/2, 1+a [(0, T ) x A ,  Rn( J ) ] ;  
For com'Jonents i E i where 9J. = ° U · VC(k ) == 0 C(k) E Cl +al2.a [(O T ) x A Rn( J» )  J" • , " I ' I  . " . 
Furthermore, in all cases cfk) and Cfk) satisfy the inequalities fj $ c�k ) $ Cj i n  [0, T] x Q x A and 
�j S; Cj(k) S; Cj in  [0, T ]  x A . 
Proof 
We first consider the case when Dj, 9Jj > ° for all }. It i s  obvious that for equations (3.6.8)-(3 .6. 1 1) ,  all  
conditions of Theorem 3 . 1 .2 except for those l isted in assumption (iv) are satisfied. Note that the function 
Cjk-I) in the boundary condition of cfk) are functions of I and Z but are independent of x. The function 
Cjk-I) therefore satisfies the Holder continuity property in I required by assumption (v) of Theorem 3 , 1 ,2 
and Z may be treated as a parameter. It is therefore enough to show that h (/ ,  x, C)k-1» E C a l2,a, a 
[[0, T) x n x A x Rn(/) , Rn( / ) ] . 
3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 67 
We have 
where, 
Also , 
where, 
I/i Ct ,  x, C)k-I ) (/, x, Z»-/i Ct: x, C)k-I) Ct: x, z» 1 � kt (fi )[I/-I ' IIX/2 +  IC)k-I) (/, x, Z)-C)k-I )Ct: x, z)IJ 
� k/ (/i )[ I/ - I ' llx/2 + kt (C;k- I» I / - I ' I( I+IX)/2 ) 
� kt (/i (I, x, C;k-I» )I/ _ I* IIX/2 , 
(3.6.19) 
1 f. ( (k- I)( » _ r.( ' (k- l)( " » 1  < k ( 1". ) [11 _ ' IIIX 1 (k-1 ) ( )_ (k-1) ( • * )IJ JI I, X, c) I, x, Z JI I, X ,  c) I, x, z _ X,Z JI X X + Cj I, x, z Cj I, x ,  z 
� kx,z (/i )[l Ix - X* IIIX + kx,z (C)k-I» (lC)k-I\1 (d(Q » l -IX l ix - x* lIcx + l iz - z* IICX )J 
< k ( 1". (1 X C(k-I » )(l Ix - x' IIIX + I Iz - z' lIlX ) - X,z Ji , , J ' 
(3.6.20) 
These two equations together show that 
(3.6.21) 
i .e., /i (t , x, e)k-I » is HOlder continuous in 1 and (x, z) with exponent a/2 and a, respectively. 
For a given z in A, it fol lows from Theorem 3 . 1 .2 that (3 .6.8)-(3.6. 1 1 ) has a unique solution cfk) , where 
cfk) (I, x; z) E C1+IX/2,2+cx [(0, T] x Q ,  Rn(l ) ] . 
To show that dk) is Htilder continuous in z with exponent a, we consider equations (3 .6.8)-(3.6. 1 1 ) with z 
and z* and look at the difference of these equations. Note that from the assumptions, 
so lhat 
lc(k-l) (1 z) - C(k-I)(t z' )1 < K (c(k-l » IC(k-I ) 1 (d(A» I-CX l l z- z* lIcx ) , ) , - z J J c1 , 
-kz (/i(/ , x, C)k-I » )lI z  _ z* l Icx � /i(t , x, c;k-I )( t , x, z» - /i (t ,  x, C)"-I )( t , x, z' » 
� kz(/i(t , x, c;k-l» )lI z _ z* IICX , 
-Kz (CY-I» IC;k-l\1 (d(A» I - CX l l z - z' lIcx � cY-I)Ct , z) - cY-1) Ct , z * )  
< K (C(k-l» IC(k-I) 1 (d( A» I -cx llz - z* IICX - z J ) cl , 
3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 68 
It then foIlows that 
Letting 
and 
where 
-Kz (CY-I » lIi ICY-l ) ICI (d(A» I-lX l lz - Z* lIlX 
< D 
d 
( (k)( ) (k)( *» (k) ) (k)( *» - i dn ci t, X, z -ci t, X, z +Hi(Ci (t, X, z -ci t, x, z 
< K (c(k-l » /-I. IC(k-I ) 1  (d(A» I-lX ll z - z* l llX - Z )  I }  cl , 
.(k) ( ) .(k) ( " ) k ( " ( (k-l) )1 1 " I III ('j [ , X, Z - (.j I , x, Z - z .lj [ , X, Cj z - z  I 
Kll z - z" l llX 
we obtain the problems 
and 
dW'2 2 0. aW2 
_1_- o .V W2 < 0 _I -'- + W2< 1 W '2 0 < 1 at 1 x 1 - , II , an 1 - , I ,  - . 1 
(3.6.22) 
(3.6.23) 
(3.6.24) 
(3.6.25) 
(3.6.26) 
(3.6.27) 
(3.6.28) 
(3.6.29) 
The problems (3 .6.28) and (3 .6.29) are equivalent and it foIlows by Theorem 3 .2.9 (Strong Comparison 
Thcorcm) for parabolic equal ions Ihat 
Wjl � - 1 ,  (3.6.30) 
and 
(3.6.31) 
-----�-
3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 69 
Therefore 
so that 
-(K + kz ( !; (t ,  x, c;k-J» )T)l I z _ z' l Ia � (dk ) (t ,  x, z) - cfk ) (t ,  x, z' » 
� (K + kz ( !; (t ,  x, cjk-J»)T)lI z - z' lIa , 
i .e. , c?) is Holder continuous in z. 
We see that (3.6.8)-(3 .6. 1 1 ) has a unique solution cfk ) , where 
cfk ) (t ,  x, Z)E C1+aI2,2+a,a [(O, T] x.Q x A , Rn(l) ] . 
(3.6.32) 
(3.6.33) 
(3.6.34) 
Note that the equation (3.6.32) could also have been obtained by integrating (3 .6.22) with boundary and 
initial conditions (3.6.23)-(3 .6.24) and noting that the corresponding Green's function is integrable. 
It is obvious that for equations (3.6. 1 2)-(3 .6. 1 5) ,  VI E Ca12,a [(O, T ] x 1\ ,  R
n l and all conditions of Theorem 
3 . 1 .2 except for those l isted in assumption (iv) are satisfied. Therefore, it is enough to show that 
F(t z C(k- I » + /I 'f C(k- l ) E CaI2,(X I I O  T l x  A x R"(.!) R"(J) l 
, ' ' j , tJ!l2 " ' 
, . 
We have 
where 
Also, 
I[ F; (/ , z, Cj(k - I ) (t , z» +  Hj f cfk-I ) (t ,  x, z)] - r F; (/� z, Cj(k- 1 ) (t :  z » + lIj f cfk-I ) (/� x, z)]1 an2 an2 
:.:; I fj (t ,  z, CY-1) ( t , z» - fj (t : z, C;k-J ) (t : z» 1 + Hjlfan2 cfk-J)(t, x, z) - cfk-
I)(t : x, z)1 
< K (F )[ I / _ I ' laI2 + IC(k - I) ( 1 z) _ c(k-I) (t ' z)l] + H'f Idk-I)(t x z)_c(k-l)(/ ' x z)1 - I , ) ' ) "  d[J2 J " I " 
< K ( F)[lt _ t ' la12 +  K (C(k-l » lt _ t ' I( 1 + a)/2 ] + Irk (C�k-I » f It _ t' I(I+ a)/2 
- , , � , j " , an2 
< K (F )I' lt _ t " laI2 + K (C (k - I » l t - l ' I(I + a)12 ] +  lI , k (c�k-I» sflt _ tO I( l+a)12 - , , � , j " , 
< K (F (t z C(k-I» + "' f C(k-I)(t x z» lt- t · la/2 - , , ' ' j  , a!l2 ' " , 
(k-I) f (k-I) . , 1 /2 (C(k - I» k « (k-I) ] K, (F;(/, z, Cj )+ Hj an2 Cj (I, x, z» = [1 + I r K, j + Hj , Cj )S¥'] . (3.6.35) 
I[ fj ( t ,  z, Cj(k- I ) (t ,  z» + lIj J cfk- 1 ) (t , x, z») - l F; ( t ,  z� Cj(k -l ) (t ,  z· » + lIjf cfk - I ) (t ,  x, z')] 1 an2 an2 
I F ( C(k-I ) ( . » F ( • C(k -I ) (  · » 1 L/ IJ (k -l ) ( ) (k -I) (  ' )1 :.:; i t , z ,  j t , z - 'j t , z ,  j I , Z  + 1' i  an2 Cj I , X, z  - Cj t , x, z 
:.:; I fj (t ,  z, cjk - l ) ( t ,  z' » - fj(l , z� C;k - I ) ( t ,  zO » 1 + Hj fan2
ICfk - I) ( t ,  x, z)-dk-1 )(t , x, z')1 
� Kz ( F; )[" z - z · "a + IC;k-I ) (t, z) - C;k-I) (t, z ' )I] + Hj kz (cfk- I» fan2 " z _ zo "u 
w here 
3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 70 
K (F(t z C(k-l»)+ II I c�k-l») = K (F)[I + K (C(k-l»)IC(k-l) I (d(A))l-a )+ /I ·k (c�k-I») �� (3 6 36) z , ' ' j  , J!l2 ' Z , . Z j j c1 , z ,  v'f . • •  
These equations together show that 
(3.6.37) 
i .e . , F; (t, z, CY-I» ) +  I-/j fa� c}k-l) is HOlder continuous in t and z w ith exponent al2 and a respectiveiy. 
It follows from Theorem 3 . 1 .2 that (3 .6. 1 2)-(3 .6. 1 5 )  has a un ique solution Cfk) ,  where Cfk) E cl+aI2.2+a 
[(0, T) x A ,  Rn(J» ) .  
T o  prove ( I )  and ( I I )  in the general case, w e  need only observe that h (I, x ,  c;k-l) E Ca12.a,a 
[[0, T] x n x A x Rn(l) , Rn(l ) ] from (i) and (ii). The proof is similar to that shown earlier. 
In the case of (\), we note that for components i, where Dj, llj > 0, 
by the same argument as above using Theorem 3 . 1 .2. 
In the case of (I I) ,  we note that for components i, where Dj = Ilj = 0, 
(k ) ( ) _ rl � ( • (k-l) (  • »)d . Cj I , X, Z - Cj,O + J o Ji t , x, Cj I , X, z t ,  
(3.6.38) 
(3.6.39) 
exists and is unique by the Holder continuity property of t�(t, X, C)k-I» in t , x and z. Also, if  h(t, x, C)k-l» 
is Holder continuous in t w i th exponent a/2, then � is HOlder continuous in t w ith exponent a/2 . dt 
Furthermore, 
Letting 
and 
-k;c z (f; (t , x, cj(k-I » ))(I Ix - x' IIa + I Iz - z' IIa ) � �(cfk) (t, x, z) - cfk)(t , x: z ' )) , at 
� k;c,z (h (t, x, C)k-l» ))(IIx _x' IIa + " z - z' "a ) ,  
Cfk ) (t ,  x, z) - cfk ) ( t ,  x, z* ) + k;c,z (h (t, x ,  c;k-I» )(l I x - x' l l a + I I z - z ' l I a )t 
kx, z (cj, o )(l I x - x' l Ia + I I z - z' l Ia ) 
(3.6.40) 
(3.6.42) 
3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 
cV ) (t ,  x, z) - dk ) (t, x, z* ) - kx.z (!;(t, x. c;k-I »)( l l x  - x· l la + l i z  - z* l Ia ) t 
kx.z (Ci. O )(l lx - x* l la + I I z - z* l Ia )  
we obtain the problems 
and 
OW'1 __ I > 0 W ' I O > - 1 
ot 
-
, '
. - , 
OWi2 < ° < 1 -- - , Wi2 0 - . 
ot . 
71 
(3.6.43) 
(3.6.44) 
(3.6.45) 
The problems (3 .6.44) and (3 .6.45) are equivalent and it fol lows by Theorem 3 .2.3 (Strong Comparison 
Theorem) for ordinary differential equations that 
Wil � -1 ,  (3.6.46) 
and 
(3.6.47) 
Therefore 
-(kx . z (Ci.O ) + kzUi(/ ,  x , (Y - l » )'J ')(l I x -· x · l IlX + I I z - z · WX )  � «(·Y ) (t ,  x, z ) - c? ) ( t ,  x, z · »  
� (kx.z (Ci.O )  + kz Ch(/,  x, c;k-I» )T)(l l x  - x· l la + l i z  - z * l Ia ) . (3.6.48) 
so that 
i .e. , cV) is HOlder continuous in x and z. 
This result also follows by integrating (3 .6.40) through w ith respect to I and applying the initial condition 
(3.6.41 ). A ltogether, these imply that 
(3.6.50) 
To prove (III)-( V) in the general case, we need only observe that F;(t , z, CY-l» + Hi Jan2 cfk-l ) E Cal2•a 
[[0, T] x A x Rn(J) , R"(J) ] from (i)-(v). 
In the case of (III), we note lhat for components i, where !lJi > 0, 
by the same arguments as above using Theorem 3 . 1 .2. 
In the case of (IV), we note that for components i, where !lJi = 0 and u ·  vcfk) = 0 .  
C(k) ( ) - C  lI F ( ' C(k-l ) I'l f ( k- l) ( ,  )d '  , I , Z - i O  + 'i I, z, J' + i ci I, x, Z I .  , • 0 aQ2 
(3.6.51) 
(3.6.52) 
3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 72 
exists and is unique by the Holder continuity property of F; (t, z, CY-I» + /-Ij Lil2 dk-
I) in t and z. By the 
same argument as for the proof of ( I I) , we see that 
(3.6.53) 
In the case of (V), we see that by (H5) and (H6), z(t, to, zo) and Yj(t, 0, YiO; zo) are unique solutions of 
(3.6. 1 6) and (3.6. 1 8), respectively, on [0, T] . Choose YjO = Cjo(zo) and note that if z = z(t , 0, ZO), then 
because of uniqueness, Zo = z(O, t, z). Also, the solution (z(t, 0, zo), Yj(t, 0, YjO; ZO» of the systems 
(3.6. 16) and (3 .6. 1 8) is a characteristic equation of (3.6. 1 2). Hence, for each solution of (3.6. 1 6) and 
(3.6. 1 8) ,  we have 
(3.6.54) 
and consequently, 
c[k)(t ,z) = li (t, 0, Cj o (z(O, t , z»; z(O, t, z» .  (3.6.55) 
Now by using assumptions (A5) and (A6) , it is easy to show that c[k) (t , z) defined by (3 .6.55) satisfies 
(3 .6 . 1 2) .  
To show uniqueness of  solutions of  (3 .6. 12), we  suppose, that cg) and cg) are two solutions of 
(3.6. 1 2) on [0 ,  T1 x if. By Theorem 3.2.9 (Strong Comparison Theorem) Cor first order partial differential 
equations, we see that cflk) � cg) � Cflk) and therefore Cflk) coincides with cg) . 
The Hlllder continuity of Cfk\t, z) is obtained by examining the characteristic equations (3.6. 16) and 
(3.6. 1 8) ,  so that 
(3.6.56) 
Finally, we show that (fj , �j ) and (Cj ,  Cj ) are lower and upper solutions of (Cj(k) ,  Cfk) .  To show that 
(Cj ,  Cj ) is an upper solution of (cfk) ,  dk) ,  we need only observe that 
D· �(C - c(k » ) + H (c· - c(k» = fJ.(C - C(k-I» > 0 on (0 T]xaf22xA I an " ' "  I I I - , . , 
a - (k) 2 - (k ) - (k) - (k) -(c- C ) - !f)V (C-C· ) + U · V(C - C· ) + /-J.d(C -C ) at " , " " ' " 
> F· (  C- · )  - F· ( C(k- I) I-I.! ( - _ . (k-I» - , t, z, J ' t , z, j + ,  c, c, (Jil2 
� 0 in (0, T]xA, 
since.t and Fj are monotone nondecreasing in Cj and Cj, respectively. 
We may therefore apply the maximum principle for the parabolic operator or Theorem 3 .2.3 (Strong 
Comparison Theorem) for ordinary differential equations or Theorem 3.2.6 (Strong Comparison Theorem) 
for iirst order partial difCerential equations to conclude that (Cj ,  Cj ) � (cfk) ,  dk) ) .  Similarly, (fj , �j ) may be 
shown to be a lower solution of (cfk) ,  dk) and the theorem is complete.D 
3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 73 
To start off the i lerative procedure, we need some continuity properties of (fj , qj ) and (Cj ,  Ej) .  The 
properties of the mapping ff, from (c;k-I ) ,  CY-I» into (cfk) ,  Cfk) are then given by the following lemma. 
Lemma 3.6.6. 
Consider the IBVP (3.6.8)-(3.6 . 15) and suppose that the assumptions (// ] )-(//4) hold. Let there exist 
(fj , qj) and (Cj ,  Ej ) which are lower and upper solutions respectively of Sn, Bn. 
Assume that 
(i) For components j E I, where Dj, IIj > 0, fj ' Cj E ct+ a/2.2+a.a r(O, T] x .Q  x A ,  Rn( l ) ] ;  
(ii) For components j E /, where D j = IIj = 0, Cj '  Cj E Cl +a/2.a.IX r (O, TI x .Q  x A ,  Rn( l) I ; 
(iii) For componenH j' E , where 9) . > 0 C ·  E E CI +ll/2.2+a [ (0 T J x A Rn(J) J '  • . , j ' -J '  J ' " 
(iv) For componentS j' E J where 9) · = 0  u · VC(k-l ) "" 0  C .  E ' E  C I+a/2. I +a [(0 T] x A  Rn(J) ] , j ' ) T- ,  -J ' J " , 
and assumptions (HS)-(/-/6) hold; 
(v) For components j E .I, where 9)j = 0, U ·  VCY- 1 ) == 0, 9.j ' Ej E CI +a/2. ll r (O, 'f' l x A ,  Rn(J) ] .  
Then the mapping ff,from (C;k- I ) ,  C;k-l» to (Cfk ) ,  C?) possesses the following properties: 
(II) !Yis a monotone operator on the intervals I fj , Cj I and I �'j , Ed . 
Proof 
We first consider the case when Dj' 9)j > 0 for all components j. The natural imbedding of ct+aI2.2+a.a 
[ (0. '!'] x D x A ,  Rn(l) 1 in to C1•2•a [(O, T l x D x A , R"( I ) 1 and C1+a/2.2+a r(O, Tl x A , Rn(J) 1 into CI•2 
1 (0, 'J' 1 x A ,  R"( .I ) I impl ies I hal (;.j ' Ej E C'·2,11 1 (0, TJ x !2 x A ,  Rn( l )  1 and 9.j ' Ej E C'·2 1 (0, 'I'J x A ,  Rn( J )  I .  
The boundedness of Q and A ,  together w ith  the fact that their boundaries belong to C2+a, shows that, 
if c;k- I) (t , x; Z)E C1•2 [(O, T l x D ,  Rn(l) ] (with A treated as a parameter space) and C;k- I ) (t , Z)E 
CI•2 [(O, T] x A , Rn(J) ] ,  then c?-I) (t ,  x; Z)E Wd·2 [(O, T] x .Q ,  Rn( l ) ]  (with A treated as a parameter space) 
and C;k-I)(t , Z) E Wd·2 [(O, T] x A ,  Rn(J) ]  for q > 1 .  From Lemma 3 . 1 .5 ,  we may take q to be identical in 
both cases. 
Thi s ,  in v iew of Theorem 3 . 1 . 1  ( Imbedding Theorem) ,  y ie lds that c;k-t) (t , x; Z)E 
C(\+a)/2.1+a [(O, T] x Q ,  Rn( l ) ]  (with A treated as a parameter space) and CY- 1 ) (t , Z) E C(\+a)/2.1+a 
[(0, T] x A , R n( J » ) .  From Lemma 3 . 1 .3 ,  a may be chosen to be identical in both cases. From arguments 
s imi lar  to that shown in the proof of Lemma 3 .6 .5 ,  we see that c;k-l ) (t ,  x, Z)E C(l+a)/2,1+a.a 
[(0, T] x D x A , Rn(l » ) . 
It is immediate that the proof of (I) fol lows from the choices (C<k- ]) C(k-I »  = (c · C . ) and J ' J -J ' -J 
( (k-I) C(k-I ) _ ( - C- ) '  L 3 6 5  Cj ' j - Cj ' j In emma . . .  
All the other possible cases are treated similarly. 
We have shown that if (c(.o) C(O» = (c· C- · )  then (c(O) C(O» ) > !Y(c(.o) C(O» ) = (c(l) CP» )  and i f  J ' J J ' J I ' I - J ' J I ' I 
(c(.o) dO» = (c C . ) then (c(O) C(O» < !Y(c(.o) C(O» = (cP) C(I) We have in fact proved that the J ' J -J '  -J I ' I - J ' J I ' I '  , 
mapping fTmaps intervals [fj ' Gj ] and [9.j '  Ej ] onto themselves. 
3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 74 
To prove (II), let Cjl ' ej2 E lej ' Ej l and Cjl ,  Cj2 E I {j ' Ej 'l where (Cjl ,  Cjl )� (Cj2 ' Cj2 )  for all components 
j. We want to show that ff(cjl ,  Cjl )  � ff(Cj2 ,  Cj2 ) .  
Let (Uj , Uj ) = ff(cjl > Cjl ) -ff(cj2 ,  Cj2 ) .  Then the monotone nondecreasing property o f  f; and Fj 
implies that 
au · 2 
-t-D/VxUj = f;(t , x, Cjl ) - fi (t, x, Cj2 )  � 0 in (0, nx.QxA, 
au· D·-I + fl u ·  = f/.(C·I - C -2 )  > 0 on (0 nxaf22xA , an " " I - , , 
(3.6.57) 
(3.6.58) 
and from the maximum principle for the parabolic operator or comparison theorems for ordinary differential 
equations or first order partial differential equations, (Uj ,  Uj )  � 0 or ff(cjl ' Cjl )  � ff(Cj2 ,  Cj2 ) .  This shows 
that .o/'is a monotone operator on the intervals [fj , Cj 1 and r�j ,  C; J . O  
The monotone operator ff will play a central role in the iteration scheme. 
Remark 3.6.5 
Iff; and Fj are strictly monotone increasing in Cj and Ci' respectively, then by Theorem 3 .2. 13  (Generalised 
Strong Comparison (Contact) Theorem) , ff(cjl ,  Cj l » ff(c:j2 ,  Ci2 ) ,  (unless ff(cj l ,  Cjl )= ff(cj2 ,  Cj2 )  in 
which case the right hand sides of (3.6.57) and (3.6.59) are identically zero; but th is happens only if 
(Cil ,  Cil ) =(ci2 ,  Cj2 ) , from the strict monotone property off; and Fj). We say that the monotone operator .o/' 
is monotone operator in the sense of COLLA TZ [76], i .e., (Cil ,  Ci l )  � (ci2 ' Cj2 )  implies that ff(Cil ' Cjl »  
.o/'(Cj2 , Cj2 ) · 
Remark 3.6.6 
If f; and Fj are monotone nonincreasing in Cj and Cj, respectively, the operator ffis alternating on the 
intervals [fj , cd and [C ,  Cjl in the sense that (Cil ,  Ci2)  � (Cj2 ,  Cjl )  implies that ff(cil >  Cj2 )$ff (cj2 ,  Cjl ) .  
To prove this, let (Uj , Uj ) = ff(cjl ,  Ci2 )  - ff(Cj2 ,  Cjl ) .  Then the monotone non increasing property of/; and 
Fj implies that 
a
a�j -DiV;uj = !; (t,  x, Cjl ) - !; (t, x, Ci2 ) � 0 in (0, 71x.QxA, 
au· D -' + f1.u · = i-J . (C·I - C-2 ) < 0  on (0 nxaf22xA ' an " " I - , , 
(3.6.60) 
(3.6.61) 
CJU· 2 f _I -g)V U +u·VU +flSIfU =F(t z C -I )-F(t z C -2)+H CI -C '2 � ° in (0, 71xA, (3.6.62) CJt I I I I I I "  J I '  ' J I an2 I I 
and from the same argument as before, using the maximum principle for the parabolic operator or comparison 
theorems for ordinary differential equations or first order partial di fferential equations, we see that 
.o/'(Cjl ,  Cj2 )�.o/'(Cj2 ' Cjl ) ·  
I t  is necessary to choose a proper initial iteration to ensure that the sequences ((C� k ) ,  Cj(k» ) are monotone 
sequences t.hat converge to the unique solution of Sm Bn and are within t.he intervals [fj , cd and [C , Cd . 
From Lemma 3 .6.6, it i s  obvious that the monotonicity of these sequences obviously depend on the 
3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 75 
monotonicity of Ii and Fj and the initial iteration is taken to be either an upper or a lower solution which is 
required to satisfy cerLain inequalities on the corresponding system. 
We may therefore use the initial iteration (c/O) , �(O» = (Cj , Cj) to construct the sequence 
( c?) ,  �(k» } from the following equations 
dC,(k) 
_' __ DS12c�k) = r et x c(k-I» in (0 T]x.Q x A  
at I x I J i ' , } ' , 
d-(k) D'�+H-c�k) = H,C,(k-l) (0 llxd.Q2xA ' an I I I I '  , 
-(k) �-95.V2C,(k) + u ' VC(k) + II 'siC.(k) = F(t z C�k-I » + ff . J C�k-I) in (0 llxA dt 1 1 1 1 1 I "  J 1 iJil2 'I " 
or the sequence (Cdk) ,  k�k» } with (dO) , k�O» = (fj , �, ) may be determined from the equations 
dC(k) _-_' __ D,V2 c(k) = r et x /k-I» in (0 T] x.Q xA  
at ' x  -I J i , ' -J ' , 
d (k) D �+  H,c(k) = H -C(k-I) (0 llxd{22xA I an I -I I -I ' , 
dC(k) -=.L_ 95.V2C(k) + u , VC(k) + ff .siC(k) = F(t z c(k-I » + H f  c(k- I) in (0 llxA. dt 1 -I -I 1 -I I "  -1 1 iJil2 -I ' 
Note that the sequences ( c?) , �(k» } and ( dk) , k�k» } may he obtained independently of each other. As in 
Lemma 3.6.5, uniqueness and existence of the sequences ( c?) ,  �(k» } and ( f�k ) ,  k�k» } fol low from 
similar arguments for uncoupled scalar systems of nonhomogeneous l inear parabolic differential equations, 
ordinary differential equations or first order partial differential equations by using the monotone properties of 
Ii and Fj. 
Definition 3.6.2. 
Th {( (k) C(k » } d ( -(k) C-(k» } ' th ( (0) C(O» - ( C ) d C-(O) C-(O» - ( - C- ) e sequences fj ' _j an Cj ,  j WI fj . _I - fj . _j an Cj ' j Cj . j .  are 
called minimal and maximal sequences, respectively. We say that (kj , () and (Cj ,  �) are minimal and 
maximal solutions respectively in the regions [fj , c, ] and [�j ,  C\ ] ,  if for any solution CCj' C) of Sn, 8n  
where fj $; Cj $; Cj and �j $; Cj $; Cj , then fj $; C j  $; Cj and kj $; Cj $; �. 
From Lemma 3 .6.6 together with the monotonicity of f!r, we shall show that the sequence 
( c(k) C(k» ) with (c(O) C(O» = (c C ) is monotone nondccreasing and the sequence {(c�k) E(k» )with _I t _I _I ' _I _I ' _I , ' I 
(c/O), �(O» = (Cj ,  Cj) is monotone non increasing. 
Furthermore, Cfj ' �j ) $; CCj ,  Cj ) results in cdk) ,  k�k» $; (C?) ,  �(k» for all k and pointwise l imits 
(fj ' kj) and (Cj ,  �)  exist. This is all done in the following lemma for Sn ' Bn , 
The only difficulty arises for components i E J, where !i'j = 0, u '  VCj(k) ¥ 0, and in this case we will 
have the following additional assumptions: 
dF; , d ' , a; eXists an IS contmuous; 
(Hg) dF; , d ' , -- eXIsts an IS contmuous; de . 1 
3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 76 
I t  has been shown in Lemma 3.2.2 and Lemma 3 .6.4 that our assumptions (H I ) of Lipschitz continuity 
properties and (H2) of Holder continuity properties for the functions Ii and Fj imply similar Lipschitz and 
Holder continuity properties for the functions L, ]; ,  Ej and F; .  It can also be shown that similar properties 
hold for our assumptions (H7)-(Hs) of differentiabil ity properties for the functionsli and Fj . 
Lemma 3.6.7 
Suppose in addition to the assumptions of Lemma 3.6.6, that (({;Y) ,  [�k» } are minimal sequences and 
((i::lk) , C;(k» } are maximal sequences. Also, assume that for components i E J, where 9>j = 0, U ·  VC;(k) , 
u · V[�k) $ 0 ,  that assumptions (H7)-(Hs) hold. Then 
�j $ [�O) $ [�I) $ . . .  $ [�k) $ �(k) $ . . .  $ �(I) $ �(O) $ Cj for (t, z) E (0, T] x A ,  
for all k = 1 ,2, . . .  and the pointwise limits 
. « k) C(k» - ( C hm fj ' _j fj '  _j ) ,  
k--'too 
I· (-(k) C-(k» - (- C-) 1m Cj ' j Cj , j ,  
k--'too 
exist , implying that 
for all k = 1 ,2, . . . 
Proof 
Let (Uj , Vj) = (Cj(O) -cP) ,  C;(O) - C;(\) = (Cj-cP) ,  Cj-�(I» , for all i. Then by definition 3.6.l of upper and 
lower solutions and definition 3 .6.2 of maximal sequences, the fol lowing inequalities will hold 
dUj 
_ D V2u .  = dej _ D.V2 c.- I'. (t X c·) > ° in (0 T]x.QxA dt 1 x 1 dt 1 x 1 J j , , J - , , 
D dUj + f-1 .u . =D �(e.-c(I» + ff . (e -c(I) = o. dej + ff . (e. - C. » O on (0 TJXd.Q2XA , an l J I an '" ", I " , ', I an I I I - , , 
dU 2 dC· 2 - - - - f _I _ �V U + u · VU +H . .9fU· = [_I - �V C· + u · VC·+ fl.srlC. ] - [F (t z C - )+H · c· ] dt 1 1 1 1 1 dt 1 1 1 1 1 I " J 1 df22 1 
� ° in (0, T]xA, 
with similar inequalities holding in the boundary and initial conditions. 
We first consider the case when Dj' 9'Jj > ° for all ). We see that from Lemma 3. 1 .8 (Maximum Principle) for 
the parabolic operator that (Uj , Vj ) � 0 ,  i .e. Cj(O) � cP) on (0, T] x.Q  x A and E,(O) � E;(\» on (0, T] x A .  
This result also fol lows from the monotonicity of fT, since i f  (Cj ,  Cj ) � (clO) ,  E;(O» , then S'(Cj , Cj )  � 
tJr(-c (O) C-.(O» = (-c(l ) E(\) and (c(O) C.(O» > (c.(I ) C(I» if we let (c.(O) E(O» = «(5 C·) . J , ' I I '  I , ' J - , '  I I ' I  P I 
3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 77 
._- -- _. __ ._-_ ... _------_._-----
Therefore, from Lemma 3 .6 . 5 , we conclude that CP)E C l +a/2, 2+a, a [(0, T] x .Q x A , R n(l) ] and 
C;(1) E C I+a/2,2+a [(0, T] x A ,  Rn(J ) ] .  
By  natural imbedding th is  impl ies that CP) E CI,2,O [(0, Tl x .Q x A ,  Rn( l ) ]  and C;(I) E  Cl,2 
[(0, T] x A ,  Rn( J) ] . 
AlI the other possible cases are treated similarly so that 
(i) 
(ii) 
(iii) 
For components i E I, where Dj = Hj = 0, CP)E CI,o,o [(O, T] x .Q  x A ,  Rn( l )] , 
For components i E I where 9>. = ° u ·  vE(I ) ,± ° E(t) E cl,t [(0 T] x A Rn( J) ]  t " , r " " ,  
For components i E I where 9>. = ° u · VC(I) == ° C,o) E Cl,o reO T] x A Rn( J » ) , I , , ' I " .  
In the case of (ii i) , we see that z is treated as a parameter. If (H7)- (Hg) are assumed, i t  then folIows from 
standard results on ordinary differential equations which depend on parameters (HARTMAN [ 1 1 9 ,  p. 95-99] 
that C;( I ) will be continuously differentiable in z. Therefore, if (H7)-(Hg) are assumed then in alI cases for 
components j E I, CP) will be continuously di fferent jab Ie in z. Furthermore, in alI cases for components 
i E I, where Dj , H j > 0, it can be demonstrated from ( 3 . 6 . 1 0) that cf) wilI also be continuously 
differentiable in z. 
We may similarly show using by the definitions of upper and lower solutions and of minimal sequences, that 
dO) ::; dl) and (;,\0) ::; �\l) . 
Now let (Uj ,  Vj ) = (cP) -dl) , C;(l )  - (;,\ 1 » . Then the monotone nondecreasing property of/; and Fj implies 
that 
aUj _ D , 02 . = � ( -�O» - �( (.0» - �( - . ) - � (  . » 0 '  ( 0  1'J �, A al · v xU. Jj I , x, C, J j I ,  x, f, Jj I, x, C, Jj I, x, f, _ to , x .. ""'.11 , 
au ac(l) ac(l) D· -j + H _u · =[D -j- + H·c�I) ] - [D- -=L.. + H -c�t) ] = H - (C,(O) - C �O» = H ·(C -C - » O on (0 TlXail2XA • an • •  • an • • • an • -. • •  -. • •  -' - , • J , 
aVj - 9>.V2v. + u . VV. +HS'IU. =F(1 z C ) - F (I z C - ) + H .J (c - c · » O in (O llxA al • . . . . .  , ' 1  . "  -1 • an2 • -. - , , 
and it follows from the monotonicity of [if, that 
(f�l) , �P » ::; (cP ) , E;(t» , 
implying that 
and 
Assume, by induction, that 
for k= I ,  . . ,m . 
3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 78 
The only difficulty arises for components i E J, where 9Jj = 0, U · VE;(k) $0 and in this case we cannot 
assume that in general ,  that the assumption (H6) will be satisfied. However, in this case we assume (H7)­
(Hs) hold so that C?-l) and CY-2) will be cOnLinuously differentiable in z for all }  by earlier arguments. 
Thus for componenLs i E I, where Dj, Hj > 0, it can be shown that c�k-l) will be continuously differentiable 
in z from (3.6. 1 0), so that by assumptions (H7)-(Hs), Fi(l, z, c;k-t» + Hj fan c?-l) will be continuously 
differenLiable in z. For components j E I, where Dj = Hj = 0, we see that in thfs case by assumptions (H7)­
(Hs), F;(l, z, C?-l» will be continuously differentiable in z. It then follows that (HARTMAN [ 1 19 ,  pp. 95-
99]) assumption (H6) will be satisfied in the general case. 
The functions (Uj ,  Vj ) = (Cj(m) - c/m+1) , E;(m) - E;(m+l» and the monotone nondecreasing property of Ii and 
Fj implies that 
aUj _ D n2 . - F ( -�m-l » _ F ( -�m» > O ' (0 1'J "" A al 1 v XUI - JI t, X, C] Jj t, x, C] _ In , X ':A.f1, 
a a-(m) a-(m+l) D . .-!i + IIu = [D -C-j -+I-J .c(m) ]- [D . Cj +f-J .c(m+I)] = If .(C(m-I) _ C(m» > 0 on (0 T]xa!hxA 1 an 1 1 1 an 1 1 1 an 1 1 1 1 1 - , , 
with similar inequalities in the boundary and initial conditions. This ensures that 
C-.<m) > c-.<m+l) and C-.(m) > C-:(m+l) , - ,  , - " 
from the monotonicity of .0/' and proves that {(c?), E;(k» } with (cj(O) , E;(O» = (Cj ' Cj) is a monotonic 
non increasing sequence. 
I t  follows from a similar induction argument that {(dk) , k�k» } with (d!l) , k�O» = (fj , �j )  is a 
monotonic nondecreasing sequence and by a similar induction argument C?-I) � ��k-I) and E;(k-I) � h�k-I) 
for k=l , . . .  ,m ensures that clm+1) � f�m+l) and E;(m+l) � k�m+l) .  The following inequalities then hold 
for all k = 1 ,2, . . .  
I t  follows from the monotonic property of  our maximal and minimal sequences, its boundedness by 
(fj ,  �j )  and (Cj ,  Cj ) and the monotone convergence theorem that the pointwise limits 
I· « (k) C(k» - ( C ) 1m fl ' -I - �j , _j , k�oo 
I · (-(k) C-(k» - (- C- ) 1m Cj ' j Cj , j ,  
k�oo 
exist, implying that 
c � ��O) � k� l) � . . .  � k�k) � . . .  � kj � E; � . . .  � c,(k) � . . .  � c,(1) � c,(O) � Cj for (t , z) E (0, 1"] X X,  
for all k = 1 ,2' 0 0 .0 
-��-----------
3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 79 
3.6.3 Existence of Solutions 
It can be shown that our minimal and maximal sequences ( dk) , ��k» } and ( e?), �(k» } converge not only 
pointwise but converge uniformly (in appropriate function spaces) as well . The fol low ing theorem is an 
existence theorem for solutions to the system Sn ' Bn . 
Theorem 3.6.1 (Generalised Existence Theorem) 
Let the assumptions of Lemma 3.6.7 hold. Then the minimal and maximal sequences ((f�k) , ��k» } and 
((c?), C;(k» ) converge monotonically and uniformly from below and above to (fj ' �) and (Cj , C;) 
respectively, where (fj '  £2j) and (ej ,  E; )  are solutions of Sn ' Bn ' Moreover (fj ' £2j) and (ej '  E; )  are 
minimal and maximal solutions respectively of Sn ' Bn in the regions [fj , Cj ] and  [qj , Cd and b y  
uniqueness (fj , C ) =(ej ,  E; ) .  
Proof 
We first consider the case with Dj, 9Jj > ° for all i. 
Let dk), Cj(k) E Cl+aI2.2+a.a[(O, T] X .Q  x A ,  Rn(/) ] and £2�k� C;(k) E Cl+aI2.2+a [(0, T] x A ,  Rn(J)] 
for k = 1 ,2, . . . and let us consider the maximal sequence ((Cj(k) , C;(k» ) . 
We note that Cl+aI2.2+a [(0, T]x.Q, Rn(I)] CW�;2 [(0, nx.Q, Rn(/)] and CI+aI2.2+a [(0, T] x A ,  R
n(J)]c 
W�;2 [(0, T] x A ,  Rn(J) ] for P I � (ml+2)/( I-a) and P2 � (m2+2)/( 1-a) .  
From Lemma 3 . 1 . 1 .  lhis implies that C1+al2,2+a[(O, T] xQ, Rn(/) ] �W;·2 [(0, T] xQ, Rn(/) ] and 
C1+a/2,2+a [(0, T] x A ,  Rn(J) ]!:;;; w�·2 [(0, T] x A ,  Rn(J) ] , where q = min (P l , P2) . 
By  Theorem 3 . 1 .4, we see, lhat e?) (with A treated as a parameter space) and E;(k) satisfy the 
fol lowing Agmon-Douglis-Nirenberg estimates (see Theorem 3 . 1 .5): 
and 
II c?) I Iw? 1 (0, T]xn. RoCI ) 1$ C(II!;(t , x, cY-1» I IU I(O, T]xn. R°(l) ] 
I IC-(k-I ) U  + j W;I2-1I2Q,I-I/q [(O, TjxBf.!I , Ro(l) j 
+ II Cj.o"W;-2/Q [n, R0(l) j ) '  
I IC-(k) 1 1  < C(II F (t C-(k-I) f -(k-I) I I  j w?[(o. TJxA. ROc/) )- j , Z, j + Bf.!2 Cj U [(o. TjxA, RoCJ» ) 
+ Ilci.l llw�/2-1/2Q'I-I/Q [(0. T)xBA1 • RO(/) )  
+ I ICj.o l lw;-2/Q [A. RO(J» ) ) '  
(3.6.63) 
(3.6.64) 
The fact that e(k-l)E [c · c )  and C�k-l)E [C - C )  i e e(k-I) and C�k-l) are bounded the continuity of 1'. J -J ' J J -J ' J '  . . , J J ' J I 
and of Fj imply by the continuity and boundedness of the Nemytskii operator (Lemma 3 . 1 .6) that the 
sequences ( h (t, x, eY-1» ) and (Fi(t, z, CY-1» + Lili e?-I») are unifonnly bounded in C[(O, 1'] x.Q ,  Rn(l) ] 
(with A treated as a parameter space) and q(O, T] x A, Rn(J) ] respectively. 
Since C[(O, T] x !2 ,  Rn(l)J and C[(O, 'J'j x A . Rn(J) J are dense in £11(0, 'J'] x !2 ,  Rn(l)J and 
Lq [(O, T] x A ,  Rn(J) J respectively, i t  fol lows by the continuity and boundedness of the Nemytskii operator 
(Lemma 3 . 1 .7) that ( h (t, x, eY-1» ) and (F;(t , z, CY-1» + Lf.! e?-I») are bounded sequences in 
U[(O, T] x .a, Rn(l)] (with A treated as a parameter space) and £1[(0, h x A ,  Rn(J)] , respectively. 
This, together with the above estimates (3.6.63) and (3.6.64) shows that the sequences Ie?») and 
{E;<k)} are uniformly bounded in W;·2[(O, T] x !2 ,  Rn(/) ] (with A trealed as a parameter space) and 
3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 80 
W �.2[(0. T] X A .  Rn(J)] respectively. Therefore by application of Theorem 3. 1 . 1  (Imbedding Theorem). we 
obtain 
(3.6.65) 
and 
(3.6.66) 
for all k = 1 .2 . . . . where C in both estimates are independent of any element of W;· 2 [ (0. T] x.Q ,  Rn(l) ] or 
W? [(O. T] x A . Rn(J) ] .  
From (3 .6.65) and (3 .6.66). we  can conclude that every uniformly bounded sequence in  W;·2 
[(0. T] x.Q .  Rn(l) ] and W? [(O, 1'] x A , Rn(J)] i s  also uniformly bounded in C( l +a)/2.1+ a [(0. T] x.Q . Rn(l ) ]  
(with A treated as a parameter space) and C(I+a)/2.1+a [(0, T]x A, Rn(J» ) ,  respectively. From Lemma 3 . 1 .3 ,  
we may take a in both cases to be identical. 
Thus the sequences (c?)} and (C;<k) }  are uniformly bounded in C(l +a)/2.1+ a [(0. T] x.Q .  Rn(l)] (with 
A treated as a parameter space) and C( 1+a)I2.I+a [(O. T] x A. Rn(J) ] respectively. Therefore, by applying 
Lemma 3 1 6 ( /'· (I x c(k-I) } and (F(I Z E�k-l» + J C�k-I)} are also bounded sequences in Ca12.a • •  , " , J ' , ' J all ' 
[(0. T]x.Q .  Rn(l) ] (with A treated as a parameter space) an�1 C a/2•a [(0, 1'] x A. Rn(J» ) ,  respectively. Hence 
by the Schauder type estimates (Theorem 3 . 1 .3). we have 
and 
I Ic? ) lIe1 +at2,2+a [(0. TJxIi. R"U) l $ C(II/; (/. x. eY-l» lIca/2.a l (O. TlxIi. /l"U) J 
+ I I C? -I) lIe(l+a)!2.1+a [(0, 1 ' ]xJfJ1 • R"(/) I 
+ I ICi.0 Ile2+a eD. R"(/» ) )  (3.6.67) 
I IC-(k ) 1 1  < C'(II F (  C-(k- l» J -(k- I) I I i cl+a12.2+a [(o. TlxX. R"()) )  - i I ,  z, j + afh Ci ca12•a [(o. l')xX. H·()) ) , 
+ IICdle(l+a)l2.1+a [(0. TjxaAI •  R"(J» )+ I I Cj•O lIe2+a [A. RO(J» ) ) (3.6.68) 
for al l  k = 1 .2. . . .  which implies the uniform boundedness of sequences (Cj<k) } and (t;<k)} in 
Cl+a/2.2 +a [(0. T] x.Q . Rn(l ) ]  (with A treated as a parameter space) and Cl+a /2.2+ a [(O, T] x A . Rn(J» ) ,  
respectively. 
We may use arguments given in Lemma 3.6.5 to show that sequences (e/k)} are Holder continuous in 
Z with exponent a and may be shown to be uniformly bounded in A. 
Therefore. by the natural compact imbedding of C l +a/2.2+a. a [(0. T] X .Q  x A .  Rn(/) ]  and C l +a /2.2+ a 
[(0. T] x A .  Rn(J)] into C I .2.O [(0. T] X.Q  x A .  Rn(/ ) ]  and CI.2 [(0. T] x A. Rn(J» ) ,  the sequences (e?)} and 
(t;(k)} are relatively compact in CI•2,0 [(0. T] x Q x A .  Rn(/) ]  and CI•2 [ (0, T] x if. Rn(J)] respectively. 
This implies that there exists subsequences of (e?)} and (C;<k) }  which converge in C 1•2•O 
[(0. T] x.Q  x A ,  Rn(/ ) ]  and C 1.2 [(0. T] x if. Rn(J)] respectively. 
Let (Ci . Ci ) where Ci E CI.2.O [(O, T] x.Q  x A ,  Rn( / ) ]  and Ci E C1.2 [(0, T] x A ,  Rn(J)] be the l imit of 
this subsequence. 
On the other hand, we have shown that the sequence ((c/k). C;(k » ) converges pointwise to (Cj .  C;) .  
Therefore, Cj = Cj on [(0. T] x .Q x A .  Rn(/) ] and Ci = C; on [(0. T] x A • Rn(J) ] .  This shows that the whole 
sequences (e? ) }  and (C;(k») converge in CI•2•O [(0. T] x.Q  x A. Rn(l ) ]  and CI•2 [(0. T]x if. Rn(J) ] to ci and 
C; respectively . �hat is lim C/k) = Ci in C I • 2.O [(O, T] xQ xA , Rn(/) l and lim C;(k) = t;  i n  C I•2 - (J) k �oo - - - -J<.�oo [(0. T] x A ,  Rn ] and fi $ Cj $ Ci on [0. T] x .Q x A and �j $ Ci $ Ci on [0, TJ x A . 
3,6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 81 
Similarly, by imitating the preceding argument relative to the minimal sequence ( f�k) , ��k» } , one can 
conclude that the sequences (dk)} and (��k)} converge monotonically in C1•2•O [(0, T] x .Q  x A ,  Rn(l) ]  and 
C1•2 [(0, T] x if, Rn(J) ]  respectively; their l imits are denoted by fj and C which belongs to 
C1•2•O [(0, T] X .Q  x A ,  Rn(l) ]  and d·2 [(0, T] x if, Rn(J) ]  respectively and satisfy the relation fj � fj � Cj on 
[0, T] X .Q  x A and C � �j � Cj on [0, T] x if ,  Thus the limits 
and 
d (k) d I ' [2L _ D.V2 (k ) ] = fj _ DP2 . I ' [ ,, ( (k-l» ]  ,, ( ) 1m :1 I x fl :1 I V xfl '  1m Jj t, x, fj = Jj t, x, fj , k�oo at at k�oo 
l im [D· df\k) + H. dk)] = D. dfj + H. c . lim [H . c(k-l) ] = l-J . c . 
k�oo I dn I -I I dn I -I ' k�oo I - I  I -I ' 
dC(k) dC l im [� - 9l:V2 c(k) + u '  VC\k) + H·s(C\k)] = -=i.. _ GlV2c . + u ' Vc . + H·s:lC ·  
k�oo dl I -, -, , -, dt -'! -, -, , -p 
d-(k ) :1-I· [ Cj D 02-(k) ] _ aCj D 02-1m --- , v  C . - - - · v  c ·  
k�oo dt I x ,  dl ' x  I ' :1-(k) :1-I· [D aCj IJ -(k)] - D aCj IJ - I '  [ I -C(k-l) ] - leI -C · 1m j --:1 + IjCj j -;- +  �jCj , 1m /-, j  j - �j " k�oo on on k�oo 
--(k) --
lim [ dCj _ �.V2c.(k) + u ·  VC.(k) + If .dc.(k» )  = dCj - 9) V2C + u ·  vE + /l dC 
k�oo dt " ' " dl " " "  
I , [F ( C--(k-l» H I -(k-l) ] _ F ( C-- ) H I -1m j I, Z' : i + j Cj - 'j I, z, -i + j Cj , k�oo v iJD2 -.J iJD2 
etc, exist uniformly on C I•2,u l (O, 'J'J x !2 x A , Rn(l) j and C l.2 l(O, 'J' j x A, Rn(J) J , respectively,  Thus we 
conclude that (fj ' �) and (Cj ,  C;) are solutions of the IBVP 
and 
dC ' 2 it' - DjVxfj = !; (/ , X, f) , 
dC · Dj d� + l-Ij fj = lIj�j ' 
d�j 2 I Tt - �V �j + u , V�j + Hjs(C = FJt, z, �) + Hj iJD2 fj ' 
dC' -D· -' + H·c· = H·C,· l an l l I '  
;C. 2-- -- -- -- f -' - $. v C + u ' V C + /I·de. = F (t Z C)  + If . c· dt I ' " " ' ' \J  ' iJfl2 " 
respectively, 
We therefore have that (fj ' �.> and (C,. , C;) are solutions of Sn ' Bn . 
3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 82 
For components j E I, where Dj = Hj = 0, the uniform convergence of sequences (c?) } to solutions f.j and 
Cj is standard (HARTMAN [ 1 1 9] ,  LAKSHIMIKANTHAM [ 1 57]). The sequence of approximations (3.6.8) are 
uniformly bounded and equicontinuous and hence possesses uniformly convergent subsequences. The rest of 
the argument follows along similar lines to that already discussed. The case for components i E J, where 
9Jj = 0 and u· VC}k) == 0 is similar as is the case for components i E J, where 9Jj = 0 and u· VC?) '1= O .  
Now we  show that (fj ' �: ) and (Cj ,  C;) are minimal and maximal solutions o f  Sn ' Bn . Let (Cj, ci) be  any 
solution of Sn ' Bn such that f.j � Cj � Cj and G � Cj � Cj . 
Since (Cj, Cj) = f/{Cj, Cj), it follows by Lemma 3 .6.6 and from the definitions of ( f�k), 
( clk� C;(k) )} , that 
implies that 
( C ) < /JT( (0) C(O» < ( . C ·) - (JIf . C ·) < trr(-(O) C-(O) < (- C- ) fi ' ,.... i - i7 fi ' -I - e" , - ",C" I - v ci ' i - Ci t  i ' 
Let us assume that for some m > 1 ,  
Then we shall show that 
( (m+l) C(m+ l» < (c . C .) = (-(m+l) C-.(m+l» f., ' _I - I, I C, ' I • 
From Lemma 3 .6.6 (II) and from the definitions of ( dk� dk))} and (Ull C;(k) )} ,  we arrive at 
(C�m+l) c�m+l» = f/(c�m) C(m» < (c ' C ·) = (JIfc ' C ·) < f/(c.(m) c.(m» = (c�m+l) C.(m+l» _, ' _I _I , _, - h ' ..,-\ h , - ' "  ' " • 
Thus, it follows by mathematical induction that 
(C�k) C\k» < (c C) = (c.(k) C.(k» ) -I ' -I - '"  , ' I ' 
for all k = 1 ,2, . . .  
Hence, we have 
proving that (fj ' �j ) and (Ci , C;) are minimal and maximal solutions of Sn ' Bn ' 
Finally, by uniqueness results demonstrated in section 3.3, (fj , �j )==(Cj ,  C;) .O 
For the general system Sn ' Bn , which may possess no monotone propertjes, the following theorem 
fol lows from Theorem 3 .4. 1 and summarises how we may set up monotone sequences for this general 
system which converges to a solution of a new systcm which rclates in some way to the solution of the 
original system Sn' Bn · 
3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 83 
Theorem 3.6.2 
The general system Sn. Bn for which /; and Fi satisfies Lipschitz continuity properties (H 1) and Holder 
continuity properties (H2). may be imbedded in a system Sin . Bin of twice the order which is coupled 
by monotone functions j;" and F/ of the new dependent variables Vi and Vi. Moreover. all the solutions 
(Ci . Ci) of the general system Sn. Bn are solutions of the new system. where 
Vi =ci. Vn( /)+i = -Ci for i= l  . . . . n(l) (3.6.69) 
and 
Vi = Ci• Vn(J)+i = -Ci for i= l . . . . n(J). (3.6.70) 
Let (Yi . Y;) and (Vi .  v,. ) be lower and upper solutions for the system Sin .  Bin with continuity 
properties given in the assumptions of Lemma 3.6.6. Let also the assumptions of Lemma 3.6.7 hold for 
the system Sin .  Bin . Then the minimal and maximal sequences {(��k) ,  1:::�k ) }  and  {(vi(k) .  v,:(k) }  of 
Sin . Bin given by Theorem 3.6.1  converge monotonically and uniformly from below and above to 
(�i ' 1:::i ) and (Vi .  v.:) respectively. where (�i ' L)  and (vi .  v.:) are solutions of Sin ' Bin satisfying the 
following inequalitites 
for all k = 1 ,2, . . .  
Wi = Vi + vn(l)+i = 0 in  QxA, at  t = O. 
a;i -DiV;Wi = f;*(I ,  x, Vj) +  fn(/)+ i (t . x, vk ) in (0, Tjx£2xA, 
aw· 
an' = 0 on (0. Ilxa.QlxA, 
aw Di an' -Hi (Wi -wi) = ° on (0, Ilxa£22xA, 
a::- 9W;Wi+u'VWi = Ft(t, z, V,· )  + Fn(J)+i (t, z, Vk ) - Hif  (Wi - Wi ) in (0, IlxA, ot . iHl2 
VI Wi + !l!  �Wi = 0 on (0. TJxaAI .  
onl 
aWi = 0 on (0, TJxaAa, a = 2, 3, ana 
(3.6.73) 
(3.6.74) 
(3.6.75) 
(3.6.76) 
(3.6.77) 
(3.6.78) 
(3.6.79) 
(3.6.80) 
for (wi , Wi) = (vi + vn(l)+i ' Wi + Wn(J)+i ) generates the unique solution (fi '  C) == (Ci , E;) == (Ci , Ci ) of the 
general system Sn .  Bn , where 
3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 84 
for (I, x, z) E (0, T] X.Q  x A ,  
for (t , Z) E (O, T] x A, for all k = 1 ,2 .. . . 
We have shown in this section that if  the reaction functions obey a monotone property . our sequence of 
approximate solutions is a monotone sequence that converges uniformly and thus our limit function is 
actually a solution of the given problem. 
Although a number of existence-comparison theorems for weakly coupled parabolic systems have 
also been known and can be established by various other methods such as the functional analytic approach of 
KUIPER ( 149) and in connection with invariant sets (BEBERNES el al. [35-37)). the monotone argument is 
more constructive and provides a simpler and straightforward proof than the other methods. The definition of 
upper and lower solutions with the monotone argument is however more restrictive (BEBERN IlS and 
SCHMITr [37]). It must be noted that it is also not the only way of obtaining a constructive existence proof. 
Our existence theorem in this section relies on solving quasilinear parabolic differential equations by 
monotone iteration. The iteration could alternatively have been done by successively applying a monotone 
integral operator with an appropriate Green's function as its kernel. This procedure also starts with a lower 
and an upper solution (BANGE l33J) . Furthermore. these same approaches can be used to obtain a similar 
existence-comparison theorem for the time independent problem simply by dropping the time derivatives and 
the initial conditions. This will be discussed in a chapter four when dealing with the existence of a steady 
state solution. 
Remark 3.6. 7 
In many cases, we will require for practical reasons that the functionsJi and Fj be redefined for Cj' Cj < ° so 
that if 
and 
{fi(/. X. Cj) 
,+ I x c '  = fi ( ,  " , ) fi (t . x, 0) 
, +  , _ 
{Fi(/ . z. C) 
Fi (/ , Z . C, ) - Fi(/ , z, 0) 
for cj :2: 0  
ror cj < O, 
for Cj :2: 0  
for Cj < 0, 
then it can be shown that these new functions have the same Lipschitz and HOlder continuity properties as 
our original functionsJi and Fj. 
Remark 3.6.8 
In this section we have shown lhal lhe imbedding resulLs of seclion 3 .4 may be used lo oblain existence 
theorems for systems of equations with nonmonotone reaction functions. The imbedding results do not need 
boundedness nor does it need the additional Holder continuity properties that we have assumed for our 
3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 85 
system. This additional continuity property was only needed in order to establish the existence of solutions of 
a corresponding l inear system. 
Suppose we prove that any solution of the system Sn, Bn must be bounded by some constant K. 
Change then the definitions of fi(t, x, Cj) and Fi(t, z, Cj) for Icj l ,  ICjl > K ,  for instance by defining 
and 
� {J:(t, X, Cj ) for I c/ : ;  K 
J:(t, x, Cj ) = 1'. ( + K) " + K ), t , X, _ lor _ Cj > , 
� {F;(t, z, Cj) for IC/::; K F(t z C . )  = 
, ' ' J  F·( + K) f + C K , t , z, _ or _ j > . 
If we are then able to prove the existence of a solution (Ci, Cj) for the problem Sn, Bn with l(t, x, Cj ) and 
F;(t, z, C) given above, then (Ci, Cj) is also a solution of the original problem Sn, Bn . The same is true of 
uniqueness. We also note that these new functions satisfy the same Holder continuity properties that are 
required in section 3 .6, as our original functions. It is common to use functions such as these new functions 
for proving uniqueness and existence theorems given that we have upper and lower bounds particularly for 
functional analytic existence proofs (FRIEDMAN [94, p.203] , BEBERNES and SCHMITT [35] ,  LADDE et. al. 
[ 1 53]) .  These bounds do not necessarily have to be constants K as in the above equations but may be 
arbitrary functions with the appropriate continuity conditions. 
Remark 3.6.9 
We can choose 
and 
C-· = C-· = AeRI " ' 
where A is a constant determined from the boundary conditions and R is, as yet an unspecified constant so 
that 
-AReR1-J:(t, x, -AeR1 ) $ O , 
AReR1-J:(t, x, -AeRI ) � O ,  
-AReR1-F; (t, z ,  - AeR1) $ O ,  
AReR1- F;U, z ,  -AeRI ) � O ,  
and we can easily satisfy the above inequality since fi and Fi are Lipschitz in Cj and Cj, respectively by 
choosing R sufficiently large. This demonstrates the existence of suitable lower and upper solutions (fi ' fi) 
and (ci ' Ci ) respectively. 
Remark 3.6.10 
The constructive methods of proving existence results for problems in this section can also provide numerical 
procedures for the computation of solutions which are of greater practical value than the theoretical existence 
results. By replacing the differential system by a suitable finite difference system (which is a discrete version 
of the continuous problem) and using an analogous definition of upper and lower solutions, it is possible to 
construct monotone sequences which converge monotonically to a solution of the finite difference problem 
(GROSSMAN [ 1 1 1 1 ,  GROSSMAN and Roos [ 1 1 0 1 .  PAO 1 2 1 9-2221) . 
3.7 NOTES AND COMMENTS 86 
The asymptotic stability and multiple steady states may also be studied in the framework of this finite 
difference system (PAO [222]) as well as error estimates between the true solution and the computed mth 
iteration (PAo[22 1 ]) .  It is obvious that these numerical procedures may be applied to the system Sn, Bn 
which may obey no monotone property. The discretised functional term in (3 .6. 1 2) is expressed as a 
summation. 
From the type of nonlinearitiesfi and Fj, an iterative scheme may be considered which accelerates the 
rate of convergence of the sequences of iterates defined in (3.6.8)-(3.6. 1 5) .  This involves solving coupled 
systems of linear equations and thus fewer equations (LADDE [ 1 5 3 ,  p . 1 7 1 ] ). It is useful in speeding up 
numerical convergence. An example of another process which is useful in speeding up numerical 
convergence will be seen in section 6.3. 
Remark 3.6.11 
Equations Sn ' Bn have been chosen with bioreactor applications in mind, but the theory can be readily 
generalised in a number of ways. Our proofs are still valid for Di'Y:cj replaced by 'V x . (Dj (x, Cj )'V xCj )  and 
9)j'V2Cj replaced by 'V .  (!l{ (z, C;)'VCj ) ,  provided that we have uniform ellipticity conditions for these more 
general equations. Furthermore, the mass transfer coefficients /-Ij could be functions of x and I,  provided that 
these functions are still positive and satisfy appropriate continuity properties, and a wider class of coupling 
functions is permissible, since fi and Fj may be permitted to depend on 'V Cj and 'VCj, respectively. However, 
it is necessary to require a Nagumo type growth condition with respect to these variables. 
Remark 3.6.12 
In this section we have assumed that the boundary conditions (2. l .4) and (2. l .7) are of the Robin type. We 
may treat the Neumann and Dirichlet type boundary conditions similarly by using appropriate theorems for 
linear parabolic equations with Neumann and Dirichlet type boundary conditions (see Remark 3 . l .2 and 
Remark 3. 1 .3) . 
3.7 Notes and Comments 
Sections 3.3-3.5 are adapted from PARSHOTAM, McNAI3B and WAKE [228] and MCNABB and PARSHOTAM 
[ 1 88 ] .  However, most of the proofs for Comparison Theorems in these papers are obtained in a different 
manner and order. 
We see from the imbedding results of section 3.4, that many results for monotone systems may be 
applied to systems which obey no monotone property. Since existence of solutions in our imbedded system 
(which obeys a monotone property) implies existence of solutions in our original system (which may not 
obey any monotone property) and vice versa, so does many qualitative properties of solutions. We have seen 
from section 3 .4 and 3.5 that uniqueness and global stability may be such properties. There are also many 
other qualitative properties such as bounds, gradient bounds and asymptotic stability which give rise to these 
properties from one system to the other. We have seen from Theorem 3 .6.2 that bounds on the solution in 
our imbedded system can give rise to bounds in our original system (it is  noted however, that upper and 
lower bounds need not exist in order for the imbedding results to apply). From these bounds it can be shown 
that asymptotic stability of the solution in our imbedded system also implies asymptotic stability for solutions 
of our original system. Other properties of solutions which give rise to these same properties from one 
system to another are perturbation solutions. It is often easier to obtain a perturbation solution for systems 
where the nonlineariLies obey a monotone properly (see section 6.5). This could be useful in parameter 
sensitivity analysis in systems of equations which do not obey a monotone property. 
3.7 NOTES AND COMMENTS 87 
There are many useful qualitative properties of scalar parabolic and elliptic equations which rely on 
the monotonicity of the nonlinear reaction functions. Some of these properties may be generalised to 
monotone systems. The imbedding results of this chapter are useful in generalising these properties to 
arbitrary systems which do not possess any monotone property. Some of these results could be to show the 
existence or nonexistence of dead cores, radially sym etric solutions which can occur if the nonlinearities are 
singular (GATICA el al. [ 1 03] ,  CASTRO and SIIIVAJI [5 1 ]) ,  degenerate solutions, nonnegative solutions 
(CHAllROWSKI [52]) and certain properties of unstable solutions. The imbedding results of section 3 .4 also 
applies to weak (or generalised) solutions (CI-IABROWSKI [52-63]) and the qualitative properties of such 
solutions. 
The imbedding results of section 3 .4 may also be applied to a much larger class of differential 
equations. It is known that these imbedding results do not hold for certain hyperbolic equations but it is not 
clear whether it would hold for larger classes of differential equations such as differential equations of the 
type we have seen which are coupled in their derivatives. It is however known that comparison theorems do 
exist for such problems (TRUDINGER [286]) and these comparison would be useful if these imbedding 
results were able to be extended to such a class of equations. It would appear as if we would have to imbed 
such a nonmonotone system that is coupled in its derivatives several times in order to obtain a monotone 
system with reaction functions monotone in all its dependent variables and derivatives. It is also not clear 
whether these imbedding results may be extended to higher order differential equations or even functional 
differential equations, stocastic differential equations and a more general class of equations. It is known 
however that comparison theorems do hold for certain classes of causal functional differential equations 
(MCNABB and WEIR [ 187J) and higher order differential equations (SPERB l27 1 J) and these comparison 
theorems would also be useful if these imbedding results were able to be extended to this class of equations. 
In section 3.5, we established some useful conditions for the global stability of the general system Sn, 
Bn where we assumed at the outset that the system Sn, Bn was a quasimonotone system, i.e. Ii and Fj are 
monotone nondecreasing in Cj and Cj respectively for j � i. These results could also have been obtained by 
assuming that Sn, Bn was a monotone system , i.e. fj and Fj are monotone nondecreasing in Cj and Cj 
respectively for all j. This is not a restriction on the stability results since if this monotone property is not 
satisfied, we may make the substitution Cj = e -Kx\ Wj and Cj = e -Kz\ Wj , (where XI  and z\ are chosen without 
loss of generality to be the first components of X and z) to obtain a system of the same type but with new 
functions that are monotone nondecreasing in Cj and C. Note that it is more appropriate in this case to make 
this substitution rather than Cj = e-K1wj and C1 = e-KI� in studying the stability of our general system. 
In  many cases, the nonlinear functions Ii and Fj need only satisfy a one-sided Lipschitz condition 
when obtaining comparison theorems and in proving monotone convergent sequences to a solution of the 
system Sn, Bn (KELLER [ 1 4 1 J ,  AMANN [5], LADDE el al. [ 1 53J , PAO l 2 1 1 ,  2 1 9J). 
In section 3.6, it can be shown that if fj and Fj satisfies a HOlder condition then locally upper and 
lower solutions can be constructed and any problem of the system SrI> Bn which has nonuniqueness locally 
has distinct maximal and minimal solutions (BEBERNES and SClIMrrr [35 J). 
4 
The Steady State Problem 
4.0 Introduction 
For the study of the stability of the solutions of parabolic initial boundary value problems. one has to have a 
good knowledge of the steady states, that is of solutions to the time independent problem. In this chapter, we 
shall look at the steady state system Sn ' Bn , that corresponds to the unsteady state system Sn, Bn. We shall 
obtain theorems that guarantee existence of solutions to the steady state system Sn . En and establish 
relationships between the unsteady state system Sn, Bn and the steady state system Sn . En . 
In section 4. 1 ,  we shall collect some notational conventions and basic definitions and give some 
general results and relationships of the spaces that are needed in order to obtain the exact result on the 
solvability of linear elliptic equati?ns. The Maximum Principle for elliptic equations which will be used 
throughout this thesis will also be defined. 
In section 4.2, we see that for the purposes of uniqueness. stability and existence theorems, we may 
assume at the outset that the system Sn . En is a quasi monotone system. i .c .  Ii and Fi are monotone 
nondecreasing in Cj and Cj respectively for j :;C i. This is not a restriction on these theorems since if this 
monotone property is not satisfied, then Sn . En with general functionsfi and Fj can be imbedded in a system 
S2n ' E2n of the same form. It can be shown that solutions of this new system generate solutions of the 
original system and therefore uniqueness. stability and existence can be implied in the original system but 
only if uniqueness is guaranteed in the system S2n . E2n . 
In section 4.3, that solutions of problem Sn ' En specified by Cj, \ exist. This is done by constructing a 
sequence of approximate solutions which converges monotonically and uniformly to a limit function which is 
a solution of the system Sn ' Bn . 
In seeton 4.4 we use relationships concerning the asymptotic behaviour of linear parabolic equations 
as t -7 00  and their corresponding linear elliptic equations to make a statement about the relationships between 
solutions of the steady state problem Sn . En and the unsteady state problem Sn, Bn. 
Finally, in section 4.5 we shall discuss some relevant literature and future work. 
4.1 Definitions, Notation and General Results 1'01' Linear Ell iptic Equations 
In this section, we shall collect some notational conventions and basic definitions. We shall also give some 
general results and relationships of the spaces that are needed in order to obtain the exact result on the 
solvability of linear elliptic equations. The Maximum Principle for elliptic equations which will be used 
throughout this thesis will also be defined. Finally. we shall look at some relationships between solutions of 
linear parabolic and linear elliptic equations. 
88 
4.1 DEFINITIONS, NOTATION AND GENERAL RESULTS FOR 
LINEAR ELLIPTIC EQUATIONS 
4.1 .1  Definitions and Notation 
The following notation will be used throughout this section 
x = (Xl . X2 • . . . . •  xm) denotes a point in Rm. 
� is a bounded. open. connected domain in Rm. 
iY9 denotes the boundary of �. 
iff denotes the closure of �. 
u E R.  
Dxu = (du/dx l  • . . . . •  du/dxm) .  
1 1 · 1 1 will denote the Euclidean norm in  Rm. 
Definition 4.1.1 
89 
A vector field Vex) = (V l (X) • . . . . •  Vm(x)) is said to be a unit outerward normal (outward normal or 
outernormal) at x E Q if x - hv E Q for small h > O. The outernormal derivative is then given by 
dU = lim u(x)-u(x-hV) . 
dV h-+O h 
ci."d .J hi) ,'s CI v,n"J- .ed-of nOf,-r>oJ +0 fi-
4.1.2 General Results and the Relationships between HOlder, Lq and Sobolev Spaces 
The following definitions of H(jJder. Lq and Sobolev spaces are adapted [rom LADYZHENSKAYA [ 154. pA-
5J and GILBARG and TRUDINGER [ 106. p. 144] .  
Hijlder Spaces 
For A�Q. a function f: A � R is  said to be Holder continuous of exponent a, where 0 < a $ 1 .  if there 
exists a constant Ii = Ii(A) such that 
If(x)- f(y)l $ lillx- ylla . 
We note that the quantity 
Ii�(f)= sup If(x)-f�y)l . x,yeA IIx-yil 
"Y 
is the smallest number H. 
We shall say thatf E Ck+a[A . RJ iff : A -7 R is continuous, the partial derivatives of f. up to order k are 
continuous on A and the kth partial derivatives are Holder continuous with exponent a. For f E C2+a[A. RJ, 
we shall use the following notation: 
where 
IIfll� "= suplf(t.x)1 +I-I�(f), 
xeA 
A 
A A 
m m d2 f IIfIl2+a= IIjll1+a+LL dX dX " 
• 
1=1 J=1 1 J a 
and 
4.1 DEFINITIONS, NOTATION AND GENERAL RESULTS FOR 
LINEAR ELLIPTIC EQUATIONS 
II 
a
if II
A 
I 
a
if I 
a
if - = sup- +H�(-) , aXj a xeA 
dXj aXj 
90 
Since iff is closed, it is required that the derivatives up to order k can be continuously continued from the 
interior of '9 to all of iff . If a = 1 ,  we also say thatfis Lipschitz continuous. 
The superscript in the foregoing notions may be deleted if there is no ambiguity on which set the above 
norms are to be determined. 
Lq Spaces 
Lq ['9, R] is the Banach space consisting of all equivalent c1ases of Lebesgue measurable functions u defined 
on n into R with a finite norm 
where q ;::: 1 .  
Sobolev Spaces 
For nonnegative integer t , W�['9, R] is the Banach space consisting of the elements Lq [" R] having 
generalised (weak or distributional) derivatives of the form Da for all 101 � t. The norm in it is defined by 
where the summation Llalsl is taken over all nonnegative integers a satisfying the condition a � t. 
Note that the space W�[', R] is in a certain sense analogous to the Cl+a [ij, R] spaces. In the W� ['9, R] 
spaces, continuous differentiability is replaced by weak differentiability and Holder continuity by q­
integrability, so that W�[', R] = {u E Wl[" R]; Dau E Lq [" R] for all lal� I] , where Wi ['9, R] is the linear 
space of t times weakly differential functions. 
General Results in HOlder, U,q and Sobolev Spaces 
We now give some general results in Holder, u·q and Sobolev spaces. Most of these results are special cases 
of those proved in section 3 . 1  and are only included here for easy reference. 
Lemma 4.1 .1 
If f, g E Ca [A, R] , then f + g E C(x [A,  R] and IIf + gl l� � IIfll� + I Igll� . 
Lemma 4.1.2 
If f, g E Ca[A, R] , then fg E Ca[A, R] and l l fgl l� � l I fl l� I Ig l l� . 
4.1 DEFINITIONS, NOTATION AND GENERAL RESULTS FOR 
LINEAR ELLIPTIC EQUATIONS 
The nexl Lhree lemmas and their corollaries are analogous in Holder, U.q and Sobolev spaces. 
Lemma 4.1.3 
Suppose 0 < 13 � a � 1 .  Then Ca (A) � cP (A) . 
Corollary 4.1.1 
Suppose 0 < a, 13 � 1 .  Then Ca (A) n cP (A) = Cr (A) where r= min ( a, 13) . 
The following lemma is stated without proof in BURKILL [44] 
Lemma 4.1.4 
Suppose 1 � P I � P2. Then LP2 [A, R) eL!'I [A, R) .  
Corollary 4.1.2 
Suppose 1 � P I ,  P2. Then L!'2 [A, R) nLPI [A, R) = L!'[A, R) where p = min {P I ,  P2 ) .  
The following lemma and its corollary follows from Lemma 4. 1 .4 and the definilion of Sobolev Spaces. 
Lemma 4.1.5 
Suppose 1 $ PI $ P2 ' Then W;2 I A, R I C;;; W� lA ,  R I .  
Corollary 4.1.3 
Suppose 1 $ P I , P2. Then W�2 [A, R) n W�I [A, R] = W�[A, R] where p = min (P I , P2 ) .  
As with sec lion 3 . 1 ,  we define the rollowing openllor which lakes a space inl<> ilSelL 
Definition 4.1 .1 
The Nemytskii operator .A(u) is defined by 
.A(u)(x) = h(x, u), x E ij,  
for u E Cl+a[ij, R] . 
We now presenL a result concerning the Nemylski i  opera lor .;1(u) which is a special case of Lemma 3. 1 .6. 
Lemma 4.1.6 
Let h E ca[ij xR, R], and let .A(u) be the Nemytskii operator .;1(u). Then 
(i) .A"E C[Cl+a[f, R] , ca[f, R) ) ; 
(ii) .A"takes bounded sets in Cl+a[ij, R] into bounded sets in ca[ij, R] . 
Remark 4.1.1 
91 
From the fact that the space Cl+a[f, R) is compaclly imbedded in CI [f, R] the Nemylskii operator belongs 
to C[ct9, R] , C[ij, R) ) .  
4.1 DEFINITIONS, NOTATION AND GENERAL RESULTS FOR 
LINEAR ELLIPTIC EQUATIONS 
92 
We present another resulL concerning the Nemytskii operator .A(u). This lemma is a special case of 
Lemma 3 . 1 .7 
Lemma 4.1 . 7  
Let h E Ca[:f x R,  R] ,  and let .A{u) be  the Nemytskii operator .A(u). Then 
(i) .kE C[Lq [:f, RJ , Lq [:f, R] ]; 
(ii) .ktakes bounded sets in Lq [�, R] into bounded sets in Lq [�, R] . 
The Relationships between these Spaces 
The spaces defined above are needed to obtain the exact result on the solvability of boundary value problems 
for linear elliptic equations in spaces Wq2[�, R] .  Note that an analogous lemma to Lemma 3 . 1 .8 holds for 
elliptic equations which relates the differential properties of the boundary values of functions from the classes 
Wi[�, R] and of their derivatives in terms of the spaces W�[a�,  R] (see LADYSI-IENSKA YA [ 154, pp.43-
44]) .  
We say how smooth fuctions in a Sobolev space are, by  imbedding Sobolev spaces continuously into Holder 
spaces. This is often called the Sobolev Lemma or the Imbedding Theorem. We firstly require the following 
defintion 
Definition 3.1 .5 
Let � be an m-dimensional domain with boundary a�. We say that a� belongs to class C2+a, if for every 
XE �, there exists a neighbourhood U of x such that � n U can be represented in the form 
for some i, 1 S; i s;  m,  where h E C2+a[a�, R] . 
We now state the following imbedding theorem. For the proof of this theorem, see LA DYZI lENSKA YA [ 1 54, 
p.60] , GILBARG and TRUDINGER [ 106, p . 1 55] or ADAMS l l j .  A similar theorem is given in TEMME [282, 
p.42] for convex domains with Coo boundaries. 
Theorem 4.1.1 (Imbedding Theorem or Sobolev Lemma) 
Let � k: Rm and let a<fj be of class C2+a. 
(i) Suppose 2q > m > q. Then Wq2l�, R] is imbedded in CI+).t l�, Rj , where 0 < Il S; 1 -2/q ;  
(ii) Suppose m = q .  Then Wq2[�, Rj is imbedded in Cl+).t[�, RJ , where 0 < J1 < 1 ;  moreover suppose 
m = q = 1 ;  then Wq2[�, R] is imbedded in cI+).t [�, R] for J1 = 1 .  
4.1.3 Solvability of Linear Elliptic Equations 
We will discuss in this section a specific example of a elliptic equation that occurs in this thesis. For the 
definitions of more general linear and quasilinear second order equations of elliptic type, as well as their 
uniformly elliptic conditions, see LADYZl-lENSKA YA [ 1 54, p. 1 1 ] . 
4.1 DEFINITIONS, NOTATION AND GENERAL RESULTS FOR 
LINEAR ELLIPTIC EQUATIONS 
93 
Let aij, bi and c belong to Ca [�, R] and let c � O. Let L be a second order differential operator defmed by 
(4.1 .1)  
Definition 4.1.3 
The differential operator L defined by (4. 1 . 1 )  is said to be elliptic at a point x E �, if the coefficient matrix 
ajj{x) is positive definite, that is there exist two functions a:. and I such that 
m 
o < a:.(x)I I�1I2 �  2>ij(X)�i�j � I(x)I I� 1I2 , 
i,j=l 
for all � E Rm - (O ) . If 
a:.(x) > 0 in QT, 
then !l! is called elliptic in �. If 
for a positive number Ao then !l! is called almost strictly elliptic in �. If 
A (X) < K 
1{x) -
, 
(4. 1.2) 
for some positive number K, then !l! is called strictly uniformly elliptic in '[), that is (4. 1 .2) can be written as 
�1I� 1I2 � faij(X)C;i� j � KII C; 112 , K . . I ' . J= 
for all � E Rm, X E �. 
Let � be a bounded domain in Rn. Letp, q E Cl+a[a�, R] be nonnegative functions which do not vanish 
simultaneously and let v(x) be the unit outward normal vector field on CJ<9 (which belongs to the class C2+�. 
Consider the l inear second order elliptic boundary value problem (BV? for short): 
where, 
Lu  = h(x) in '[), } 
B u = qJ(x) on a,[), 
du -Bu = p(x)u + q(x)- for u E Cl['[) , R] . 
dv 
(4.1.3) 
(4.1 .4) 
. e. Let us now state the c lassical existence and umquenss theorem whose proof can be found in 
� 
LADYSHENSKAYA [ 154, p. l37] and GILBARG AND TRUDINGER [ 1 06, Ch 6]. 
Theorem 4.1.2 Assume that 
(i) ajj, bj, c E ca['f) , R] , c(x) � 0 and L is strictly uniformly elliptic in '[); 
(ii) p, q E CIHl[Jn, R] for p and q nonnegative functions and there exists J..Ll>O such that p 2: J..LI for 
all x E CJ<9; 
(iii) CJ<9 belongs to class c2+a; 
4.1 
(iv) h E ca[�, R] ; 
DEFINITIONS, NOTATION AND GENERAL RESULTS FOR 
LINEAR ELLIPTIC EQUATIONS 
(v) l/J E Cl+a[a�,R] . 
Then the linear elliptic BVP (4. 1 .3)-(4. 1 .4) has a unique solution u such that u E C2+a[�, R] . 
We note that u and all its first and second partial derivatives are bounded in � . 
94 
The following result provides the global a priori Schauder-type estimates for classical solutions of linear 
elliptic BVP (4. 1 .3)-(4. 1 .4). For the proof, see AGMON-DOUGLIS-NIRENllERG [3, p.668]). 
Theorem 4.1 .3 Assume that the hypotheses of Theorem 4.1 .2 hold. Then for any u E C2+a[�, R] , there 
exists a positive constant 
which is independent of u such that 
l Iu l lf+a � C(II Lul l!+ I I Bullr!a ) '  
(4. 1.5) 
(4.1.6) 
Moreover, if u is the classical solution of the linear elliptic BVP (4. 1 .3)-(4. 1 .4), then (4. 1 .6) reduces to 
l Iul lf+a � C(I I hl l!+ 1 I l/JI lr!a ) '  (4.1 .7) 
Remark 4.1.2 
If either if the conditions p � J1.1>O or c � 0 is not satisfied in Theorem 4. 1 .2, unique solvability is no longer 
assured. However, unique solvability holds under conditions p � 0, c � 0 and either p $ 0 or c $ 0 
(GILllARG and TRUDINGER [ 1 06, p. l 30l. 
Remark 4.1.3 
Analogous theorems to Theorem 4. 1 .2 hold for the linear elliptic BVP (4. 1 .3)-(4 . 1 .4) with Dirichlet 
conditions (GILBARG and TRUDINGER 1 106, p.107]) .  In this case, it is required that l/J E c2+�a�, R] 
We shall state some results for solutions in the Sobolev spaces W.i[�, R] , q > 1 analogous to the Schauder 
results in the Holder spaces C2+a[f, R] . Let us state the following uniqueness and existence theorem that 
provides us with generalised (weak) solutions of the linear elliptic BVP (4. 1 .3)-(4. 1 .4). Its proof may be 
found in LADYZHENSAYA [ l 54, p. 160j . 
Theorem 4.1.4 Assume that 
(i) 
(ii) 
(iii) 
(iv) 
(v) 
aij, bi, c E C[�, R] , c(x) � 0 and L is strictly uniformly elliptic in �; 
p, q E Cl[�, RJ for p and q nonnegative functions and there exists ).L ( >O such that p�J1.I !or all 
x E ()�; 
� belongs to class C2+a; 
h E Lq[f, R] for q > 1 ;  
1 -l/J E C  [� , R] . 
Then the linear elliptic BVP (4. 1 .3)-(4. 1 .4) has a unique solution u such that u E Wq2[f, RJ . 
4.1 DEFINITIONS, NOTATION AND GENERAL RESULTS FOR 
LINEAR ELLIPTIC EQUATIONS 
95 
The following result gives the global a priori Agmon-Douglis-Nirenberg estimates (or Lq estimates) 
for generalised solutions of the linear elliptic B VP (4. 1 .3)-(4. 1 .4). The proof may be found in AGMON­
DOUGLIS-NIRENBERG [3, p.704] and LADYZHENSAYA [ 1 54, p. 162] 
Theorem 4.1.5 Assume that the hypotheses of Theorem 4.1 .4 hold. Thenfor any u E Wq2 [�, R] , there is 
a constant 
C = C(m, K, q, CY9, the modulus of continuity of au and norms bj and c), (4.1.8) 
which is independent of u such that 
(4.1.9) 
Moreover, ifu is a generalized solution belonging to Wi[�, R] , then 
(4.1.10) 
Remark 4.1.4 
Note that the assumptions (i), (ii), :(iii) and (v) of Theorem 4. 1 .4 are satisfied by the assumptions (i), (ii) , 
(iii) and (v) of Theorem 4. 1 .2. The uniqueness, existence and Agmon-Douglis-Nirenberg estimates of 
generalised (weak) solutions of the linear elliptic BVP (4. 1 .3)-(4. 1 .4) are orten given with the assumptions 
of Theorem 4. 1 .4 and h E Lq[ifj , Rj for q > 1 (TEMME [282, p.G5 ]). 
Remark 4.1.5 
Analogous theorems to Theorem 4. 1 .4 and Theorem 4. 1 .5 hold for the linear elliptic B VP (4. 1 .3)-(4 . 1 .4) 
with Dirichlct conditions (LADYZIIENSAYA [ 1 54, p. 149j , GILBARG and TRUDINGER [ 1 06 ,  p.24 1 ]) .  
We shall now look at the maximum principle for elliptic equations 
4.1.4 The Maximum Principle for Elliptic Equations 
Throughout this thesis, we will use various forms of the maximum principle for the elliptic operator to obtain 
information about the solutions of our equations. A simple form of the maximum principle that we will find 
useful is the following 
Lemma 4. 1.8 (Maximum Principle) 
Let II' E  C2 [�, R] be such that -LII'5, 0 in '9 and B V/ 5,  0 on if9. Then 11'5,  () on � .  
Other forms of the maximum principle for the elliptic operator are given by PROTTER and WEINBERGER 
[234] and SPERB [27 1 ] .  
4.1 .5 Relationships between Solutions o f  Linear Elliptic and Linear Parabol ic Equations 
In order to show the relationships between unsteady state solutions and steady stale solutions, we shall 
require some theorems concerning the asymptotic behaviour of linear parabolic equations as t � 00. The 
following theorem is proved by FRIEDMAN [94, Ch. 61  and CARTER [50J. The nOLation used is  from section 
3 . 1 .  
4.1 DEFINITIONS, NOTATION AND GENERAL RESULTS FOR 
LINEAR ELLIPTIC EQUATIONS 
Theorem 4.1.6 Suppose that 
96 
(i) aij , bi, c in !l!u are uniformly continuous and bounded in QT and !l! is strictly uniformly parabolic; 
(ii) p and q in fiJu are nonnegative functions which are uniformly continuous and bounded in �. and 
for some J.l.1 > 0, pet, x) � J.l.1 for all (t, x) E rT; 
(iii) u(t, x) satisfies the dif erential equation 
!l!u = h(t, x) in QT, 
together with the boundary condition 
fiJu = ¢J(t, x) on rT, 
where h is continuous on QT and ¢J is continuous on rT. 
If lim h(t, x) = ° uniformly on QT ' lim ¢J(t , x) = ° uniformly on 71· and lim c(t, x) $ ° uniformly on 
1-+00 t�oo 1-+00 
QT' then lim u(t, x) = ° uniformly on QT ' 
t�oo 
The following theorem shows the relationships concerning the asymptotic behaviour of linear parabolic 
equations as t � 00 and their corresponding linear elliptic equations. 
Theorem 4.1 . 7  Suppose that 
(i) aij , hi, c in !l!u are uniformly continuous and bounded in Qr and !l! is strictly uniformly parabolic; 
(ii) p and q in fiJu are nonnegative functions which are uniformly continuous and bounded in rT and 
for some J.l.1 > 0, pet, x) � J.l.l for all (t, x) E 71· ; 
(iii) aU ' bi , C E ca lf, R] in Lu and L is strictly uniformly elliptic; 
(iv) p and q in Bu are nonnegative functions in CI+lX [J�, RJ and for some {L\ > 0, p(x) � {Ll for all 
X E  J�; 
(iv) au (t, x) � aij (X) , bij (t, x)� bi (x) ,  c(t, x) � c(x) as t � 00, uniformly in Qr ; 
pet, x) � p(x) , and q(t, x) � q(x) as t � 00 ,  uniformly in �.; 
h(t ,  x) � hex) as t � 00, uniformly in (IT ; 
¢J(t, x) � �(x) as t � 00, uniformly in �. ; 
If u(t, x) satisfies the differential equation 
!l!u = hU, x) in QT, 
together with the boundary condition 
fiJu = ¢J(t, x) on rT, 
where h is continuous on Qr and ¢J is continuous on rT and if u(x) is the unique solution of the 
boundary value problem 
Lu= hex) in �, 
Bu = �(x) on �, 
where h E Ca[�, R] and � E Cl+a[�, RJ , then 
u(t, x) � u(x) as t � 00 uniformly on (1 . .  
Proof 
4.2 IMBEDDING RESULTS 
Put W(/ , x) = u(t, x)-u(x), for (I, X)E 0. Then: 
fl!w = 9!u- 9!u+Lu-Lu 
= h(t, x)-h(x)+ (L- 9!)u in QT, 
3Jw = [iJu-[iJu+Bu-Bu 
�I/J(/, x)- �(x)+(B-[iJ)u on rT. 
97 
From the assumptions of this theorem, the boundedness of u(x) and its first and second partial derivatives on 
� (see Theorem 3 . 1 .2), we may apply Theorem 4. 1 .6 and conclude that lim W(/ , x) = O, uniformly on Qr , 
I�� 
implying that u(t, x) � u(x) as t � 00 uniformly on QT ' O  
Remark 4.1.6 
Analogous theorems to Theorem 4. 1 .6 hold for the linear BVP (4. 1 .3)-(4. 1 .4) with Dirichlet and Neumann 
type boundary conditions (FRIEDMAN [94, Ch.6]). 
In section 4.2, we shall present some imbedding results for the system Sn ' Bn . 
4.2 Imbedding Results 
For the purposes of uniqueness, stability and existence theorems of the steady state system, we may consider 
the quasimonotone system Sn ' En where Ii and Fj are monotone nondecreasing in Cj and Cj respectively for 
j � i .  This is not a restriction on these theorems of this chapter since if this monotone property is not 
satisfied, then the system Sn ' Bn with general functions Ii and Fj can be imbedded in a system S2n ' E2n of 
the same form where !; (x, () is replaced by l (x, f.k ' c, ) for t.he first 11(/) dependent variables Cj and by 
f.; (x, fk ' e, ) for the next ll(l) dependent variables f..j. Also, F; (z, Cj ) is replaced by �(z, �k ' <'7;) for the 
first n(1) dependent variables C; and by f.j (z, fk ' <'7;) for the next n(J) dependent variables Ci . It can be 
shown that solutions of this  new system may generate solutions of the original system and therefore 
uniqueness, stability and existence can be implied in the original system. 
We consider the new system S2n ' B2n of up to twice the order satisfied by fj ' Cj ' fj and C; in the 
following equations: 
aC · ae -=1.=0  _I = 0  on aD xA-an ' an l '  
aC ae -D -=1. = H (C - c · ) 0 -' = H· (C, - e )  on aQ2xA-
� an I -I -I ' I an I I ' 
4.2 IMBEDDING RESULTS 
where L, h Ei and F;are defined in (3.2.9)-(3.2. 1 2) .  
98 
Note that the functions (h ,  -[) and (F;, -fJ obey a mixed quasimonotone property (mqmp) in 
the sense of LADDE et al. [ 153 ,  p. 107] , Le, the functions }; and -f ,  are monotone nondecreasing in CI and -I 
monotone nonincreasing in £k for all i � k, I and the functions F; and -L are monotone nondecreasing in 
CI and monotone nonincreasing in �k for all i � k, I . 
If we therefore slightly modify the system S2n ,  B2n as suggested by McNABB [ 1 86] ,  by introducing 
new variables Vi ::= Ci , V n(/)+i ::= -f.i for i ::=  1, .. , n (I) and Vi ::=  � ,  Vn(J)+j -C for i ::=  1, .. , n(J) and if we set 
ft ::=  Ij , in*(I)+j ::=  -L for i ::=  1, .. , n (l) and F/,' ::= F; ,  Fn(J)+j ::= -f..i for i ::=  1 , .. , n (J) then we obtain a new 
system Sin '  Bin for which f/ and F/ are nondecreasing functions of Vj and Vj, respectively , for all j � i. It 
can be shown by Lemma 3 .2.2, that these new functions have the Lipschitz properties that were imposed on 
the original functions!; and Fj• 
Every solution (Cj ,  Cj )  of Sn , Bn generates a solution Vj ::= Cj ,  vn(l)+i ::= -fj and Vj = Cj, Vn(J)+j ::= -�j 
of the new system Sin '  Bin with the special property that for i � n(l), Vj + vn(/ )+j = 0 in Q x A and for i � 
n(J), Vi + Vn(J)+j = 0  in A .  Conversely, any solution (Vi, Vj) of the new system Sin' Bin with the special 
property that for j � n(J) we have Vi,l + Vn(J)+j,1 == 0 on JAi l  may in some cases be shown by the following 
theorem to give rise to a solution of the system '�n '  Bn . 
Theorem 4.2.1 .  
The general system Sn, Bn for whichJi and Fi are Lipschitz continuous in Cj and Cj respectively, may be 
imbedded in a system Sin '  Bin of twice the order which is coupled by monotone functions ft and Fi· of 
the new dependent variables Vj and Vi. Moreover, all the solutions (ci ' Ci ) of the system Sn , Bn are 
solutions of the new system, where 
Vi = Ci , Vn(/)+i ::= -Ci for i ::= 1, . .  ,n(J) 
and 
Vi = Ci, Vn(J)+i ::= -Ci for i = I, . .  ,n(J) 
and all the solutions (Vi, Vj) of Sin '  Bin for which 
Jw _I ::= O on JQl xA; dn 
Jw D -I -H(W - w ) ::= O  on Jf22 x A '  I an I I I , 
(4.2.1) 
(4.2.2) 
(4.2.3) 
(4.2.4) 
(4.2.5) 
(4.2.6) 
(4.2.7) 
(4.2.8) 
4.2 IMBEDDING RESULTS 99 
with Wj = Vj + Vn(l)+j for i = 1 ,  . .  ,n(l) and "'i = Vi + Vn(J)+j for i = 1, . .  ,n(J), generate solutions (Cj , Cj ) of 
the system Sn, Bn , provided that solutions (Vj , Vi) of Sin '  Bin are unique. 
Proof 
We first note that if we set fj == Cj == Cj and C == � == Cj where (Cj , Cj ) is a solution of Sn, Bn , then we have 
a solution of the new system Sin' B2n , so that the solution set of this new system contains all of the 
solutions of the original system Sn , Bn . In this system, we make the variable change 
Vj =Cj ,  vn(l)+j = -f.j , for i= l , . . ,n(I) , (4.2.9) 
Vi =Cj , Vn(J)+j = -L, for i= l , .. ,n(J), (4.2.10) 
so that the coupling functions ft and Pi·are nondecreasing functions of all the new dependent variables Vj 
and Vj' respectively, for all j � i. Denote this system by S�n ' B�n ' 
The solutions of Sn ' Bn generate solutions in Sin ' Bin for which Wj = Vj + vn(l)+j = 0 in .a x A  and 
Wj = Vi + Vn(J)+j = 0 in A .  
Suppose we  have a solution (Vj ,  Vi)  of  S�n ' Bin for which (Wj , Wi) = (Vj + vn( l)+j , Vi + Vn(J)+j ) 
satisfies (4.2.3)-(4.2.8). We then obtain the following system of equations, S; for (Wj , Wi):  
-DjV;Wj = /;*(x, Vj ) + In*(/)+j (x, vk )  
= �(x, fj ' ck ) - L (x, fj ' Ck ) 
= �(x, Cj - Wj '  ck ) - L(x, Cj - Wj '  Ck )  in .QxA, 
= F*(z V. ) + F*(J) · (z Vk ) - H. J  (W - w ) , ' J n +" ' rJ!J2 " 
= �(z, f;..j ' Ck ) -!!.j (Z, �i '  Ck ) - lljLn
2 
("'i - Wj ) 
= F;(z, Cj - Wj ,  Ck ) -!!.JZ, Cj - Wj '  Ck ) - HjLn2 (Wj - Wj ) in A, 
aw 
D -' = !-f . (W - w ) on a.a2 x A  , an ' "  . 
In addition, (Wj , W;) satisfies the boundary conditions B; given by Bn with Wj,! = Vi,l + Vn(J)+j,1 = O .  
(4.2.1 1) 
(4.2.12) 
(4.2.13) 
But � (x, Cj-Wj ' Ck )-L(x, Ci-Wj ' ck )  and �(z, Cj-Wi ,  Ck )-!!.;(Z, Cj-Wj ' Cd-Hjfa ("'i-Wj ) 
vanish when Wj == wi == 0 for all ), and since (Wj, W;) == 0 is a solution of this boundary value pr�lem and 
by uniqueness it is the only solution, we conclude that Wj == 0 in D x A and Wj == 0 in A. The conclusion of 
our theorem must fol low.O 
An immediate consequence of these imbedding results is that existence, uniqueness and stabil i ty 
results for the system S�n ' B�n imply existence, uniqueness and stabil i ty for the corresponding solution of 
Sn ' Bn . Of course, solutions of Sn ' Bn may be stable in Sn ' Bn ' but unstable in the larger setting Sin ' Bin . 
This of course can imply that there may be uniqueness of solutions in Sn, Bn and nonuniqueness of solutions 
in the larger setting Sin '  Bin and if there are multiple solutions in the larger selling Sin '  B�n ' this implies that 
there can be no greater number of solutions in the original system Sn ' En . The implications of this is that 
nonexistence of solutions in the system Sin ' Bin must imply nonexistence of solutions in Sn '  Bn . As with 
the time dependent problem, UlCre arc many OIlIer implications of these imbedding results (sec section 3 .7). 
MASSEY UNIVERSITY 
LIBRARY 
4.3 EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM 1 00 
Remark 4.2.1 
Note that the functions 
and 
F(z cJ, - w  Ck ) - F . (z CJ· - w. Ck ) - /-I ·f (W - w) I '  J '  -I ' . J ' 1 (J[).2 1 1 
in the right hand sides of (4.2 . l 1 )  and (4.2. 12) are monotone nondecreasing in Wj and Wj, respectively. 
In section 4.3 we shall study the existence of solutions to the steady state system Sn ' En ' 
4.3 Existence of Solutions to the Steady State Problem 
In this section we shall be looking for steady state solutions of Sn ' Bn which are solutions of the time 
independent problem Sn ' En . We demonstrate in this section that solutions of problem Sn ' En specified by 
Cj.l exist. By a solution, we shall understand a classical solution (Cj, CD of Sn ' Bn , where 
(i) For components i E I where Dj, Hj > 0, Cj are continuous in .Q xA , have continuous first order Xj 
derivatives in .Q xA and continuous second order Xj derivatives in .QxA . In this case, we shall look 
for classical solutions of the form Cj (x, z) E C2•O [.Q x A ,  Rn( / ) ] .  
(ii) For components i E I where Dj = Hj = 0, Cj arc continuous in !) x A .  In this case, we shall look for 
classical solutions of the form Cj (x, z) E Co.o[.Q x A ,  R n( l) ] .  
(iii) For components i E J where !1Jj > 0, C j arc continuous in X, have continuous first order Zj 
derivatives in if and continuous second order Zj derivatives in A .  In this case we shall look for 
classical solutions of the form Cj (z) E C2[A ,  Rn(J» ) . 
(iv) For components i E J where 9)j = 0, U · 'lCj $ 0, Cj are continuous in if and have continuous first 
order Zj derivatives in A .  In this case we shall look for classical solutions of the form 
Cj(Z) E C1 [A, Rn( J) ] .  
(v) For components i E J where 9)j = 0, U ·  'lCj == 0, Cj are continuous in X. In this case we shall look 
for classical solutions of the form Cj (z) E CO rA, Rn( J) ] . 
Comparison theorems arc used in this section in conjunction with theorems on a priori estimates and 
existence of linear elliptic equations to derive estimates of the system Sn ' Bn and to prove the existence of 
solutions to this system. There may be some cases where Dj or q)j may be zero and these cases are treated by 
using standard results. 
As for the system Sn ' Bn , we shall need some additional continuity properties to establish existence of 
the corresponding l inear system and make the fol lowing assumptions on !; (x, c) and F;(z, Cj) .  
(H2) (i) hex, C) E CCt[.Q x Rn( / )  , Rn( / » ) .  i .e . ,  hex, c) i s  HOlder continuous in x with exponent a, for 
each fixed value of G'j. 
4.3 EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM 1 01 
(ii) F; (z, Cj) E Ca [A X Rn(J) , Rn(J) ] ,  i .e., F; (z, Cj) is HOlder continuous in z with exponent a, for 
each fixed value of Cj. 
Note that these assumptions are satisfied by assumption (H2) iffi and Fi are independent of I. By Lemma 
4. 1 .3 ,  we see that the exponent a in both cases may be assumed to be identical. 
4.3.1 The Monotone System Sn ' fin 
We may assume at the outset that the system Sn ' En is a monotone system, in the sense that h(X, Cj) is 
monotone nondecreasing in Ci and F; (z, Cj) is monotone nondecreasing in Ci for all i .  This is not a 
restriction on the theorems of this section since if this monotone property is not satisfied we may make the 
following substitution to obtain a system of the same type but with new functions that are monotone 
nondecreasing in Cj and Ci . We first observe that if (ci' Cj) is a solution of Sn ' En and XI , Zl and U I (Z) are 
chosen without loss of generality to be the first components of x, z and u(z) respectively, then (wi ' �) 
dermed to be  where 
and 
satisfies the following system of equations: 
-D. (V2w - 2K aWi ) = K2Dw. + eKX1 " (x e-KX1 w .) in ,axA , x ,  ') , , J j ' ] ' oXl 
�e-Kxlwi = 0 on (}.alxA, an 
D.�e-Kxl w. = IJ . (e-Kz1 W. _ e-KX1 w) on a!J2xA I an I I , I ' 
-Kz1 W � (} -Kz1W -Kz1C "lA  vie . + :v. -e . = vie . I on a;1 1 I I J I " ' ani 
a -Kzl W - 0 "l A - 2 3 �e i - on ana, a , . ana 
(4.3.1) 
(4.3.2) 
(4.3.3) 
(4.3.4) 
(4.3.5) 
(4.3.6) 
(4.3.7) 
(4.3.8) 
The system is similar to the original system Sn ' Bn in that uniform ellipticity conditions are still preserved in 
(4.3.3) and (4.3 .6). The nonlinear coupling function h(x, Cj ) for Cj components in these equations is 
replaced by 
(4.3.9) 
the nonlinear coupling function F;(z, Cj) for Cj components in these equations is replaced by 
(4.3.10) 
and the functional term 
(4.3.11)  
4.3 EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM 102 
is replaced by 
(4.3.12) 
The functions K2Djwj +eKX1 hex, e-KX1wj) and !ljK2Wj +uI (z)KW; +eKZ1 F;(z, e-KZ1 W) satisfy a monotone 
property given by the following lemma: 
Lemma 4.3.1 
Our assumptions (H I ) of Lipschitz continuity properties for the functions /; and Fj with respect to Cj and 
Cj imply that the functions K2Djwj + eKX1 h(x, e-KX1 wj ) and !ljK2W; + ul (z)KWj + eKZ1 F;(z, e-KZ1 W) 
with Cj = e-KX1 wj and Cj = e-KZ1W;,  are monotone nondecreasing in Wj and Wj, respectively. 
Proof 
Assume that Wj $ wj so that Cj $ cj . 
From (HI), we see that 
and therefore, 
[K2DjWj + eKX1 h ex, e-KX1 wj)] - [K2Djw; + eKX1 h ex, e-KX1 wj )] 
= K2DjeKxl (Cj - c;) + eKX1 [hex, Cj ) - h ex, cj )J 
< _eKx1 [K2 D· (c� - c·) + K(c· - C ·)] - I I I I J J 
$ 0, 
if K is chosen to be large enough. This shows that 
so that this new coupling function K2Djwj +eKx1 hex, e-Kx1 Wj) is monotone nondecreasing in Wi. A similar 
argument holds if we have to show that �K2W; +uI (z)KW; +eKz1 F;(z, e-KZ1 Wj) is  monotone 
nondecreasing in Wi. This may be shown irrespective of the sign of UI(Z).O 
It  can also be shown that these new functions also satisfy the same Lipschitz and Holder continuity 
properties as our original functions. 
Lemma 4.3.2 
Our assumptions (H I ) of Lipschitz continuity properties for the functions h and Fj with respect to the 
v a r i ab l e s  Cj and Cj imply similar Lipschitz continuity properties for the funct ions 
K2D·w- +eKx1 " (x e-Kx1 w · ) and $.K2W. +uI (z)KW +eKz1 F(z e-Kz1 W·) with respect to the variables I I Jj , J I I I I '  J 
Wj and Wj,for Cj = e -Kxl Wi and Cj = e -Kzl W; . 
Proof 
We need to only show that 
where, 
4.3 EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM 103 
< K2D lw _w* l+eKxl k- l e-Kxl lw '  -w* 1  - I I I I J J 
< K2D' lw ' -w� l+k lw ' -w� 1  - I I I I J J 
� k sup IWj -wj l ,  
J 
k = max (K2Dj , kj ) .  j 
and the first part of the proof follows. The rest of the proof follows similarly.O 
Lemma 4.3.2 
Our assumptions (H t> of Lipschitz continuity properties for the functions /;, with respect to the 
variables Cj and assumptions (H2) of Holder continuity properties for the functions /;' with respect to x 
with cJfixed imply similar Holder continuity properties for the functions K2Djwj + eKX1 /; (x, e-KX1wj ) 
with respect to x with Wj fixed, where Cj = e-KX1 Wj .  Similarly, our assumptions (H I ) of Lipschitz 
continuity properties for the functions Fj, with respect to the variables Cj and assumptions (/12) of 
Holder continuity properties for the functions Fj, with respect to z with CJfixed imply similar Holder 
continuity properties for the functions �K2lti +UI (z)Klti +eKZ1 F; (z, e-KZ1 Wj) with respect to z with Wj 
fixed, where Cj = e-KZ1Wj .  
Proof 
We shall only show that fi (x, e-KX1wj) is Holder continuous in x. The rest of the proof is simi lar and 
follows from Lemma 4. 1 . 1  and Lemma 4. 1 .2. 
where 
l fi (x, e-KX1wj ) _ fi(x: e-Kx: wj)1 
= I/; (x, e-KX1 wj ) _  fi(x; e-KX1wj ) + fi (x: e-KX1wj) - /;(x: e-Kx: wj)1 
� l fi (x, e-KX1 wj ) _ fi (x: e-KX1wj )I+lfi (x: e-KX1wj ) _ fi(x: e-Kx: wj)1 
� kx (fi)l lx - x· lIlX + kj le-Kx1wj _ e-Kxj wj l 
� kx (fi)l I x  - x* lllX + kj lwj l le-Kxl 
_ e-Kxj I 
� kx (fi)l I x  - X* lIlX +kj lwj lKl ixl - x; I I
I-lX II xl - X; l IlX 
::; kx Cfi (x, e-KX1wj»l Ix - x· l IlX, 
We may assume that the substitution (4.3 . 1 )-(4.3 .2) has been made and that fi(x, Cj) is monotone 
nondecreasing in Cj and F; (z, Cj) is monotone nondecreasing i� Cj� If, on the other hand the monotone 
property is not satisfied by all the other variables, then the syslem Sn ' Bn with general functionsJi and Fj can 
. . -
be imbedded in a system S2n '  B2n of the same form where fi (x, C j ) is replaced by /; (x, fk ' (;/ ) for the firSl 
n(I) dependent variables Cj and by t (x, fk ' (;/ ) for the next n(I) dependent variables �j . Also, F;(z, Cj) is 
replaced by F;(z, {;,k ' �) for the first n(J) dependent variables C; and by !!.j (z, {;,k ' �) for the next n(J) 
dependent variables �j . The exislence resulLs obtained for lhis new syslem of lwice the order salisfied by 
-----------
4.3 EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM 104 
(fi ' �i)  and (Ci ' G;) then implies a solution of our original system Sn. Bn by the imbedding results of 
section 4.2 if uniqueness is guaranteed in the monotone system S2n . B2n . 
It has been shown in section 3.2 that the functions L. l .  Ei and F; satisfy the same Lipschitz 
continuity properties as our original functionsJi and Fi in the system Sn . Bn . It can also be shown that these 
new functions satisfy the same Holder continuity properties as our original functions and the fol lowing 
lemma is a spccial case of Lemma 3.6.4. 
Lemma 4.3.3 
Our assumptions (H2) of Holder continuity properties for the functions fj. with respect to x with Cj 
fixed imply similar Holder continuity properties for L and l .  with respect 10 x with fie and (;, fixed 
and so there are constants kx such that 
l�i (X. fie ' �)-�i(:: f" .
_
C, )1 � kx�[)IIX-:*�ICX .} 
I.t;(x. fie ' cl ) - .t;(x. fie ' c,)1 � kx (.t;)lIx - x  II . 
(4.3.13) 
Similarly. our assumptions (H2J of Holder continuity properties for the functions Fi with respect to Z 
with Cj/ixed. imply similar Holder continuity properties for Ei and F; with respect 10 Z with �" and 
� fixed and so there are constants Kz• such that 
IEj (z. k" . � )-Ej (z : k" . � )I � Kz (E; )Il z - z* l IlX .  } 
1F;(t. z. �k ' Cr)- F;(t. z: �k ' Cr )1 � Kz (F;)lI z - z* lIcx . 
(4.3.14) 
For the purposes of our ex istence proof. we firstly look at the monotone system '�n . Bn where we assume 
our coupling functionsJi and Fj are monotone nondecreasing in Cj and Cj. respectively for all }. 
4.3.1 Upper and Lower Solutions and Monotone Iteration 
We shall now introduce the concepts of upper and lower solutions relative to the monotone system Sn . Bn . 
Definition 4.3.1 .  
Assume that 
(i) For components i E I. where Di > O. fj and Cj are continuous functions in .Q x A with continuous 
first order Xj derivatives in .Q x A and continuous second order Xj derivatives in .Q x A ; 
(ii) For components i E I. where Dj = II i = O. fi and C, arc continuous functions in £2 x A ; 
(iii) For components i E 1. where �j > O. C and Cj are continuous functions in if. with continuous first 
order Zj derivatives in if and continuous second order Zj derivatives in A; 
(iv) For components i E 1. where �i = O. u ·  '\lCi $0.  �i and Ci are continuous functions in A . with 
continuous first order Zj derivatives in A; 
(v) For components i E 1. where �j = O. u ·  '\lCj == O. �j and Cj are continuous functions in A ; 
4.3 EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM 105 
The ordered pair of functions (fi ' �i ) and (ci ' Ci )  with fi � Ci on n x A and �i � Ci on If arc said to be 
lower and upper solutions of Sn ' Bn respectively, if Ihey satisfy: 
and 
-DiV; fi � h(x, c;.;) in ,axA, 
dC· -=:!.. � 0 on d,alXA, dn 
dc · D -=:!.. < H (C·- c· )  on d,a2XA I an - ' -' -I ' 
- !llV2 c. + u ·  VC. + H.J (C.-c - ) � F(  z, (;:J' )  in A, I _I  _ _I I aDz _I -I I _ 
-D;V; Ci � h (x, Cj ) in .(lxA, 
de· 
d� � 0 on d,alxA, 
dc· - _  D·-' > I-f . (C.- c· · ) on d,a2XA I dn - I I I , 
_ !�:v2 C· + U ·  V C+ l-f . J (E - c » F ( z  C )  in A I I I I aD2 I I - I ' J ' 
- dE VI C·+ 9). __ , > VIC. I on dAI I I dnl - I. ' 
dC· 
__ , � o  on dAa, a =  2, 3 ,  dna 
respectively. 
We nOLe from the counterexample at !he end of section 3 . 1 ,  !hat comparison !heorems analogous to Theorems 
3 .2 . 1 1  and 3 .2. 1 2  do not hold in general in the case of the corresponding steady state or time independelll 
problem Sn ' Bn . Hence, if there exist (fi ' �i ) and (ci ' C;) which are lower and upper solutions of the steady 
stale problem Sn ' Bn and (Ci, Ci) is a solution of Sn ' Bn , then in contrast to the unsteady state problem Sn, Bn, 
we cannot assert that fi � ci � Ci and �i � Ci � Ci . 
However, the method of monotone i teration is still applicable and shows !he existence of at least one 
solution (Ci, Ci) of Sn ' Bn lying between (fi ' �i ) and (ci ' Ci ) · 
Lower and upper solutions may not always exist for elliptic equations. Therefore, as a result of this, 
certain unstable solutions cannot be obtained by monotone i teration (PARTER [230] , KELLER and COHEN 
[ 1 39 ] ,  AMANN [9]) .  However, it must be noted that as with parabolic equations (PAO [222]), there are 
geometric conditions which the nonlinear reaction functions f; and Fi may satisfy which guarantee the 
existence of ei ther lower or upper solutions for elliptic equations (AMANN [9]) .  These lower and upper 
solutions may not necessarily exist simultaneously. 
As for the system Sn, Bn, it is important to note that the lower and upper solutions provide lower and 
upper bounds for solutions of ,�'" Bn which can be improved by monotone iteralive procedures. 
4.3 EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM 106 
In order to establish an existence theorem for Sn ' Bn in terms of upper and lower solutions, we define 
a transformation !!i, by 
and consider the sequences {(c�k) , Cfk» } where c�k) is obtained from the l inear system 
and Cj(k) is obtained from the linear system 
:v-(k) C(k) rill Q\.,j - C a A VI ' + ;v.-- VI ' I on .Ill ,  , ' ani ' " 
ac(k) -a ' = 0 on aAa, a=2, 3 ,  na 
' th < (k-I) < - n A .  d C < C(k-I) < C- A f' k - 1 WI £j _ Cj _ Cj on .U X .ll an _j _ j _ j On .ll or , . . . .  
(4.6.15) 
(4.3.16) 
(4.3.17) 
(4.3.18) 
(4.3.19) 
(4.3.20) 
(4.3.21) 
For each k, the system (4.3 . 16) consists of n(J) l inear, completely uncoupled boundary value problems with 
boundary conditions g iven by (4.3 . 1 7)-(4.3 . 1 8) and this system is uncoupled from the system (4.3. 1 9) 
which also consists of n(J) l inear, completely uncoupled boundary valuc problems with boundary conditions 
g iven by (4.3 .20)-(4.3 .21) .  
Since cfk) (x, z) is not  differentiated with respect to z in (4.3 . 1 6) ,  A may be considered to be a 
parameter space in (4.3 . 1 6)-(4.3. 18). For functions c[k) (x, z) where Dj = Hj = 0, n may also be considered 
to be a paramctcr spacc and for functions C[k)(z) , where 9)j = 0, U ·  vcfk-1) == 0, wc may similarly u'cat A as 
a parameter space. The cxistence and uniqucness of sequences {(c?) , Cj(k» } may therefore fol low from 
solving standard scalar systems of l inear ell iptic equations (LADYSHENKA YA [ 154] or GILBARG and 
TRUDINGER [ 1 06]) which may or may not depend on parameters, systems of first order partial differential 
equations which may depend on parameters (LAKSI IMIKANTHAM et al. I I GO]) and systcms of algebraic 
equations in many variables. 
The nonlinear algebraic equations h (x, Cj ) = 0 obtained when Dj = Hj = ° may be expressed as 
Cj = fi (x, C j ) + Cj in order to perfonn functional iterations to find Cj in terms of Cj. We shall develop a general 
theory for a broad class of monotone iterations involving such algebraic equations. This class of i terations 
includes Newton's method as wcll as a family of mCI,hods, which arc Ncwton-Gauss-Seidel processes 
(ORTEGA and RI IElNBOLT 1 206, 207 J ,  LADDE et al. 1 1 53 ,  p. 36 1). It is the Lipschitz propcrty that may be 
used instead of differentiability in many other iterative methods. Note that fi (x, Cj ) + Cj satisfy the Lipschitz 
and HOlder continuity properties that are assumed on our original fi (x, Cj) as well as the monotonicity in Cj 
(see Lemma 4. 1 . 1 ) . We may therefore rewrite (4.3 . 16) as Cfk)= h(X, C;k-l» + cfk- 1) when D, = Hj = O .  The 
existence and uniqueness of sequences ( c?) } follows from the uniquely dcfined solutions to algebraic 
equations. 
4.3 EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM 107 
The rest of these theorems will require Holder continuity properties on the functions !; (x, C)k-I» and 
F (z dk-I» +H·f C�k-I) which are satisfied if either C(k-I) E Cl+a,a [{), x A Rn(l) ] C(k-I) E Ca,a I , J I a I J '  ' J  
[{), x A Rn(l) ] c(k�h E CI+a [A Rn(J)] or C(k- I) E Ca [A Rn(J) ] , , J ' J , . 
These theorems will also require Holder continuity properLies in the boundary conditions and so we 
make the following assumptions on Ci,l ' We will assume that dQ and dA belong to class C2+a. 
(H' ) C. E CI+a [A Rn(J) ] 3 1,1 ' • 
The velocity distribution vector function u(z) is also required to satisfy the following Holder continuity 
property 
where UI(Z) is chosen without loss of generality to be the first component that is nonzero. For components 
i E J, where 9Ji = 0, u·VCi ¥'O, we shall also need the following additional assumptions 
(Hs) Assume that A == A I X A n-I ,  where A I E R .  
(i) For each (ZIO , zo) E A I X A n-l , there exists a unique solution Z(Z I ,  Z IO, zo) of 
dz �z) -
- == -- ,  Z(Z IO) = zo, on A I ,  dZI ul (z) 
where ZI correspondes to the nonzero component U I (Z); 
(i i) Z(Z I o  Z IO, zo) is continuously differentiable with respect to (Z IO, zo); 
(iii) The relationship 
holds. 
(H6) Assume that A I is the interval [a, b] 
(i) For each Zo E A n-I and YiO E Rn(J), there exists a unique solution Yi(Z \ ,  a, YiO; zo) of 
F ( ( ) C(k-I» H sYY. H J (k-I) (  ( » } dJi == I Z) , Z z) , zlO , zo '  J - I I + I aaz ci zl o X, Z Zi o  zlO , Zo , dZ) UI (z) 
Yi(a ) = YIO , 
on A I , where Z(Z I , Z I O, zo) is the unique solution of (4.3.22); 
(ii) Yi(Z I , a, YiO; zo) is  cont.inuously differentiable with respect to (YiO, zo). 
(4.3.22) 
(4.3.23) 
(4.3.24) 
4.3 EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM 108 
Note that assumptions (H3)-(H6) will  hold in either our original system 5n , Bn or the monotone system 
S2,, ' B2n and (Hf;) can be shown to hold in the monotone system 52" ,  B2" if it holds in our original system 
Sn ' Bn . 
Lemma 4.3.5 
Consider the BVP (4.3.16)-(4.3.21 )  and suppose that the assumptions (H I ), (H1 ) -(/-I:") hold. Let there ex i s t  (fj , !,;;j )  a n d  (Cj '  Cj ) which are lower and upper solutions respectively of Sn , Bn w i t  h (k I) - (k I) - -£j $. Cj - $. Cj on .Q x  A and c;,j $. Cj - $. Cj on A .  
Assume that 
(i) 
(ii) 
(iii) 
(iv) 
(v) 
For components } E I, where Dj, /-Ij > 0, C)k-I) E Cl+CX,cx[.Q x A, R"(I » ) ;  
For components j' E 1 where D ·  = fl · = 0 c(.*-I) E CCX,CX [!2 x A R"( I » ) · , J J ' J " 
For components } E 1, where 9)j > 0, Clk-I) E CI+cx [A ,  R"(J) ] ;  
For components j' E 1 w he r e  9) .  = 0 u ·  VC(k-I) ,.,O  C(k-I) E CCX [ A  R"(J) ]  and assumptions , J '  J T' ,  J ' 
(fls )-(H;' )  hold; 
For components j' E l  where 9) , = 0  u · VC(k-l)= O  C(.*-l) E Ccx[A Rn(J) ] , J ' J ' J , .  
Then the BV? (4.3 .16)-(4.3.2 1)  possesses a unique solution (cfk) , C? » ,  where 
(1) 
(II) 
(III) 
(IV) 
(V) 
For components i E 1 where D ·  H ·  > 0 c(k) E C2+cx.cx [.Q x A R"( I » ) · , h i ' , " 
For components i E 1 where 0. = /-I. = 0 c(k) E CI+cx,cx [.Q x A R,,( I » ) ·  , ' "  , " 
For components i E 1, where 9)j > 0, cfk) E C2+cx r A ,  R,,(J) 1 ;  
For componcflIs i E J where �' = 0 U ·  VC<k) $0 C<k) E c 1 + lX l A  R,,(J ) J ' , ' t I " " 
For components i E 1 where 9). = 0 u · VC<k) = 0 C<k) E Ca[A Rn( J ) ]  , " I ' I  ' • 
Furthermore, in all cases c?) and Cj(k ) satisfy the inequalities fj $. Cfk) $. Cj in .Q x J\ and c;,j $. C?) $ Cj 
in J\ .  
Proof 
We first consider the case when Dj, 9)j > 0 for all }. It is obvious that for equations (4.3 . 16)-(4.3. 1 8), all 
conditions of Theorem 4. 1 .2 except for those listed in assumption (iv) are satisfied. Note that the function 
Cjk-I) in the boundary condition of cfk) arc functions of z but arc independent of x. The function Cj*-I) 
therefore satisfies the Holder continuity property required by assumption (v) of Theorem 4. l .2 and z may be 
treated as a parameter. It is therefore enough to show that fi(x, cjk-l» E Ccx,a [.Q x A x Rn( l ) , Rn( l ) ] and this 
will foil ow as a special case of (3.6.2 1). 
For a given z in A, i t  fol lows from Theorem 4. 1 .2 thut (4.3 . 1 6)-(4.3. 1 H) hus a un ique solution cfk ) ,  where 
C�k) (X; Z) E  C2+a [!2 , Rn( l ) ] . 
To show that cfk) is Holder continuous in Z with exponent a, we consider equations (4.3 . 16)-(4.3 . 1 8) with 
Z and z· and look at the difference of these equations. Note that from the assumptions, 
so that 
4.3 EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM 109 
It then fol lows that 
Letting 
and 
where 
-k ( f. ( (k-I» )" _ ' " a < �« (k) ( )_ (k) ( ' » _D .n2 ( (k) ( )_ (k) ( ' » % J , X, C, Z Z - at c, x, z ci X, z , v  X c, x, z c, x, z 
-KZ(CY-I » lli I C;k-I\1 (d(AW-IX "Z-Z ' "a 
wil = 
Wj2 
� q :n (c?) (x, z) - c�k) (x, Z' » + Hi(C�k) (x, z) - C�k) (X, z · »  
� Kz(cy-l» Hi 'C�k-l\l (d(A» l-a l i z - z· l Ia . 
(k) ( ) (k) ( ' ) k ( f. ( (k-l» )" ' " a 1 2 Cj t, X, z - Cj / , X, z - z Jj /, x, Cj z - z 2l5i"XI 
K l l z  - z' "a 
(k) ( ) (k) ( ' ) k ( r( (k-l» )" ' " a  1 2 Cj t, X, z - Cj / ,  x, z + z .Ij t, x, Cj z - z  2l5i"XI 
K" z - z ' "a 
we obtain the problems 
and 
(4.3.25) 
(4.3.26) 
(4.3.27) 
(4.3.28) 
(4.3.29) 
(4.3.30) 
(4.3.31) 
4.3 EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM 1 10 
The problems (4.3 .30) and (4.3 . 3 1 ) arc equivalent and it follows by well known results (sec KEADY and 
McNABB [ 1 37]) that 
and 
wl � -( I + d(.o ) + H. (d(.o » 2 ) , 2D. 
Therefore 
so that 
-K(I + d(.o) + H. (d(.o»2)l I z - z· l Ia :5 (cfk) (x, z) - c�k) (x, z· » 2D. 
:5 K(1 + d(.Q ) + H. (d(.o»2)l l z  _ z· l Ia , 2D. 
i .e., cfk) is Hl>lder continuous in z.  
We see Lhat (4.3 . 16)-(4.3 . 1 8) has a unique solution c?) , where 
c}k) (x, z) E C2+a,a [.0 x A ,  Rn(l» ) .  
(4.3.32) 
(4.3.33) 
(4.3.34) 
(4.3.35) 
(4,3.36) 
Note that Lhe equation (4.3 . 34) could also have been obtained by integrating (4.3 .25) with boundary 
conditions (4.3.26) and noting that the corresponding Green's function is integrable. 
It is obvious Lhat for equations (4.3 . 1 9)-(4.3 .2 1 ) , VI E Ca [A ,  Rn) and all conditions of Theorem 4. 1 .2 
except for those l isted in assumption ( iv) are satisfied. Therefore, it is enough to show that 
F;(z, CY-I» +H.J cfk-I) E Ca [A x Rn(J) , Rn(J» ) and Lhis follows as a special case of (3 .6.37). It follows 
from Theorem 4. 1�fLhat (4.6. 1 9)-(4.6.2 1)  has a unique solution C?) , where C?) E C2+a [A , Rn(J) ] .  
To  prove (1)  and ( I I )  i n  the general case, w e  need on ly observe that h(x, c?-I» E Ca,a 
[.(2 x A x Rn(l) , Rn(l ) ] from (i) and (ii). The proof is similar to that shown earlier. 
In the case of (1), we note that for components i, where D., H. > 0, 
dk) E C2+a,a [.0 x A Rn(l) ] I " 
by the same argument as above using Theorem 4. 1 .2. 
In Lhe case of (II), we note that for components i, where D. = 1-1. = 0, 
(4.3.37) 
exists and is unique since it is uniquely defined. Furthermore, it follows directly that if h(x, C)k-I» and 
4.3 EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM 1 1 1  
Cfk-I) i s  Holder continuous in x and z with exponent a, then cfk) is also Holder continuous in x and z with 
exponent a, implying that cfk) E Ca,a [Q x /1 ,  Rn(l» ) .  
To prove (III)-(V) in  the general case, we need only observe that F; (z, C)k-I» + Hi !a.a2 cfk
-I) E 
Ca [/1 x Rn(J) , Rn(J)] from (i)-(v). 
In the case of (III), we note that for components i ,  where !i'i > 0, 
by the same arguments as above using Theorem 4. 1 .2. 
In the case of (IV), we note that for components i, where �i = 0, u ·  vcy) == 0 ,  
dk) = I 
F (z C(k-I » + H·I c(k-I) I ' J I a.a2 I 
(4.3.38) 
(4.3.39) 
exists and is unique since it is uniquely defined. By the same argument as for the proof of (II), we see that 
(4.3.40) 
In the case of (V), we see that by (Hs) and (HiJ ,  z(z I ,  z 1 0, zo) ancl Yi(Z I ,  a, YiO; ZO) are unique sol utions of 
(4.3 .22) and (4.3 .24), respectively, on A I . Choose YiO = Ci, I (ZO) and note that if Z = Z(Z I , a ,  zo) , then 
because of uniqueness, Zo = z(a, Z I , z). Also, the solution (Z(Z I , a, zo), Yi(Z I ,  a, YiO; zo» of the systems 
(4.3 .22) and (4.3 .24) is a characteristic equation of (4.3 . 1 9). Hence, for each solution of (4.3 .22) and 
(4.3.24), we have 
(4.3.41) 
and consequent! y , 
(4.3.42) 
Now by using assumptions (Hs) and (H6 ) ,  it is easy to show that Cfk) (z) defined by (4.3 .42) satisfies 
(4. 3 . 19) .  
To show uniqueness of solutions of  (4. 3 . 19) ,  we suppose, that Ci(lk ) and cg) are two solutions of 
(4.3 . 1 9) on A = A I X A n-l . By Theorem 3 .2.9 (Strong Comparison Theorem) for fi rst order partial 
differential equations, we see that cflk) $ cg ) $ cft) �U1d therefore cg) coinc ides willI cg). 
The Holder continuity of C? ) O , z) is obtained by examining the characteristic equations (4.3 .22) and 
(4.3.24), so that 
(4.3.43) 
Finally, we show that (fi ' fi ) and (ci ' Cj) are lower and upper solutions of (cfk) , CY» . To show that 
(Cj , Cj ) is an upper solution of (cfk) , C[k» , we need to only observe iliat 
D o2 ( - (k» > F. ( -: ) F. ( (k-l » > () . r\.., A - i V x  Ci- Ci - Ji x, Cj - Ji x, Cj _ ill .loolA", 
4.3 EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM 1 1 2  
D.�(c.- C(k» + H · (c·- c(k» = fi. (c.-dk-I» > 0 on dQ2xA ' an I I ' "  ' " - , 
- 9jv2(cj-cfk» + u ·  V(cj-cfk» + Hjs( cj-cfk» 
� F;(z, C)- F;(z, cfk-1» + HjJ (cj- cfk-1» iJn2 
� ° in A, 
since Ii and Fj are monotone in Cj and Cj' respectively. 
We may therefore apply Lemma 4. 1 .8  (Maximum Principle) for the elliptic operator or Theorem 3 .2.6 
(Strong Comparison Theorem) for first order partial differential equations or algebraic inequalities to 
conclude that (Cj ' Cj ) �  (cfk ) ,  cfk» . Note that in the case for components i E 1 ,  where Dj  = Hj = 0,  we 
have 
- (k) > 1'. ( - ) 1'. ( (k-I) - (k- I) > () . r�IA Cj - Cj - Jj x, Cj - Jj X. Cj + Cj- Cj _ m .u", • 
and in the case for components i E J, where 9)j = U · V(Cj-Cfk» = 0, we need only observe that 
H·S( C·- C(k» > F(z C · ) - F(z C(k-l» + HJ (c·- c(k-l» > ° in A I I I - I ' J  I ' J I (Jil2 I I - , 
since fj and Fj are monotone in Cj and Cj' respectively and therefore (Cj ,  Cj)�(cfk ) ,  cfk» . Similarly, 
(fj . C)may be shown to be a lower solution of (cfk) . C?' » and the theorem is complete.O 
To start off the iterative procedure, we IIced some continu ity properties of (fj ,  �j ) and (Cj ' Ci ) .  The 
properties of the mapping !!J, from (cfk-I ) ,  C?-l »  to (Cfk) , Cfk» are then given by thefollowing lemma. 
Lemma 4.3.6. 
Consider the BVP (4.3 .16)-(4 .3.21) and suppose that the assumptions (J-J I ), (J-J2) -(J-J�J hold. Let there 
exist (fi ' �i ) and (ci '  Ci ) which are lower and upper solutions of Sn ' En ' 
Assume that 
(i) For components j E 1. where Dj• Hj > O. £j . Cj E C2+a.a[Q X A .  Rn( /) ] ;  
(ii) For components j E 1, where Dj = Hj = 0, £j ' Cj E Ca.a [Q x A ,  Rn(/ ) ] ;  
(iii) For components j E J, where 9Jj > 0, (dj ' Ej E C2+a lA,  RII( J) ] ;  
(iv) For components j E J, where 9Jj = 0, U ·  VCY-I) $0, (dj ' Cj E C1+a [A,  RII( J ) ] ,  and assumptions 
(Hs )-(H;') hold; 
(v) For components j E J. where 9)j = 0, U ·  VC;k-l) == 0, (dj ' Ej E Ca [ A .  Rn(J) ] .  
Then the mapping !!J, from (e;k-I) , cjk-I» to (eV), dk» possesses the following properties: 
(II) fTis a monotone operator on the intervals I.fj , cd  and l�j ,  Cd ·  
4.3 EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM 1 13 
Proof 
We first consider !.he case when Dj, g}j > 0 for all components j. The natural imbedding of C2+a.a 
[.Q x A ,  Rn(l) ]  into C2.a [.Q x A ,  Rn(l) ]  and C2+a [A , Rn(J) ] into C2 [A, Rn(J) ] implies !.hat £j '  Cj E 
C2.a [.Q X A , Rn(l) ] and Sj' Ej E C2 [A , Rn(J) ] .  
The boundedness of .Q and A, together wiLh the fact that their boundaries belong to C2+a, shows that, 
if c;k-I ) (x; Z) E C2 [,Q , Rn(l) l  (with A treated as a parameter space) and C?-I) (Z) E C2 [A , Rn(J) ] ,  then 
c;k-l) (x; Z) E Wi [.Q , Rn(l) l  (with A treated as a parameter space) and CY-I) (z) E Wi [A , Rn(J) l for q > 1 .  
From Lemma 4. 1 .5, we may take q to be identical in both cases. 
This, in view of Theorem 4. 1 . 1  (Imbedding Theorem), yields that c;k-I) (x; z) E CI+a [.Q, Rn(l) l  
(wiLh A treated as a parameter spaee) and Cyk-l) (z) E CI+a [A ,  Rn(J) ] .  From Lemma 4. 1 .3 ,  a may be chosen 
to be identical in both cases. From arguments similar to that shown in Lhe proof of Lemma 4.3.5, we see Lhat 
C;k-l) (X, z) E CI+a•a[.f2 x A ,  Rn(l) ] . 
I t  is immediate that the proof of (I) fol lows from the choices (c;k-l) ,  CY-I» = (£j '  9.j ) and 
( (k-l) C(k-l » - ( - C- ) ' L 4 3  5 c j ' j - c j '  j In emma . . . 
All the oLher possible cases are treated similarly. 
We have shown that if (c(O) C(O» = (c· E )  then (dO) dO» > fI'(c(O) C(O» = (d1) C� l) and i f  J ' J J ' J , ' , - J ' J ' "  
(c(O) C(O» = (c - C) then (c(O) C�O» < fI'(c(O) C(O» = (c(1) C( I » We have in fac t  proved that the J ' J -J ' -J " ' , - J ' J ', . , ' 
• 
mapping fl'maps intervals I £j . C:j I and ISj '  Ej J onlO themselves. 
To prove (II), let Cjl > Cj2 E [£j '  cj l and Cjl , Cj2 E [9.j ' Ejl where (Cjl , Cjl)'?, (cj2 , Cj2) for all components 
j. We want to show that fIf(cj l .  Cjl ) ?fIf(cj2 .  Cj2 ) .  
Let (ui . Ui )=fI'(Cjl .  Cjl )-fI'(Cj2. Cj2 ) .  Then the monotone nondecreasing property of Ii and Fi 
implies tliat 
au· D· -' + f-I.u · = i-J . (C1 -C2) > 0 on aQ2xA , on " " , , - . 
, 2 f '  -�V U· + u · VU + IJ ·S'/U· = F(z C1 ) - F(z C2) + H· c'l - c2 > O m A  , " " "  • J " J ' (Jil2 ' , - ,
(4.3.44) 
(4.3.45) 
(4.3.46) 
and from Lemma 4. 1 .8 (Maximum Principle) for Lhe elliptic operator, or Theorem 3 .2.6 (Strong Comparison 
Theorem) for first order partial d ifferential equations or from algebraic inequalities, we see that CUj, Vj) � 0 or 
fIf(Cil ,CiI ) ?'  fIf(Ci2 .Cj2 ) · This shows that S'is a monotone operator on the intervals [£j ' Cj ] and Le,  Cd .O 
The monotone operator fIf will play a central role in  Lhe iteration scheme. 
Remark 4.3.1 
As with the unsteady state system Sn, Bn, we see that if Ii and Fj are strictly monotone increasing in Cj and 
Cj' respectively, then by Theorem 3 .2. 1 3  (Generalised S trong Comparison (Contact) Theorem), we see Lhat 
fI'(Cj l '  Cj1 » fI'(Cj2 , Cj2) ,  (unless S'(Cj l ,  Cjl ) =fI'(Cj2, Cj2 )  in which case the right hand sides of 
(4.3 .44) and (4.3 .46) are identically zero; but this happens only if (Cj l ,  Cjl ) =(Cj2 ,  Cj2 ) ,  from the strict 
monotone property ofli and F;). We say that the monotone operator fl'is monotone operator in the sense of 
COLLATZ [76] , i.e., (Cj t ,  Cjd ?' (Cj2 , Cj2 ) impl ies Lhat fI'(Cjl , Cjl » fI'(Cj2, Cj2 ) ·  
4.3 EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM 1 14 
--- ----- --
Remark 4.3.2 
As with the unsleady Slate problem, Sn, Bn, we see that iff; and Fj are monotone nonincreasing in Cj and Cj, 
respectively , the operator S' is alternating on the intervals [fj , c';] and [C , C\ ] i n  the sense that 
(Cjb Cj2) � (Cj2 , Cjl ) implies that S'(Cjl , Cj2)�S'(Cj2 ' Cjl ) .  To prove this, let (Uj , Uj ) = S'(Cjl , Cj2)­
S'(Cj2 , Cjl ) .  Then the monotone non increasing property off; and Fj implies that 
Ju · D·_I +H·u· =fJ. (C-I -C-2 ) < O  on Jfl2XA ' an " I I I - t 
-9J..V2U· + U · VU + fJ..%U· = F(z C -I ) - F (z C -2 ) + H·f (C·I - C·2) > 0 in A I I I I I I '  J I '  J I aD2 I I - , 
and from the same arguments as in Lemma 4.3.6, !!kjl � !!kj2 and PJCjl � PJCj2. 
It is necessary to choose a proper initial i teration to ensure that the sequences {(c�k) , Cfk» }  are monotone 
sequences that converge to a solution of Sn , Bn and are within the intervals [fi ' cd and [�j ,  C\ ] .  From 
Lemma 4.3.6, it is obvious that the monotonicity of these sequences obviously depend on the monotonicity 
off; and Fj and the initial iteration is taken to be either an upper or a lower solution which is required to 
satisfy certain inequali ties on the corresponding system. 
We may therefore use the in itial iteration (Ci(O) , G(O» = (Cj , CI ) to construct the sequence 
{(c?) , G(k» }  from the following equations 
-!llV2C.(k ) + U · VC(k) + f-f .S'lC.(k ) = F(z C�k-I» + H· f e.(k-I ) in A I I I I I I '  J I aDz I ' 
or the sequence {(dk) , ��k» }  with (dO) , ��O» = (fi ' C) may be determined from the equations 
JC(k) . D --'-+Hc(k) = H·C(k-l) on Jfl2xA I an I -I I -I ' 
_!ll.V2C(k) + U ·  VC(k) + If .S'lcCk) = F(z, C(k-I » + lI f c(k-l) in A. I -I -I I -I I -J I an2 -1 
Note that the sequences {(elk) , G(k» }  and {Cdk) , ��k» }  may be obtained independently of each other. As for 
before, uniqueness and existence of sequences {(ej(k) , C;(k» }  and {(dk) , ��k» }  fol low from similar 
arguments for uncoupled scalar systems of nonhomogeneous linear elliptic differential equations, first order 
partial differential equations or algebraic equations by using the monotonicity properties off; and Fj. 
Definition 4.3.2. 
The sequences {(c(k) C�k» }  and {(e(k) E(k» }  with (c(O) C�O» = (c C-) and (e(O) E(O» = (c· C )  are -I ' -I I ' I -I ' -I _I ' _I I '  I , ' " 
called minimal and maximal sequences, respectively. We say that (fj, C)  and (ci ' E; )  are minimal and 
maximal solutions respectively in the regions [fj , cd and [�j, Cd ,  if for any solution (cl, C i) of Sn , Bn 
where fi � Cj � ci and (;j � Ci � Ci , then fi � ci � ci and �i � Cj � E;. 
4.3 EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM 1 15  
From Lemma 4.3.6 together with the monotonicity o f  ff, we  shall show that the sequence 
( dk), ��k» } with (f�O) ,  ��O» = (fi '  fi) is monotone nondecreasing and the sequence ( c?) ,  C;(k» } with 
(S(O) , �(O» = (Cj ,  C:j ) is monotone nonincreasing. 
Furthermore, (fj , fj ) � (Cj , C:j ) results in (dk) , ��k» � (c?), E;(k» for all k and pointwise limits 
(fi ' C) and (ci '  C;) exist. This is all done in the following lemma for S n ' Bn · 
The only difficulty arises for components i E J, where 9Jj = 0, U · VCfk) $. 0, and in this case we will 
assume in addition that the assumptions (H7) and (Hg) hold. 
Lemma 4.3. 7. 
Suppose in addition to the assumptions of Lemma 4.3.6, that ( dk), ��k» } are minimal sequences and 
( c?) ,  �(k» }  are maximal sequences of Sn , Bn . Also, assume that for components i E J, where 9Jj = 0, 
u .VC;<k) , u · V��k) $. 0 , that the assumptions Uh)-(J-/g) hold. Then 
for all k = 1 ,2, . . .  and the pointwise limits 
I · « k) C(k» - ( C ) 1m f.j , -i - fi '  _j , k-+oo 
lim (lY) C(k» = (c C) I ' I  ' 1 '  I '  k-+oo 
exist, implying that 
for all k = 1 ,2, . . . 
Proof 
Let (u U )  = (c�O) - c(1) C(O) - C(l» = (c -c( 1) c -C(1» for all i Then by definition 4 6 1 of upper and ' "  , ' "  , " "  I "  • • 
lower solutions and definition 4.6.2 of maximal sequences, the following inequalities will hold 
-DiV;ui = -DiV; Ci - hex, c) � ° in iliA, 
D aUj +Hu =D �(c - c(I) +H(c - c(1) =D. aCj +H(c -C» O on an2xA , an " ' an " ' " , an " , - , 
with similar inequalities holding in the boundary conditions. 
We first consider the case when Dj, 9Jj > ° for all j. We see that from Lemma 4. 1 .8 (Maximum Principle) for 
the elliptic operator that (u
" 
Uj ) � 0 ,  i .e. cP» � c?) on !2 x A and E;(O) � E;(I) on if. 
This result also follows from the monotonicity of  ff, since if  (e" Cj ) � (ciO) , C;(O» , then ff (ci ' Cj ) � 
rrr(c-�O) C-.(O» = (c�l ) C(1» and (c�O) C(O» > (c(l ) C.( l) if we let (c�O) C(O» = (c C ) . v I " I ' ,  I " - , ' I I " P I 
4.3 EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM 1 1 6  
Therefore, from Lemma 4.6.5 , we conclude that c(1) E C2+a•a [.Q X A ,  Rn(l) ] and �(l) E C2+a 
[A , Rn(J) ] .  By natural imbedding this implies that cP) E C2•O [.Q x A ,  Rn(l ) ] and �(1)E C2[A , Rn(J) ] . 
All the other possible cases are treated similarly so that 
(i) For components i E I, where Dj = IIj = D, C?)E Co.o [.Q x A ,  Rn(l) ] ,  
(ii) For components i E J, where 9lj = D, U ·  V�(I) $. D, �(l) E cI [A , Rn(J) ] 
(iii) For components i E J, where 9lj = D, U ·  V�(l) == D, �(l) E CO [A , Rn(J) ] .  
In the case of (iii), we see that if (H7)-(Hs) are assumed, then from properties of aglebraic equations we see 
that CP) will be continuously differentiable in z. Therefore, if (H7)-(Hs) are assumed then in all cases for 
components j E J, CP) will be continuously differentiable in z. Furthermore, in all cases for components 
i E J, where Dj, H j > 0, it can be demonstrated from (4.3 . 1 8) that c?) will also be continuously 
differentiable in z. 
We may similarly show using by the definitions of upper and lower solutions and of minimal sequences, that 
c(O) � cq) and C�O) � C�l) • -I -I -I -I 
Now let (Uj ,  Uj )  = (cP) -d1) , �(l ) -�\1» . Then the monotone nondecreasing property of/; and Fj implies 
that 
-DjV;Uj = J;(x, cjD» - J;(x, f}O» = J;(x, Cj ) - J;(x, �) 2: () in f2xA, 
()u. dCP) oc( l ) - -V. _I + II -u .  = r v. _1_ + If.c� I) I- r v. -- I- + lI - c( l ) l = II (C.(O) _C(O» = Il · (c.-c. ) > 0 on o.Q2xA, 1 an 1 1 1 an 1 "I 1 all 1 �I 1 1 -I 1 1 _I -
-9J·V2U· + U · VU· + fl .sf.U. =F(z C · )-F (z C - ) + J-J.f (c·-c · ) > 0 in A 1 1 1 1 1 I '  J I '  -J 1 aD2 "I _I - , 
and it follows from the monotonicity of fY, that 
implying that 
and 
Assume, by induction, that 
C-.(k )  < -c�k-l) and C-(k) < C�(k-l) I - I I - I ' 
for k= l ,  . .  ,m . 
---------------------
4.3 EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM 1 1 7  
The only difficulty arises for components i E ./, where 9Jj = 0, u · VE;(k) $O and i n  this case we  cannot 
assume that in general, that the assumption (H(;) will be satisfied. However, in this case we can assume that 
(H1)-(Hg) hold so that CY-l) and CY-2) will be continuously differentiable in z for all } by earlier 
arguments. Thus for components i E I, where Dj, Hj > 0, it can be shown that c�k-l) will be continuously 
, differentiable in z from (4.3 . 1 8) ,  so that by assumptions (H1)-(Hg), F;(z, CY:-1» +HjJa C�k-l) will be 
continuously differentiable in z. For components j E I, where Dj = Hj = 0, we see thaf1n this case by 
assumptions (H1)-(Hg), F; (z, CY-l» w il l  be continuously differentiable in z. IL then fol lows that 
(HARTMAN [ 1 19 ,  pp. 95-99]) assumption (H(;) will be satisfied in the general case. 
The functions (Uj ,  Uj ) = (cj(m) - c/m+1) , E;(m) - E;(m+l» and the monotone nondecreasing property of fi and 
Fj implies that 
D n2 - F( -(m-l» F ( -(m» > O ' r>., A - jV  xUj - Jj X, cj - Jj x, Cj _ In �"I\.I1, 
a a-(m) a-(m+l) D,-.!!i + H u, = [D _Cj_+Hc�m) ] - [D Cj +11,c�m+l) ] = H(C,(m-l) -C,(m» > O  on af22xl1 , an " ' an ' , ' an ' , " , - , 
-!lJ.V2U,  + U · V U. + Hal).= F (z c(m» - F (z c,(m» + H f (c(m) - c,(m» > 0 in 11 " " " " I ' I I aD2 I I - , 
with similar inequalities in the boundary conditions. This ensures that 
from the mono tonicity of ffand proves that ((S(k) , E;(k» }  with (clO) , E;(O» = (Cj , Cj) is a monotonic 
non increasing sequence. 
It follows from a similar induction argument that ((f�k ) ,  ��k» } with (dO) , ��O» = (fj , �j) is a 
monotonic nondccreasing sequence and by a similar induction argument cl-1) � f�k-l) and E;(k-I) � ��k-l) 
for k= 1 ,  . . .  ,m ensures that crm+1) � dm+') and E;(m+I) � ��m+l ) . The following inequalities then hold 
for all k = 1 , 2, . . . 
It follows from the monotonic property of our maximal and minimal sequences, its boundedness by 
(fj ,  �j ) and (Cj ' Cj ) and the monotone convergence theorem that the pointwise limits 
I· « k) C(k» - ( .  C )  1m f, ' _j - fj ,  _j ,k�� 
I · (-(k) C-(k» - (- C- ) 1m Cj , j Cj , j ,  
k�� 
exist. Therefore, 
for all k = 1 ,2 ,  . . .  U 
4.3 EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM 1 18 
4.3.3 Existence of Solutions of the monotone system Sn ' En 
It  can be shown that our minimal and maximal sequences ((dk ) .  ��k » } and ((c?) . C;(k» } converge not only 
pointwise but converge uniformly (in appropriate function spaces) as well. The following theorem is an 
existence theorem for solutions to the monotone system Sn . Bn . This theorem shows the existence of at least 
one solution (Ci. Ci) of Sn . En lying between (fi .  �i) and (Ci . Ci) 
Theorem 4.3.1 (Generalised Existence Theorem) 
Let the assumptions of Lemma 4.3.7 hold. Then the minimal and maximal sequences ((f\k) . �\k» } and 
fCc?) . C;(k » } converge monotonically and uniformly from below and above 10 (fi ' � ; )  and (ci ' E; )  
respectively. where (fi ' � i )  and (ci ' C;) are solutions of Sn . En . Moreover (fi ' � i )  and (ci ' C;)  are 
minimal and maximal solutions respectively of Sn. B n in the regions [fi t  cd and [�i '  Cd . 
Proof 
We first consider the case with D i. 9)i > 0 for all i. 
Let f�k). c?) E C2+ a.a [ Q X A .  Rn(l) ] and ��k � E;(k) E C2+ a [ A .  Rn(J) ] for k = 1 .2 . . . .  and let us 
consider the maximal sequence ( ci(k) . C;(k» } . 
We note that C2+a [Q. Rn(l) h;;: W� [ Q. Rn( l ) ]  a n d  C2 + a [ A . Rn(J ) ] c; W� [ A . Rn( J ) J  for P I  � 
(ml+2)/( l-a) and p2 � (m2+2)/( I-a) .  
From Lemma 4. 1 .5 .  this impl ies that C2+a [ Q. Rn( l ) ] c; W.i [Q . Rn( l ) ] and C 2+a [ A . Rn( J» ) c;  
W,i [ A .  Rn( J» ) .  where q = min {P I .  P2} .  
B y  Theorem 4. 1 .4. we see that c?) (with A treated as a parameter space) and E;<k) satisfy the 
following Agmon-Douglis-Nirenberg estimates (see Theorem 4. 1 .5): 
(4.3.47) 
and 
(4.3.48) 
The fact that cY-I ) E  I£j ' Cj J and CY-1)E I(j' Cj ] .  i.e .•  cY-I) and C,?-l) arc bounded. the continuity off; 
and of Fi imply by the continuity and boundedness of the Nemytskii operator (Lemma 4. 1 .6) that the 
sequences (fi(x. cY- 1) }  and (F; (z. CY- I » + fa.a C?-I)} are uniformly bounded in C[Q . Rn( l ) ] (with A 
treated as a parameter space) and C[A .  Rn(J) ] resp�ctively. 
Since qQ . Rn( l ) ]  and C r A .  Rn( J ) ]  are dense in Lq [.Q . R"( I ) ]  and U t A .  R"(J ) ] ,  it follows by the 
continuity and boundedness of the Nemytskii operator (Lemma 4. 1 .7) that (fi(x. cY - 1 ) }  and 
(F;(z. CY-I» + fan c?-I )} are bounded sequences in L"1'(2 . R,,(I) J (with A treated as a parameter space) 
and U t A .  R"(J» ) .  r�spectively . •  
This. together with the above estimates (4.3.47) and (4.3.48) shows that the sequences (C;<k)} and 
(E;<k)} are uniformly bounded in Wq2 [ Q . Rn( l » ) (with A treated as a parameter space) and W.i [ A .  Rn(J» )  
respectively. Therefore by  application of Theorem 4. 1 . 1  (Imbedding Theorem). we  oblain 
(4.3.49) 
and 
(4.3.50) 
--- - -- ---------
4.3 EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM 1 1 9  
for all k = 1 ,2, . .  , where C in both estimates are independent o f  any element of  W'; l!..? , Rn( l ) ] or 
W'; [ A ,  Rn( J) ] .  
From (4.3 .49) and (4.3.50), we  can conclude that every uniformly bounded sequence in 
W'; [.Q , Rn( l ) ]  and W'; [ A ,  Rn( J ) ]  is also uniformly bounded in C1+a [!..? , Rn(I) ] (with A treated as a parameter 
space) and C1+a [ A ,  R n( J) ] ,  respectively. From Lemma 4. 1 .3, we may take a in both cases to be identical. 
Thus the sequences (Cj<k») and (E;<k » )  are uniformly bounded in C l +cx l!..? , Rn(l) ] (with A treated as a 
parameter space) and C1+ a [ A ,  Rn( J» ) .  Therefore, by applying Lemma 4. 1 .6 (.t;(x, c? -l» ) and 
(F;(z, C?- l» + La c?-l» ) are also bounded sequences in Ca [.Q , Rn( l » ) (with A treated as a parameter 
space) and e a [ A , �n( J» ) , respectively. Hence by the Schauder type estimates (Theorem 4. 1 .3), we have 
(4.3.51) 
and 
(4.3.52) 
for all k = 1 ,2, . . , which implies the uniform boundedness of sequences (c? » )  and (E;<k » )  in e2+a [!..? , Rn( l ) ]  
Cwith A treated as a parameter space) and e2+ a [A, Rn( J » ) ,  respectively. 
We may use arguments given in Lemma 4.3.5 to show that sequences (c/ k » )  are Holder continuous in 
z with exponent a and may be shown to be uniformly bounded in A 
Therefore, by the natural compact imbedding of e2+a.lX l.Q x A ,  Rn( l ) ]  and e2+ a [ A ,  Rn( J ) ]  into 
e 2•O [.(2 x A ,  Rn( l ) ]  and e2 [ A, Rn(J» ) ,  the sequences (c/k » )  and (E;<k») are relatively compact in 
e2• O [.Q x A , Rn( l ) ]  and e 2 [ A ,  R n(J) ] respectively. 
This implies that there exists subsequences of (Cj(k» ) and (E;<k») which converge in 
e2•O [.Q x A , Rn( l ) ]  and e2 [ A, Rn( J ) ]  respectively. 
Let (Cj ,  Cj)  where Ci E e2•O[.(2 x A ,  Rn( l )  J and Ci E e l •2 [ A .  Rn( J )  J be the limit of this subsequence. 
On the other hand, we have shown that the sequence (Cc? ) ,  C;(k» ) converges pointwise to CCi ' E; ) .  
Therefore c · = c· o n  e2•O [!..? x A Rn( l ) ]  and C = C on e2 [ A  Rn( J ) ]  This shows that the whole , I  I , I ,  , .  
sequences (c? » )  and (E;<k » )  converge in e2•O [.Q x A ,  Rn(l) ] and e2 [A, Rn(J) ] to Cj and E; respectively, 
that is lim c?) = Cj in e2•O [.Q x A ,  Rn( l) ] and lim C;(k) = C; in e2[A, Rn(J) ] and fj � Cj � Cj on .Q x A 
k � _ k -) ... and �j � ej � Cj on A .  
Similarly, by imitating the preceding argument relative to the minimal sequence (Cdk ) , ��k» ) ,  one can 
conclude that the sequences (f�k» ) and (��k » ) converge monotonically in e2.O [.Q x A , Rn( l ) ]  and 
e 2 [A, Rn(J) ] respectively; their limits are denoted by fi and �j which belong to e2•O [.o x A ,  Rn( l ) ]  and 
e 2 [ A, Rn( J ) ]  respectively and satisfy the relation f'j � f.; � Cj on !..? x A and �j � �j � Cj on A. Thus the 
limits 
and 
4.3 EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM 120 
I · [ F ( C-( k-l» ) LI J .-(k- l ) ] _ F ( C- ) H J  -1m j z, j + l� j Cj - j z, j + j Cj , 
k �.. dQ2 aQ2 
etc. exist unifonnly on C2•0 [.Q x A ,  Rn( l ) ] and C 2 [.I\, Rn( J ) ] , respectively. Thus we conclude that (fj ' f) 
and (Cj ' �)  are solutions of the B VP 
and 
- $.V2C· + u · VE + H ·sfC· = F (z C. ) + H . J C I I I I I I ' J I dQ2 I ' 
respectively. 
We therefore have that (fj ' �) and (Cj '  �) are solutions of Sn , Bn . 
For components i, where Dj = Hj = 0, the uniform convergence of sequences (clk» ) to solutions fj and Cj 
follows from standard theory on nonlinear algebraic equations. The sequence of approximations (4.3 . 16) are 
unifonnly bounded and equicontinuous and hence possesses uniformly convergent subsequences. The rest of 
the argument follows along similar lines to that already discussed. The case for components i, where 9)j = 0, 
u ·  VCfk) == 0 is similar as is the case for components i, where 9)i = 0, U · VC?) '$ O . 
Now we show that (fj ' �i ) and (Cj ,  Cj) are minimal and maximal solutions of  Sn ' Bn ' Let  (Cj, Cj) be  any 
solution of Sn ' Bn such that fj � ci � Cj and fi � Cj � Cj . 
Since (Ci, CD = fi(Cj, C,), it follows by Lemma 4.3.6 and from the definitions of ( dk), dk) )} and 
( clk! C;(k) )) . that 
(fj , fj ) �  (Ci, Cj) � (Cj ' ('j )  , 
implies that 
( C ) < tJT( (0) C(O» < ( . C ·) - fJ/T . C ·) < tJT( -(O) C-(O) < ( -. c";. ) fi ' _i - v L ' _j - C" I - ",C" I - v ci ' i - C" I ' 
Let us assume that for some m > 1 ,  
4.3 EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM 121 
(C(m) C(m» < (c C) = (c�m) C(m» _I ' -I - i t ' , ' I  . 
Then we shall show that 
(C(m+l) C(m+l» < (C C) = (C�m+l) C.(m+l» _I ' _I - It ,  , ' I . 
From Lemma 4.3.6 (II) and from the definitions of {(dk� f�k) )  and fCc?! C;(k) ) . we arrive at 
(c(m+l) C(m+l» = ff(c(m) C(m» < (c - C) = UlT c · C) < ff(c�m) C.(m» = (c�m+l) C.(m+l» _I t _I -I ' _J - '" '-'\ " I - I "  I "  • 
Thus, it follows by mathematical induction that 
(C(k ) C(k» < (c C) = (C.(k) C.(k» -I ' -I - ' t I I '  I ' 
for all k = 1 ,2, . . . 
Hence, we have 
(�j '  �j ) � (Cj, Ci) � (ci ' C;) ,  
Remark 4.3.3 
Note that unlike the unsteady slate system Sn, Bn, we cannot assume that for the steady slate system Sn ' Bn 
that (fi ' �)=(Cj , C; )=(Ci, Cj) unless we have uniqueness of solutions of the system Sn ' Bn . We have given 
some uniqueness criteria in section 3 .3 for the system Sn ' Bn and these can be used to show that 
(�j ,  fi )=(C; , C;)=(Ci, Ci). Other uniqueness conditions for systems of elliptic equations are given by LADDE 
I 
et al. [ 153 , p. 1 2 1 ]  and CARL and GROSSMAN [47] and may also be applied to our problem. These 
conditions arc used to show that our minimal and maximal solutions obtained by monotone iteration satisfy 
the inequalities (fj ' �j )?: (Cj '  C; )  and hence uniqueness is achieved. 
For the general system Sn ' Bn , which may possess no monotone properties, the following theorem follows 
from Theorem 4.2. 1 and summarises how we may set up monotone sequences for this general system which 
converge to a solution of a new system which may relate in some way to the solution of the original system 
Sn' Bn · 
Theorem 4.3.2 
The general system Sn ' Bn for which Ii and Fj satisfies Lipschitz cOfl/inuity properties (Hi) and Holder 
continuity properties (H5 J , may be imbedded in a system S;n , B;n of twice the order which is coupled 
by monotone functions .f and Fj" 0/ the new dependent variables Vj and Vi. Moreover, all the solutions 
(Cj, Cj) of the general system Sn ' Bn are solutions of the new system, where 
Vj =Cj, Vn(l)+i = -Ci for i= ] , .. ,n(f) (4.3.53) 
and 
Vj = Cj, Vn(J)+i = -Ci for i= l ,  . .  ,n(J). (4.3.54) 
Let (�j , Xi) and (Vj , �) be lower and upper solutions for the system S;n , B�n with continuity properties 
given in the assumptions of Lemma 4.3.6. Let also the assumptions of Lemma 4.3 .7  hold/or the system 
Sin ' S;n . Then the minimal and maximal sequences ( ��k) ,  .!:::�k ) }  and ( v?) ,  V;(k) }  of Sin ' Bin given by 
4.3 EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM 1 22 
Theorem 4.3.1 converge monotonically and uniformly from below and above to (.!:j ' !::j ) and (Vj , V;) 
respectively. where (}!.j. L )  and (vj • \1;) are solutions of s�" . iii" sat isfying the following inequalitites 
(4.3.55) 
(4.3.56) 
for all k = 1 .2 • . . .  
If furthermore, the system Si" ,  Bi" has a unique solution. then (.!:j , !::J=(Vj , V;) =( Vj, Vj) is the unique 
solution of Si" ,  iIin and all solutions (Vj, Vj) of Sin . Bin for which 
aw 
an' = 0 on a.f2lxA., 
aw 0.-' - /J (W - w )  = 0 on rJ!22xA. ' an I '  , ' 
(4.3.57) 
(4.3.58) 
(4.3.59) 
(4.3.60) 
(4.3.61) 
(4.3.62) 
for (Wj . ltj) = ( Vj + Vn(l) +j . W; + Wn(1)+j ) generates the unique solution (fj ' �j ) = (ej . C;) = (Cj . Cj ) of the 
general system S" , Bn . where 
. < (0) < ( I) < < (k) < . < . < - < -(k) < < -( I) < -(0) < -.!!, _ }!., _ }!., _ . . . _ }!., . . . _�, _ c, _ v, . . .  _ v, _ . . .  _ v, _ v, _ v, .  
< (0) < ( I) < < (k) < < < - <-(k) < <-(I ) < -(0) <-�(I)+j - .!:n(l) +j - .!:n(l)+j -· · · - Y.n(l)+j · " - Y.n(l)+j _ -Cj - vn(l)+j . . .  - vn( )+j -. . .  - vn(l) +j - vn(/)+j - vn( l) +j , 
for (x, Z) E .f2 X A. , 
Y <V(O) <V( I )  < <V(k) < <v <-C· <y- .<  <V-(k ) < <V-( l ) <\7(0) <\I . _n(1)+j --n(1)+j --,,( 1 ) +j -" --n(1)+j -" --n(1)+j - , - '1(1 )+, _ .. - n(.l)+j -··- n(.l)+j - n( 1)+j - '1(1)+1 '  
for Z E If ,for all k = 1 .2 • . . .  
We have seen that for the purposes of our exislence proof, we may look al the monolone syslem Sn ' Bn 
where we assume our coupling functions Ji and Fj are monotone nondecreasing in Cj and Cj• respectively for 
aUi. If  this monotone property is not satisfied then we may imbed our general system '�'" Bn into a system of 
twice the order which is a monotone system. The existence resulls obtained for this new system of twice \:he 
order satisfied by (fj ' �j) and (ej , C;) then implies a solution of our original system Sn . Bn by the 
imbedding results of section 4.2 but only if uniqueness is guaranteed in the monotone system S2,. , B2n · 
4.3 EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM 1 23 
If uniqueness is not guaranteed in the monotone system S2n '  B2n , the following section gives a 
general existence theorem for the system Sn' Bn where .fi and Fj may not necessarily satisfy any monotone 
property. Such solutions may or may not be obtained by monotone i teration. 
4.3.4 Coupled Lower and Upper Solutions and General Existence of Solutions of the 
Nonmonotone system S" , ij" 
We shall now introduce the concept of coupled lower and upper solutions relative to the system Sn ' Bn which 
may not necessarily possess any m�notone property. 
Definition 4.3.3 
Assume that 
(i) For components j E I, where Dj > 0, fj and Cj are continuous functions in n x A with continuous 
first order Xj derivatives in !2 x A and continuous second order Xj derivatives in !2 x A ; 
(ii) For components i E I, where Dj = Hj = 0, fj and cj are continuous functions in !2 x A ;  
(iii) For components i E J, where q'Jj > 0, �j and Cj are continuous functions in If,  with continuous first 
order Zj derivatives in If and continuous second order Zj derivatives in A; 
(iv) For components i E J, where 9)j = 0, U ·  VCj $0,  �j and Cj are continuous functions in A ,  with 
continuous first order Zj derivatives in A; 
(v) For components i E J where 9)j = 0, U ·  VCj ;: 0, C and Cj are continuous functions in If.  
The ordered pair o f  functions (fj , C )  and (Cj '  Cj )  with fi :s; Cj on !2 x A and (;;'j :s; (:, 011 If arc said to be 
coupled lower and upper solutions of Sn' Bn respectively, if they satisfy: 
and 
de 
d� :s; 0 on d!2lxA, 
(Jc. D· --=!... < H· (C - e ) on dQ2xA , an - I - ' _I , 
_!llV2 C +  U ·  V c +  HI (C- e ) :S; F (z (;k ' C/ )  in A, , -, -" aQ2 -, -, -" .-
dC 
VI C +  q) � <  VIC I on dAI _I 1 dnl - I , ' 
dC 
J 
_I :s; ° on dAa, a = 2, 3 ,  ana 
de 
_I � O on d,al xA, dn 
dC· -D· _I > H · (C - c·)  on d!22xA 1 dn - 1 1 1 ' 
4.3 EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM 124 
-91.V2 C. + u . VC. + J-J. J (C·- c·» F(z Ck ' CJ ) in A, " " Bn2 " - , ' .-
- dC  VI C+ 91 --' > vIC I on dA I , ' :1  - I ,  ' ani 
dC, 
__ I ;:::: 0 on (fAa, a = 2, 3 ,  dna 
where L. i; E and � arc defined in (3.2.9)-(3 .2. 1 2) .  
Note that these coupled lower and upper solutions may be uncoupled into lower and upper solutions by the 
following substitution 
and 
v. = C· Vn(J)+,· = -C V = C V = -C · , , ' -, ' -' -" _ n(1)+i , . 
All the other properties of lower and upper solutions discussed earlier in this section are also valid for 
coupled lower and upper solutions although it must be noted that coupled upper and lower solutions must 
occur simultaneously. 
A A 
In order to establish an existence theorem for SrI '  Bn in terms of coupled upper and lower solutions, 
we define a transformation f7, by 
(C�k ) dk» = :!i(c(k- I ) dk- I» , ' I J ' J ' (4.3.63) 
and consider the sequences ( c}k) .  cfk » } given in (4.3 . 16)-(4.3 .2 1 )  with c}k) satisfying the inequalities 
. < (k) < -. '  n A '  d C(k) " f ' th · I" C· < C(k )  < C-· · A f k - 1 f, - c, _ C, ill J..I: X an , salis ymg e mequa l lIes _, _ , _ , m  or • . . . .  
The prop�rties of sequences ( c}k) , Cfk » } are similar to that discussed earlier. However, in the present case, 
the transformation f7, may not be a monotone operator and so although we may obtain bounds for the 
solutions, these bounds may not necessarily be improved by monotone iteration. 
Lemma 4.3.8 
Consider the BVP (4 .3.16)-(4.3.21 )  and suppose that the assumptions (H I) ,  (I-J!J- (H4) hold. Let there 
exist (fi '  �i ) and (Ci .  Cj) which are coupled lower and upper solutions respectively of Sn ' Bn wi t h  
< (k-I) < - f2 A d C < C(k-I) < C- A fj - Cj - Cj on J. x an _j _ j _ j on . 
Assume that 
(i) 
(ii) 
(iii) 
(iv) 
For componentS j' E I where D ·  /1 · > 0  C(k-I) E CI+a,a [D x A  Rn(l) ] . , J' J ' J  , . , 
For componenll' j' E I where D ·  = I-I · = ° c(k- I) E Ca,a [.Q x A R,,( I) I '  � , j J ' J ' . , 
For components j E J, where �j > 0, CY-1) E C1+a [A ,  R"(J) J ;  
For components j' E 1 where q; .  = 0 u ·  VC(k-I) ,.., O C(k-l ) E Ca [ A  Rn(1) ] and assumptions 
• , J ' J � .  J ' 
(HS )-(I/(, ) hold; 
(v) For components j E J, where �j = 0, U ·  VCY-1 ) =  0, CY-l) E C a I A ,  Rn(J) I .  
4.3 EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM 125 
Then the BVP (4.3 . 16)-(4.3 .21)  possesses a unique solution (dk) , clk» , where 
(I) 
(II) 
(III) 
(IV) 
(V) 
For components i E I, where Di, IIi > 0, cfk) E e2+11, a[!2 x A ,  Rn(l) ] ;  
For components i E I, where Di = Hi = 0 ,  cfk) E el+ a,a [!2 x A ,  Rn(l ) ] ;  
For components i E J where �i > 0, efk) E C2+a [ A ,  Rn(J) ] ;  
For components i E J where �, = ° u ·  VC<k) ""-0 e�k) E el+ a [ A  Rn(J) j '  , " , r , I " 
For components i E J where �, = ° u ·  VC<k) == ° C<k) E ea[A Rn(J) ] , I ,  I " ' • 
Furthermore, in all cases c? ) and e? ) satisfy the inequalities fi � Cfk) � Ci in ,(2 x A  and �i � e? ) � Ci 
in A .  
Proof 
Most of this proof is identical to that given in Lemma 4.3.5 . However, we need to only observe that 
and 
f (  - ) < j' (  (k- J » ) < j-' (  - )  -i X, fk ' C, - i X, cj - i X, fk , C, ,
in order to show that cfk) and e?) satisfy the inequalities fi � dk) � ci in .f2 x A and �i � efk) � Ci in A .0  
The following lemma follows from Lemma 4.3.6 and Lemma 4.3 .8 
Lemma 4.3.9 
Consider the BVP (4.3 .16)-(4.3.2 1 )  and suppose that the assumptions (H I), (H5 ) - (/-/4) hold. Let there 
exist (fi , C) and (Cj ,  Ci ) which are coupled lower and upper solutions of Sn , En . 
Assume that 
(i) 
(ii) 
(iii) 
(iv) 
(v) 
For components J' E 1 where D '  H '  > 0 c '  [, E C2+a, a [!2 X A Rn(l) ] . , J ' ) ' -} ' ) " 
For components j E I, where Dj = IIj = 0, fj ' Cj E CCX,Cl [!2 x A ,  R"(l) 1 ;  
For components j E J ,  where 9Jj > 0 ,  c;.j ' Ej E C2+a [ A ,  Rn(J) ] ;  
For components j E J, where �j = 0, U ·  veY- I) ;;=0, c;.j' Ej E C I+I1 [ A , Rn(J) ] ,  and assumptions 
(/-/5 )-(/-/;' ) hold; 
For components j E J, where 9)j = 0, u '  \lCY-I) == 0, c;.j ' Ej E e Cl [ A ,  Rn(J) ] . 
Then the mapping ff,from (clk-I) , Clk-I» ) to (cfk ) , Ci(k) ) possesses the following properties: 
(II) ffis an operator which maps the intervals [fj , cd and [(I ' Cd onto themselves. 
The following theorem is an existence theorem for solutions to the general system Sn , En . 
4.3 EX ISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM 126 
Theorem 4.3.3 (Generalised Existence Theorem) 
Let there exist (fj , �, ) and  (Cj , Cj ) which are coupled lower and upper solutions of Sn ' Bn w i t h  
continuity properties given in Lemma 4.3.9. Then there exists a solution (Cj , Ci ) of Sn ' Bn satisfying 
the inequalities (fi ' �j) � (Cj , Cj) � (Ci '  Ci) .  
Proof 
We first consider the case with Dj, !lJj > O. 
It has been shown by standard continuity arguments in Theorem 4.3. 1 and by the Agmon-Doulis­
N irenberg and Schauder estimates that (c? ) } and (Ci(k ) }  are uniformly bounded sequences and hence are 
relatively compact in C2•O [Q xA ,  Rn( l ) ]  and C2[A. Rn( J) ]  respectively. 
This implies that there exists subsequences of (c�k) } and (C�k)} which converge in 
C2•O [Q x A , Rn( l ) ]  and C2[/f, Rn(J) ] ,  respectively. 
The other cases follow similarly along the lines of Theorem 4.3. 1 .0 
We note from the counterexample at the end of section 3 . 1 , that comparison thoerems analogous to Theorems 
3 .2. 1 1  and 3.2. 12  do not hold in general in the case of the corresponding steady state or time independent 
problem Sn . Bn . Hence, if there exist (fj , �i ) and (ci '  Ci ) which are lower and upper solutions of the steady 
state monotone system Sn ' Bn or if there exist (fj , C) and (Cj '  Cj ) which are coupled lower and upper 
solutions of the steady state system Sn ' Bn which may possess no monotone property and (Cj, C) is a 
solution of Sn . Bn , then in contrast to the unsteady state problem S n, B n , we cannot assert that (hi ,  �i ) � 
(ci '  Ci) � (ci '  Ci) · 
We have shown in Theorem 4.3 . 1  that if Sn ' Bn is a monotone system, the method of monotone 
iteration is still applicable and shows the existence of at least one solution (Cj, Cj) of Sn ' Bn lying between 
(fi ' �i ) and (Ci '  Ci ) .  We have also shown in Theorem 4.3.3 that if ,5n , Bn docs not possess any monOLOne 
property, then we can stil l  show the existence of at least one solution (Ci, CD of Sn ' Bn lying between (fi ' �i ) 
and (li '  Ci) .  
However, if (Ci, Ci) is a unique solution of Sn , Bn . then comparison thoerems analogous to Theorems 
3 .2. 1 1  and 3.2. 12  may hold in general in the case of the corresponding steady state or time independent 
problem Sn . Bn · Hence. if there exist (fi . C) and (ei '  C) which are lower and upper solutions of a steady 
slate monotone system Sn . Bn or if there exist (fi . �J and (ei .  CJ which are are coupled lower and upper 
solutions of a steady state system Sn ' Bn which may possess no monOLOne property and (Ci. Ci) is a solution 
of Sn . En . then we can assert that (fi . �i ) � (Ci ' CJ � (ei .  Ci ) ·  
Note that in practice, it may only be necessary to find the existence of coupled lower and upper 
solutions when proving uniqueness and existence. Note also that we may simi larly define coupled lower and 
upper solutions to the time dependent problem Sn. Bn and prove existence along the lines of Theorem 4.3.3 
(Generalised Existence Theorem) for nonmonotone systems. An example of this will be seen in section 6.6. 
Remark 4.3.4 
In this section we have assumed that the boundary conditions (2. 1 04) and (2. 1 .7) are of the Robin type. We 
may treat the Neumann and Dirichlet type boundary conditions similarly by using appropriate theorems for 
linear elliptic equations with Neumann and Dirichlet type boundary conditions (see Remark 4. 1 . 1 .  Remark 
4. 1 .2 and Remark 4. 1 .4). 
In section 404 we shall discuss the relationships between the unsteady state and steady state problems and 
study the stability of solutions obtained by monotone iteration. 
4.4 RELA TIONSHIPS BETWEEN SOLUTIONS OF THE STEADY STATE 127 
AND UNSTEADY STATE PROBLEMS 
4.4 Relatio�ships between Solutions of the Steady State and Unsteady State Problems 
In this section we use relationships concerning the asymptotic behaviour of l inear parabolic equations as 
t � 00 and their corresponding linear elliptic equations to make a statement about the relationships between 
solutions of the steady state problem Sn ' Bn and the unsteady state problem Sn, Bn. 
We may assume at the outset that the systems Sn, Bn and Sn ' Bn are monotone systems, in the sense 
that .fi(t, x, c) and .fi(x, Cj) are monotone nondecreasing in Cj and Fj(t, z, Cj) and Fj(z, C) are monotone 
nondecreasing in Cj for all i. This is not a restriction on the theorems of this section since if this monotone 
property is not satisfied then the functions (Wj ' M'i) defined by Cj = e-KX1 wj and Cj = e-Kzqiti in (4. 3 . 1 )­
(4.3.2), satisfies new systems of the same type but with new functions that are monotone nondeereasing in � 
and VIj. 
If, on the other hand the monotone property is not satisfied by all the other variables, then the systems 
Sn, Bn and Sn ' Bn with general functions .fi and Fj may be imbedded in systems S2n, B2n and S2n ' B2n , 
respectively of the same form where.fi(t, x, Cj) and .fi(x, Cj) are replaced by �(t, X, fk ' c/) and � (x, fk ' c/ ) , 
respectively for the first n(J) dependent variables Cj and by t (t, x, fk ' Cl ) andL (x, fk ' Cl ) ,  respectively 
for the next n(J) dependent variables fj . Also, Fj(t, z, Cj) and Fj(z, Cj) are replaced by F;(/, z, � k ' G;) and 
�(z, �k ' G;) , respectively for the first n (J) dependent variables � and by [j(/, Z, �k ' G;) and f..j (z, �k ' G;), respectively for the next n(J) dependent variables �j 
It has been shown by the imbedding results of section 3 .2 and section 4.2 that solutions of these new 
monotone systems may generate solutions of the original systems and therefore uniqueness, stability and 
existence may be implied in the original systems. We shall furthermore show in this section that such 
monotone systems are also useful in establishing relationships between solutions of the general steady state 
problem Sn ' Bn and the general unsteady state problem Sn, Bn which may possess no monotone property. 
s 
4.4.1 Aymptotic Behaviour of the System Sn. Bn as t � 00. 
Firslly, �e shall assume that the systems Sn ' Bn and '�n ' Bn are monotone. Suppose that we have lower and 
upper solutions (fj , �j ) and (Cj ,  Cj) with (fj , �j )::; (Cj ,  Cj ) for the unsteady state problem Sn ' Bn , for all 
T >  0, and lower and upper solutions (fj , �j ) and (�j ,  Cj) ,  respectively wi� (fj , �j )::;(�, Cj ) for the 
steady state problem Sn ' Bn , such that (fj , �j) � (fj , �j) and (Cj '  Cj) � (S , Gj) as 1 � 00, uniformly for 
(x, z) E {), x A  and Z E if .  Suppose also, that ( dk ) ,  �� k» } and ( c?) , qk » }  are minimal and maximal 
sequences respectively of the system Sn ' Bn and ( f� k) ,  ��k» } and ((alk ) ,  C,? » } are minimal and maximal 
sequences respectively of the system Sn ' Bn . Also, assume that relevant continuity properties are satisfied. 
It is very clear that from Theorem 3 .6.2 and Theorem 4.3 .2, we can by using induction, apply 
Theorem 4. 1 .6 concerning asymptotic behaviours of parabolic equations as 1 � 00 (or results for asymptotic 
behaviour of ordinary differential equations as 1 � 00 (LAKSIIMIIIANTHAM and LEELA [ 157,  pp. 108, 229]) 
if D ·  = H· = 0 or 9). = 0) to the function pairs (c�k ) C� k» (C(k ) C(k » and (C� k ) C.(k » ( 2.(k ) C.(k » and I " -I ' -I ' -I ' -I I ' I ' I ' , ' 
deduce that for all positive integers k: 
and 
( C.(k ) c.(k » -7 (2.(k ) (;<k » , " I ' , ' 
as 1 -7 00 ,  uniformly for (x, z) E Q x A and Z E X. 
Suppose further, that (2i ,  0) is the only solution of Sn ' En lying between (fi ' fi ) and (2i ,  Ci ) .  Since - -;;- " " " - -;: ( k ) " (k ) (Cj , Ci ) is the only solution of Sn ' Bn between (fi ' �i) and (Ci ' Ci) , and the sequences ( f\ ' �j )} and 
4.4 RELA TIONSHIPS BETWEEN SOLUTIONS OF THE STEADY STATE 128 
AND UNSTEADY STATE PROBLEMS 
( a?), C;(k» } ,  both converge unifonnly to solutions of Sf" Bn lying between (fj , �j ) and (aj ,  Cj ) ,  we see 
that by Theorem 4. 1 .7 (or results for the asymptotic behaviour of ordinary differential equations as 1 -+ 00  if 
Dj = Hj = ° or 9Jj = 0), it follows that 
and 
as k-+oo, uniformly for all (x, z) E .Q x A and z E A ,  and we know that 
C-(k) < C", < C",(k) -I - , - , ' 
for all (x, z) E .Q x Ii and Z E If and all positive integers k. 
Thus, given any £ >0, there exists positive integers n(£) and N(£), independent of (x, z) E .Q x A and 
z E If ,  such that 
I C(, k) (X, z) - c,, (x, z)1< E ,  - - 2 
-(.I: ) - e IC ,  ( z )  -C .  (z)1< --, -, 2 ' 
";;"(.1: ) ";;" £ ICj (x, z) - Cj (x, z)I <- ,  
- - £ IC?) (z) -Cj (z)1< 2 '  
whenever k � 1'1(£), N(£). 
2 
Further, there exists r(£, n(£» independent of (x, z) E .Q x A and r(£, N(£» independent of z E lf, such that 
I (n) ( ) -(n) ( )1 £ C,' t, x, z - C,' x, z <- ,  - - 2 
(N) - (N) £ IC,' (I, z) - C,' (z)I <- ,  - - 2 
I -(n) ( ) ";;"(n) ( )1 £ Cj I , X, Z - Cj X, Z <- ,  
2 
1c,(N) (t z) - C(N)(z)1< E , ' , 2 '  
whenever 1 > r(£, 1'1(£» , r(£, N(£». 
Therefore, 
I (n) ( ) ";;" ) fj I, x, z - Cj (X, z I< £ ,  
I��N) (I, z) - Cj (z)I< £ , 
I-(n) ( ) ";;" ( )I Cj I, x, z - Cj x, Z < £ , 
-(N) -ICj (I, z) - Cj(z)I < £ , 
whenever t > r(£, n(£» , r(£, N(£» , which implies that 
4.4 RELATIONSHIPS BETWEEN SOLUTIONS OF THE STEADY STATE 129 
AND UNSTEADY STATE PROBLEMS 
-;; ( ) (n) ( ) < - ( ) < -(n) ( ) -;; ( ) -E + Cj x, Z < fj I, x, Z _ Cj I ,  x, Z _ Cj I, x, Z < E + Cj x, Z , 
whenever t >  -r(E, n(E» , 1(10, N(E» , where (Cj '  G;) is the solution of the system Sn, Bn obtained by  
monotone iteration in Theorem 3.6. 1 (Generalised Existence Theorem). 
By  applying Theorem 3 .6. 1 for arbitrarily large T, we can show that (Cj ,  C;) exists for all t � O. 
Therefore 
as t -700, uniformly for (x, z) E [). x A and Z E X. 
Thus, under the given condition�, the existence of  exactly one solution to the steady state problem Sn ' En 
lying between (£j , �j) and (� , Cj) implies that, for any initial value (Cj.o ,  Cj,o ) lying between (fj , �j ) and 
(Cj , Cj) at t = 0, the unique solution (Cj, Cj) of Sn, Bn (for arbitrarily large 'J) will tend to the steady slate 
solution (Cj, Cj ) of Sn ' Bn as t -7 00, uniformly for (x, z) E [). x A and Z E X. 
From the uniqueness of  the steady slate problem we see that the imbedding results of  section 3 .4 and 
section 4.2 apply and this shows that this relationship will hold for the general systems Sn, Bn and Sn ' En 
which may possess no monotone property. 
4.4.2 The Stability of Solutions Obtained by Monotone Iteration 
We shall consider the stability properties of the solutions of the system Sn ' En obtained by monotone 
iteration. Let us consider the following system in the general system Sn, Bn, where the fluid velocity 
distribution u(z) and the inlet fluid concentration Cj, 1 are independent of Lime. 
aCj _ D;'V;cj = !;(x, Cj) in (0, 1lx[).xA, at 
�� = ° on (0, TJxa.f2lxA, 
Dje;; = /lj(Cj - Cj) 011  (0, 'J'lxJ.f.l2xA, 
aCj _ 9J:V2c. + u · vc. + J D· aCj = F (z C -) in (0 TlxA at I I I an2 I an I '  J ' • J , 
vICj + fJ)jdfj = VICj I on (0, TjxaAI , anI ' 
�j = ° on (0, TJxaAa, a = 2, 3,  ana 
Cj = Cj,O in [).xA, at 1= 0, 
Cj = Cj,o in A, at t= 0. 
(4.4.1) 
(4.4,2) 
(4.4.3) 
(4.4.4) 
(4.4.5) 
(4.4.6) 
(4.4.7) 
(4.4.8) 
Note that the solutions of Sn ' Bn are time independent solutions of the system (4.4. 1)-(4.4.8). We shall 
show that with the monotone iteration methods given in section 4.3, it is not possible to obtain unstable 
solutions. On the other hand. each solution obtained by monotone iteration is asymptotically stable at least 
from above or below and if solutions of Sn ' Bn are unique, the solution obtained by monotone iteration is 
4.4 RELA TIONSHIPS BETWEEN SOLUTIONS OF THE STEADY STATE 130 
AND UNSTEADY STATE PROBLEMS 
asymptotically stable both from above and from below. As earlier. we shall assume that the systems Sn . Bn 
and (4.4. 1 )-(4.4.8) are monotone systems. 
We have seen in section 3.6. that there may be geometric conditions on Ii and Fj which guarantee the 
existence of lower and upper solutions to the general system Sn. Bn. We have also seen in section 4.3 that 
there are geometric conditions onli and Fj which guarantee the existence of lower and upper solutions to the 
system Sn . En ' We now show that the existence of lower and upper solutions to the system Sn. En can imply 
the existence of lower and upper solutions to the special slstem (4.4. 1 )-(4.4.8) in Sno Bn. 
Suppose that (fj . qj) is a lower solution and (�. Cj) is an upper solution with fj � 2j on .Q x A and 
qj � Cj on A o�the system Sn. Bn and (Cj . Cj ) is a solution of the system (4.4. 1)-(4.4.8). where fj �Cj,O �� 
and qj � Cj,o � Cj . The functions 
and 
satisfy: 
and 
dfj 2 -
--at-D/Vx fj � h (x. £j ) III (0. 11x.QxA. 
dc-
d� � 0 on (0. Tjxd.(JjxA. 
dc D---=-� H· (C-c· )  on (0 11xdf22xA I an , -, _I ' , 
dC -=-- 9JV2 C+ u . VC+ lf .j (C- c ·) � F(z C . ) in (0 TJxA dr • _. _. .  iJQ2 _. • . , -) • • 
dCj D V2 - > � ( - ) - (0 11 r>., --at - j x Cj - Jj X. Cj III • x')"AA. 
d� - _ 
. 
Dj dn � Hj (Cj- Cj) on (0, 11xdf22xA, 
de 2 - - J - --' - 9JV C+ u - VC+ H (C- c »  F (z C .) in (0 T]xA at " " i)il2 " - , ' J • • 
4.4 RELATIONSHIPS BETWEEN SOLUTIONS OF THE STEADY STATE 131 
AND UNSTEADY STATE PROBLEMS 
�
Ej 
� ° on (0, nxaAa, a = 2, 3 ,  
ona 
respectively, so by Theorem 3 .2. 12  (Generalised Strong Comparison Theorem), since fj ,  (,j , aj and Cj are 
independent of t, 
dC' d('j da· dC, --=.!.. = _ = _I = _1 = 0 
at at at dt 
' (4.4.9) 
and the inequalities 
(fj , ('j) $  (Cj , Cj) $ (aj ,  Cj ) , (4.4.10) 
hold for all t > O. 
Note that in general, the existence of both lower and upper solutions to the special system (4.4. 1)-(4.4.8) in 
Sn, Bn does not imply the existence of both lower and upper solutions to the system Sn ' Bn ' 
In particular, if Cj 0 = Cj in nxA at t = 0 and Cj 0 = Cj in A at t = 0 or if Cj 0 = aj in .QxA at t = ° and 
_ t - t - I 
Cj,o = Cj in A at t = 0, we have Ihe following theorem. 
Theorem 4.4.1 
If (fj , ('j ) is a lower solution of the system Sn ' Bn , then the solution (Cj ' Cj ) of the system (4.4.�)­
(4.4.8) with initial data Cj.o = fi in UxA and Cj,o = �'j in A is monotone nondecreasing in I. If (
ai ' Ci )  
is a upper solution of the system Sn ' E,!) , then the solution (ci '  Ci) of the system (4.4. 1 )-(4.4.8) with 
initial data Ci,O =aj in .QxA and Ci,O =
Ci in A is monotone nonincreasing in t . If (fi ' ('i )  is a lower 
solution and (ai ,  Ci ) is a upper solution of the system Sn ' Bn and (Cj , Cj ) is the solution of the system 
(4.4. 1t(4.4.8) with initial data either Cj,O = fi in .QxA an!! Cj,o = �j in A or ci,O = Ci in .QxA and Ci,o = Ci in A,  then the inequalities (fi '  �j ) $  (ci ' CJ $ (ai '  Ci )  holdfor all t .  
Proof 
Firstly consider the case where (fj , �i ) is a lower solution of the system S n ' En ' and (Cj ' Cj ) is a solution of 
the system (4.4. 1 )-(4.4.8) with initial data Cj,O = fj in nx/\ and Cj,o = �j in A. We see that if (Cj , Cj ) exists 
then it satisfies 
dCj 
_ D/'V;Ci = Ii (x, c,' )  in (0, l1x.QxA, dt 
�� = ° on (0, TJXd.QIXA, 
Dj�� = Hj(Cj - c) on (0, 11 x d.Q2xA , 
aCj - !D.'V2c + u ''Vc +f o. dCj =F(z C) in (0 nxA dt 1 1 1 a!l2 1 dn I ' ,  ' , 
VtCj + flJ/fj = VtCi t on (0, T]XdAt , on l ' 
(4.4.11) 
(4.4.12) 
(4.4.13) 
(4.4.14) 
(4.4.15) 
4.4 RELATIONSHIPS BETWEEN SOLUTIONS OF THE STEADY STATE 132 
AND UNSTEADY STATE PROBLEMS 
ac. � = ° on (0, T]xaAa, a = 2, 3 ,  ana 
Cj = fj in .axA, at t = 0, 
Cj = �j in A, at t = 0. 
(4.4.16) 
(4.4.17) 
(4.4.18) 
The equations (4.4. 1 1)-(4.4. 18) are invariant under the transformation t -7 t + h, so let us consider the 
functions Cjh (l, x, Z) = Cj (1 + h, X, z) and Cjh (l, z) = Cj (l + h, z) with initial data Cjh (O, x, z) = cj (h, x, z) ? fj 
and Cih (O, z) = Cj(h, z) ? �i ' We therefore have the following equations 
aCjh 
_ DjV�Cjh = h ex, Cjh ) in (0, l1x.axA, at 
aCjh _ Dj an - Hj(Cjh - Cjh) on (0, T]xa.a2xA, 
aCih _ �V2Ch + u ·  VCh + J D· aCjh = F (z Ch) in (0 T] x A al I I I u!12 I an I '  ) ' , 
(4.4.19) 
(4.4.20) 
(4.4.21) 
(4.4.22) 
(4.4.23) 
(4.4.24) 
(4.4.25) 
(4.4.26) 
If we can show that Cjh (l, x, z) ? Cj (l, x, z) in !2 x A and Cjh (l, z) ? Cj (l, z) in If at or near 1 =  0, then the 
inequality holds for all greater I by Theorem 3 .2. 1 2  (Generalised Strong Comparison Theorem) noting that!; 
is assumed to be monotone and therefore need not be redefined. The continuity properties of the functions 
Cjh (t, x, z) and Cjh (t, z) near I = ° are satisfied by studying the behaviour of the solution near 1 = 0. 
Therefore Cj(1 + It, x, z) ? Cj (l, X, z) in !2 x A and Cj (1 + h, z) ? Cj (l, z) in If and hence C j is 
monotonically nondecreasin�Jn I and Cj is monotonically nondecreasing in t .  
The case where (2j ,  Cj )  is  an upper solution of the system �n ' Bn , and (Cj , Cj) is  the solution of the 
system (4.4. 1 )-(4.4.8) with initial data Cj O = � in.axA and Cj O = Cj in A is treated similarly. . , -
We see from (4.4. 10) that if (fj , �j) is a lower solution and (� , Cj )  is a upper solution of the 
system Sn ' En and (Cj ' Cj ) is the solution of the systel!! (4.4. 1 )-(4.4.8) with initial data either Cj,O = fj in 
.axA and Cj,o = �j in A or Cj,O = 2j in .axA and Cj,o = Cj in A, then the inequalities (fj , �j ) � (Cj , Cj ) � 
(2j ,  Cj )  hold for all 1 .0 
We shall now prove the fol lowing theorem. There are analogous proofs given in TEMME [282, p.67] and 
SATTINGER [258, p.36] for scalar parabolic and elliptic equations. 
4.4 RELATIONSHIPS BETWEEN SOLUTIONS OF THE STEADY STATE 133 
AND UNSTEADY STATE PROBLEMS 
Theorem 4.4.2 
Consider the system (4.4. 1 )-(4.4.8) .  If (fj , qj )  is a lower solution and (aj ,  Cj ) is a upper solution of 
the system Sn, En ' then under the assumptions of Theorem 3.6. 1 ,  
lim Cj (t, x ,  z) = Cj (x, z )  and lim Cj (t, z) = Cj (z) exist ,  
t�co t�oo 
where 
Furthermore, (Cj , Cj ) is equal a.e. to a classical solution of the steady state system Sn, En ' 
Proof 
By the monotone iteration procedure we can construct, for any T >  0, a regular solution to the system 
(4.4. 1 )-(4.4.8) for 0 � t � T, so the solution exists for all t > 0 and satisfies (4.4. 1 0). We may start the 
iterations with a lower solution (fj , qj )  and from Theorem 4.4. 1 ,  since Cj (t, -:: z) and Cj (t, z) are monotone 
nondecreasing with t and are bounded below and above by (fj , qj )  and (aj , Cj) respectively, 
lim Cj (t, x, z) = Cj (x, z) 
t�oo 
and 
lim Cj (t, z) = Cj(z) , 
t�co 
exist, where 
For the rest of this proof, we shall only look at the case when Dj, �j > 0 for all i. The other cases follow 
from standard results on ordinary differential equations and first order partial differential equations. 
First we prove that (Cj , Cj ) is a weak solution of the system Sn , En ' i .e. , 
and 
C E U[A Rn(J)] v2C· - u ·  V C· E U[A Rn(J)] 1 , , , I , , 
(4.4.27) 
(4.4.28) 
(4.4.29) 
(4.4.30) 
where we shall assume that A is considered to be a parameter space in (4.4.27) and (4.4.29) and where we 
use the inner product notation (-,) to denote the usual L2(D) inner product [or real functions, i.e., 
(u,V) = fDuvdx . 
Equations (4.4.29) and (4.4.30) arc equivalent to 
4.4 RELATIONSHIPS BETWEEN SOLUTIONS OF THE STEADY STATE 134 
AND UNSTEADY STATE PROBLEMS 
(Cj ,-V;*�j )-(fi (x, �), �j) = 0, 
(Cj ,  (_V2 + u · V + Hj.9l)* Ej-(F;(z, G)+HjJ  Cj , Ej) = 0, u an2 
where -V;* is the adjoint operator of -V; and (-V2 + u · V +Hj.9l)* is the adjoint operator of  
-V2 + u ' V +Hj.9l. Note that the Laplacian operators -V; and -V2 are self adjoint. 
The function (Cj ' Cj ) satisfies the system Sn ' Bn . Taking the inner product w ith the functions �j E 
CO'[.Qx A, Rn(/)] and Ej E CO'[A , Rn(J)] , we obtain 
aCj 2 (Tt, �j ) +(-V xCj ,  �j)-(fi(x, CJ ) ,  �j) = 0, 
( a�j , Ej)+(-V2Cj + u ·  VCj+Hj.9lCj, Ej)-(F;(z, S )+ Hj Jan2 Cj ,  Ej) = 0, 
and partial integration of the second terms results in 
aCj 1= 2* 1= 1= (ar' ,:> j ) +(Cj, -Vx ,:>j )-(fi (x, u)' ,:>j) = 0, 
( aa
Cj , EJ+(Cj, (_V2 + u · V + Hj.9l)* Ej )-(F; (z, C;j)+Hi f Cj , Ei) = 0,  I v an2 
This inequality holds for ali i >  O. So we have 
1 JT ()Cj 1 JT 2* 1 JI' - - . 1 -- C -V  . t-- . x c ·  . t= ° T 0 ( at ' �I)d T 0 ( . . X �I)d T 0 (fi(  , (1) '  �I)d , 
1 1' ()C . 1 1' 1 1' _ r  (_I B)dt+- r (C- (-V2 +u ·V+J-I. .9I)*B)dt=- r  (F (z C- )+H'f c ·  B)dt = O . TJo at ' 1 TJo " 1 1 TJo 1 ' J  1 anz " 1 
Now let T --700; then 
1 IT 2 * ' 2 * - (C (-V + u · V + J-I.SlI) ;:;'·)dt --7 (C. (-V + u · V + H SlI) ;:;'. ) T 0 I ' £ -" , , -" 
[rom the Lebesgue dominated convergence theorem and the uniform boundedness on the solution (Cj, Cj). 
Similarly, 
and 
..!. rT( aCj 1= .)dt = ..!. rT�(c 1= ')dt = (cj (T, x, z), Sj ) - (Cj (O, x, z), Sj ) --7 0  T Jo at ' �I T Jo at I '  �I T ' 
4.4 RELATIONSHIPS BETWEEN SOLUTIONS OF THE STEADY STATE 135 
AND UNSTEADY STATE PROBLEMS 
Therefore, we get in the limit, as T � 00, that 
( Cj , (-V2 + u . V + f(.9'I)* Ej)-(F; (z, Ci)+ l1jf cj , Ej) = O, 
oJ a{� 
for each �j E CO' [.Q x A ,  Rn(l) ]  and Ej E Co [A , Rn(J) ] .  
Now we  have to show that (Cj , Cj ) i s  a regular solution o f  of the system Sn, Bn . First, we  note that Cj is 
uniformly bounded in .QxA and Cj is uniformly bounded in A ,  thus CjE U [.Q x A ,  Rn(/ ) ]  and 
Cj E Lq [ A ,  Rn(J» ) .  Then by Lemma 4. 1 .7 ,  hex, Cj )E Lq [.Q x A  x Rn(/) , Rn(/» ) and F; (z, Cj)+HjS Cj E aD2 Lq [A x Rn(J) , Rn(J» ) .  From Lemma 4. 1 .4, q may be chosen to be identical in both cases. 
Consider the boundary value problem 
- DjV;Wj = hex, Cj) in .QxA, 
oW _I = 0 on o.QlxA, 
on 
ow ' D -' + H w· = C  on o.Q2xA 1 on 1 1 1 , 
- 9>.V2W + u · VW + /-J.sf.W. = F (z 6. )+ 11 · 1  (;. in A 1 1 1 1 1 I '  J 1 aD2 1 , 
oW ' VIWj + 9! � = Cj I on OA l , ani ' 
oW � = O  on oAa , a = 2, 3. 
ana 
(4.4.31) 
(4.4.32) 
(4.4.33) 
(4.4.34) 
(4 . •  35) 
(4 . •  36) 
By Theorem 4. 1 .4, the boundary value problem (4.4.3 1) -(4. 4-.36) has a unique solution (Wj ,  Wi),  where 
Wj E Wq2 [.Q , R
n(/ » )  (with z treated as a parameter) and w,. E Wq2 [ A ,  R
n(J) ] .  From Lemma 4. 1 .5 ,  q may be 
chosen to be identical in both cases. 
Let g be the mapping (hat associates with each right hand term of (4 . l/.3 1 )  the unique solution Wj and 
G be the mapping that associates with each right hand term of (4.4.34) the unique solution Wj. The solution 
of (4.3.3 1)-(4, 4..33) may be written as 
Wj = -gfi(x, c) 
and the solution of (4.4.34)-(4.)).36) as 
Wj = -G( F;(z, Ci )+ l1j J Cj ) . 
v ail2 
(4.4.37) 
(4.4.38) 
For Cj fixed, the operator g:U � Wi associating with each admissible right-hand term /; the uniqu�ly 
determined solution Wj, is the sum of a constant operator and a bounded linear operator. Similarly, for Cj, 1 
fixed, the operator G:Lq � Wq2 associating with each admissible right-hand term Fj + Hj fa Cj , the uniquely . D2 
4.4 RELATIONSHIPS BETWEEN SOLUTIONS OF THE STEADY STATE 136 
AND UNSTEADY STATE PROBLEMS 
detennined solution Wj, is the sum of a constant operator and a bounded l inear operator. Thus g and G are 
continuous. 
We denote these operators by g and G because of i ts connection with the Green's function of 
(4.4.3 1)-(4.3.36); G is the inverse of the elliptic operator L with boundary conditions. We will simply call G 
the inverse of the elliptic operator L. 
For q = 2/( I-a) , Wj E Wq2[.Q , Rn(l)] (where A treated as a parameter space) and \-Vi E Wq2[A , Rn(J) ] 
and we may apply Theorem 4.1 . 1 (Imbedding Theorem) to obtain Wj E C1+a [.Q , Rn(l) ]  (where A is treated 
as a parameter space) and Cj = Wj E C1+a [A ,  Rn(J) ] . From Lemma 4. l .3, a may be chosen to be identical 
in both cases. Thus we have 
and 
= (G(F (z C ) + H· f  C. ) (-V2 + u , V + H..9f)* ;::· )  , ' J  , aDz ' , , -, 
= «F;(z, Cj) + Hj faDz Cj ), G* (_V2 + U ·  V + Hj..9f)* Ej » 
= «F;(z, CJ. ) + HjJ Cj ), Ej ) aD2 
- 2 * -= (Cj ,  (-V + u · V + Hj..9f) .!:.j ) ,  
where g *  and G* are the inverses of -V; and _V2 + U ·  V + Hj..9f respectively (see RIESZ AND NAGY, [248, 
p. l 04]) .  Hence 
and thus 
( - -. _t72* * , )  - ( ,_ -, _ , ) - 0 1.-1 , COO [ n  A Rn(J) ]  W, c" v x g 17, - w, c" 17, V 17, E 0 �.;: x ,  , 
because of the invertiblity of -V; and _V2 + U ·  V +Hj.s1 and thus of -V;* and (_V2 + u ·  V + Hjs()* . 
Thus Cj = Wj and Cj = Wj almost everywhere and (Cj , Cj) is a weak solution of the system Sn ' Bn . Bul Wj 
and Wj are continuous, so modifying Cj and Cj on a set of measure zero if necessary, we get Cj = W j E 
C1+a [.Q , Rn(l) ]  and Cj = Wj E C1+a[A , Rn(J) ] . Again pulling 
Wj = -gfj(x , 9 (4.4.39) 
and 
(4.4.40) 
where g:Ci+a[.Q ,  Rn(J) ] � C2+a[.Q , Rn(J ) ]  (with A treated as a parameter space) is the sum of a constant 
operator and a linear operalor which is bounded by the Schauder eSlimales in Theorem 4. l .3, thus g is 
continuous. Similarly, G: CI+a [A, Rn(J) ] � C2+1X [A, Rn(J)] is Lhe sum of a consLanL opera Lor and a linear 
operalor which is bounded by the Schauder estimales in Theorem 4. 1 .3, thus G is continuous. 
4.4 RELATIONSHIPS BETWEEN SOLUTIONS OF THE STEADY STATE 137 
AND UNSTEADY STATE PROBLEMS 
Therefore by Theorem 4.1 .2, we obtain that Wi E C2+a [.Q ,  Rn(l») and "'i E C2+ a [ J\ ,  Rn(J») and the 
same argument that Ci == Wi and Ci =="'i  finally results in Ci E C2+a[.Q , Rn(l») (with J\ treated as a parameter 
space) and Cj E C2+a [ J\ , Rn(J» ) .  By  the same arguments in Lemma 4.3.5, it can be shown that Cj E 
C2+ a• a [.Q X J\ ,  Rn(l») and thus (Cj, Ci) is equal a.e. to a c lassical solution of of the system Sn' Bn as 
required. 
_ 
A similar proof may be obtained if we start the iterations with an upper solution (2j ,  Cj ) .0  
We now show that the solution (fj ' �) o f  Sn' Bn obtained bl.. starting the iteration a t  (fj , �j) ,  is 
asymptotically stable from below and similarly, the solution (2; , Cj ) of Sn ' Bn obtained by starting the 
iteration at (2j ,  Cj ) ,  an upper solutio� of Sn ' Bn , is asymptotically stable froEI above. 
In the special case that (S , Cj ) = (fi ' �j ) '  we will have that (2j ,  Cj) ,  (fj ' �j ) are asymptotically 
stable both from above and from below. 
We have seen that each solution (s., Cj ) of the system (4.4. 1 )-(4.4.8), where fj � ci,O � 2j and �i � Cj.o � Cj 
satisfies (fj , �i )� (Cj '  Cj ) � (2j ,  Cj ) . 
Let (Cj ,  C; )  be the solution of the system (4.4. 1 )-(4.4.8) with initial data (C(O, x, z), C(O, z» = 
(2j ,  Cj ) and let (fj ' �) be the solution of the system (4.4. 1 )-(4.4.8) with initial data (f(O, x, z), QQ, z» = 
(fj , �j ) .  By Theorem 3 .2. 12 (Generalised Strong Comparison theorem), each solution (Cj , Cj ) of the system 
(4.4. 1 )-(4.4.8) satisfies 
In particular, (fj ' f), satisfies 
(fj (X, z), �j (z» � (fj (t , x, z), fj (t , z»� (UX, z) , �j(z» , 
where (fj (x, z) , �j(z» is the solution of the system Sn' Bn , which is obtained by starting the monotone 
iteration at (fj (x, z), �j (z» since (L (x, z), �j (z» also satisfies the system (4.4. 1 )-(4.4.8). 
So (fj ' fj ) is bounded above and monotonically nondecreasing in t from Theorem 4.4. 1 ;  thus 
lim (fj (t,x,Z),hj (t, Z» exists and is a solution of the system Sn ' Bn by Theorem 4.4.2. From Theorem 4.3 . 1  I�oo ... 
(Generalised Existence Theorem), we see that (fj (x, z) , fj (z» is a minimal solution. Therefore, we have, 
Each solution (Cj ,  Cj ) with initial data (fj , �j )$ (Cj,O, Cj,o) $ (fi ' �j ) satisfies (fj ' � ; ) $  (Cj , Cj ) $ (fi ' �i ) 
and so we have proved: 
Lemma 4.4.1 
If (Ci ' Ci ) is a solution of (4.4. 1 ) -(4.4. 8) with initial data (fi' �j ) $ (Cj.o , Cj,o ) $ (fj ' �),  t hen  
If (Cj , Cj ) is a solution of (4.4. 1 ) -(4.4.8) with initial data (2j ,  Cj) � (Cj,O ' Cj, o )  $ (2j , Ci ) ,  t h en  
lim (Ci ' Ci ) = (2i , Ci ) · 
I�oo 
I  (fj ' f) = (aj ,  Ci) ,  then (fi ' fi ) '  (2j , Ci ) is asymptotically stable from above and from below. 
Corollary 4.4.1 
4.5 NOTES AND COMMENTS 138 
If there exists a lower solution (fj, �j) a!!..d an upper solution (2j ,  Cj) of the system Sn, En ' and if 
there is only one solution (fj , �) == {2j ,  Cj ) of the system Sn, En such that (fj , tj ) � (fj , �j ) and 
(aj ,  C;) � (aj ,  Cj ) ,  then this solution is an asymptotically stable equilibrium solution of the syst!.m 
(4.4. 1 )-(4.4.8) and ea.£.h solution of (4.4. 1)-(4.4.8) with initial data (fj , �j ) � (Cj,O , Ci,o ) � (2j ,  Cj ) 
tends to (fj , �) , (2j ,  Cj) as t � 00 .  
So, with the monotone i teration methods given i n  section 4.3, i t  i s  not possible to obtain unstable solutions. 
On the other hand, each solution obtained by monotone iteration is asymptotically stable at least from above 
or below. 
If (fj , �j) == (2j ,  Cj ) ,  the unique solution obtained by monotone iteration is asymptotically stable both 
from above and from below. We have seen in section 4.3 that given an upper and a lower solution of the 
system Sn , En ' there exists a solution of the system Sn , En between these upper and lower solutions which 
can be constructed by monotone iteration. On the other hand given an upper and a lower solution and that 
there exists a unique solution of the system Sn, En between these upper and lower solutions, this unique 
solution can be constructed by monotone iteration and this unique solution obtained by monotone iteration is 
asymptotically stable both from above and from below. We have shown that uniqueness implies asymptotic 
stability for these class of problems. Note that this will not be true in general (LACEY [�O]) unless both 
upper and lower solutions exist. Also, from the boundedness of our solution (fj , �), (ij ,  Cj) of the system 
Sn , En and the boundedness of (Cj, Cj) of solutio�s of (4.4. 1 )-(4.4.8) for all time, it fol1ow�that the 
asymptotic stability and uniqueness of (£j , �j ) ' (2; , Cj) imply the global stability of (fj , �) , (ij ,  Cj ) in the 
system Sn, En ' 
Remark 4.4.1 
In this section we have assumed that the boundary conditions (2 . 1 .4) and (2. 1 .7) arc of the Robin type. We 
may treat the Neumann and Dirichlet type boundary conditions similarly by using appropriate theorems for 
linear relationships concerning the asymptotic behaviour of linear parabolic equations as t � 00 and their 
corresponding linear elliptic equations with Neumann and Dirichlet boundary conditions (see Remark 4.1 .6). 
In all the cases of this section which involve the uniqueness of the steady state problem, we see that the 
imbedding results of section 3 .2 and section 4.2 apply and therefore all these results may be shown to hold 
for the general systems Sn, Bn and Sn, En which may possess no monotone property. 
4.5 Notes and Comments 
As with Chapter 3, the imbedding results in section 4.2 may give a lot of useful information about solutions. 
However as we see in this chapter this information is more restrictive and relies on proving the uniqueness of 
solutions to the imbedded system. 
To obtain existence results in section 4.3 , where the nonlinearities obeyed no monotone property, we 
redefined our nonlinear reaction functions so as to obtain a priori bounds on our solutions and classical 
results were employed in order to obtain existence. This is a simple way of obtaining a priori bounds on our 
solutions but it is certainly not the only way. We could alternatively have employed functions that control the 
growth of the nonlinear reaction functions, thereby obtaining a priori bounds on our solutions (FI1ZGIBBON 
and MORGAN [9 1 ]) .  As in our case, classical results could then also be employed in order to obtain 
existence. 
4.5 NOTES AND COMMENTS 139 
In section 4.3, we obtained the existence of solutions to the system Sn ' Bn by monotone iteration. 
This could alternatively have been established by embedding this system as steady state solutions of the 
system (4.4. 1)-(4.4.8) and using results of section 4.4 as t --) 00 to show that solutions of the system Sn ' En 
also exist. This also gives an alternative to the monotone iterative methods for obtaining minimal and 
maximal solutions (CHAN [64]). 
It is worth noting that in the definition of lower and upper solutions for scalar elliptic equations, if the 
nonlinearity term/is monotone decreasing in c, then the restriction f � C is not required but is a consequence 
of the differential inequalities (PROTIER and WEINBERGER [234] and VARMA and STREIDER [ 1985]). It is 
not clear how this relates to systems of elliptic equations. 
Systems of nonlinear elliptic boundary value problems arise in many applications such as multiple 
chemical reactions that take place in an isothermal or nonisothermal catalyst pellet and simple models of 
tubular chemical reactors (COHEN [73 ,  75] and COHEN and LAETSCH [74]). Some of these systems of 
elliptic boundary value problems possess multiple solutions. This thesis does not examine mUltiple solutions. 
However there are many situations in particle reactors where this does occur (CHI et al. [70], LUss and 
AMUNDSON [ 173] ) and interesting cases occur with the problem Sn ' En if it has several distinct solutions or 
if the micro structures are in the presence of several distinct steady states (ARONSON and PELETIER [23]). 
Simple criteria for the existence of lower and upper solutions for elliptic scalar equations is discussed by 
AMANN [6, 9], as is nonexistence and general uniqueness theorems and this theory has also been extended in 
deriving multiplicity results, namely a criterion guaranteeing the existence of at least for example three distinct 
solutions. Some of these results may be generalised to the system Sn' Bn . 
5 
The Linear Problem 
5.0 Introduction 
The nonlinearities in the system Sn, Bn generally reside in the chemical reaction terms, but at dilute 
concentrations, l inear approximations can be good enough. The validity of such linear approximations was 
tested against solutions derived numerically by orthogonal collocation techniques for a system governed by 
nonlinear Michaelis-Menten type kinetics (PARSHOTAM, et al. [226]). 
In this chapter, it is shown that linear systems of convection reaction-diffusion equations for particle 
reactors described in this thesis are shown to be amenable to certain geometrical factorization techniques 
which dramatically reduce the dimensionality of the system. These equations are also amenable to algebraic 
uncoupling transformations which simplify the tasks of obtaining analytic and numerical solutions or 
estimates. 
These same factorization and uncoupling techniques may be also applied to an associated linear 
system for vectors composed of the mean action time variable for each chemical component. These vector 
functions give the time lags for the various chemical outputs of the system during its transition from one 
steady output mode to another, and the mean first passage times and mean particle residence times 
corresponding to tracer pulse inputs of the chemicals. 
In section 5 . 1 ,  a general system of linear equations is described using matrix notation, dimensionless 
parameters and general undefined geometric configurations in order to focus attention on certain structural 
aspects of the equations. Section 5 .2 is concerned with a dominant transitory aspect of the system described 
by a mean action time variable defined for each chemical component in the micro and the macro systems. 
These variables satisfy an associated linear system of equations coupled to the system for the ultimate steady 
state solution and give a measure of the time for the Lransitions from one steady state to another. The ultimate 
steady outputs are given in terms of the final steady state solutions and the time lag constants for the various 
outputs are given in terms of the mean action time functions. 
A geomeLric factorization technique for these general linear systems is developed in section 5.3. For a 
general class of time dependent problems with factorised initial conditions, this factorization is best revealed 
in the Laplace transfonn domain but can be developed in t-space using convolution integrals. 
For steady state problems and for the systems defining the mean action times, a more obvious product 
factorization holds. These equations are also amenable to matrix transfonnations which uncouple the systems 
algebraically when the coupling maLrices are quasisymmetric in that they are the product db of a positive 
nonsingular diagonal matrix d with a symmetric matrix b. This is a common structural feature of linearised 
chemical kinetic cquations (MCNAl3l3 and BASS l189J). 
1 40 
5.1 MATRIX NOTATION 141 
The macroscopic equations are assumed to be convection dominated here. This leads to two 
simplifications. The diffusivity 9J in the macroscopic equations is dominated by dispersion effects and so its 
diagonal elements are the same for all of the chemical components. In these circumstances, we may treat !J) as 
a scalar in the convection-diffusion equations. Secondly, the boundary conditions at output surfaces of the 
bioreactor may then be assumed to be of the Danckwerts type where the normal gradients of chemical 
concentration are all taken to be zero. 
These simplifications allow the macroscopic equations for steady state concentrations and mean action 
times to be algebraically uncoupled by matrix transformations. All boundary conditions except one are 
similarly uncoupled. On the outer boundary of the particles, a linear boundary condition of the fonn (5. 1 .3) 
below is assumed. This equates the flux out of the particles, defined by a diagonal diffusivity matrix D and 
concentration gradients normal to the surface, to the flux across a fluid boundary layer. In the common 
situations where I-I is a scalar multiple of D ,  this equation too remains uncoupled by the matrix 
transfonnations which uncouple the rest of the system. 
5.1 Matrix Notation 
Our reactor occupies a region .1\, z denotes a poinL in this region and u(z) is the solenoidal fluid velocity 
distribution in A The n(J)-vector C(I , z) with components Cj (l, z) describes the concentration of the iLh 
chemical component. In the neighbourhood of any point z in .1\, there is a distribution of particles of various 
sizes and shapes and an active layer described by a region n attached to these particles. 
Points in n are denoted by x and the vector C(I, x, z) has n(J)-components Cj (l , x, z) describing the 
concentrations at time 1 at x in n, which is in the neighbourhood of the point z in A Each Cj is associated 
with the same chemical component as Cj (l, z) is outside (2 if Cj (l, z) exists. 
The diagonal matrix D has positive elements Dj in the ith row and column representing the diffusion 
constant for component i in Q. The reaction-diffusion behaviour in Q for a time span [0, Tj is assumed to be 
given by 
�� - DV ;c - ac = O  in (0, Tjx.Qxi\, (5.1.1a) 
where a is an n(l)xn(l) constant matrix describing the rate of generation of chemical components Cj due to 
reactions with the other components. 
The region {2 is assumed to have an inner boundary a{2! defining the inert particle cores of our 
representative sample, and on this inner boundary we suppose there to be no flux of chemical and hence 
�C = ° on (0, Tjxa{2lxA Oil (5.1 .2) 
The outer boundary a{22 of n permits chemical exchange between the bioparticles and the reactor fluid. A 
boundary layer attached to dQ2 is assumed to allow a chemical flux H(C-c) between .1\ and {l where H is a 
diagonal matrix with terms describing the diffusive boundary layer flux for each component. It is not 
unreasonable in many cases to assume H to be proportional to D and when this is so, we write H = yD 
where r is the scalar constant of proportionality. The linear boundary cOl1lliLion dcrivcd by cl}uaLing Lhese 
fluxes to those in {2 near a{22 is 
dC D an = H(C-c) on (0,T]Xd{22xA (5.1 .3) 
Concentrations C in the macroscopic system are assumed LO satisfy the linear matrix system 
5.2 MEAN ACTION TIMES, TIME LAG CONSTANTS AND 
MEAN RESIDENCE TIMES 
J,C - qJV2C+ u . VC -BC+ I D�C = 0  in (0, T]xA, 
ot aDz on 
142 
(5.1.4) 
where qJ is a diagonal matrix representing diffusion and dispersion effects in A, V2 and U ·  V are scalar 
operators on functions of z, B is an n(J)xn(J) constant matrix describing linear chemical kinetics and the 
integral term gives the chemical flux into .Q at (t, z). 
Let JAl , JA2 and JA3 denote three sections of the boundary of A. Fluid flows in through JA I 
carrying chemical components at given concentrations CI ,  so that 
(5.1.5) 
where VI = -u'nl is the scalar normal fluid velocity into A. Over dA2, there is no transport of fluid or 
chemical, so that 
dC 
J� 
= ° on (0, T]xJA2, (5.1.6) 
and the fluid flows out through the system over JA3 , where DANCKWERTS [79] type boundary conditions 
prevailing at high Peelet numbers are assumed, so that 
�C = ° on (0, TJxdA3. 
on3 
(5.1.7) 
At any time t, the chemical output is q(t, JA3) over dA3, where q is an n-vector whose ith term qj, gives the 
flux of component i over JA3' If VJ = u'n3 is the fluid velocity on JA3, then 
At each point z in A, the inputs to .Q are likewise given by the flux vector q(t, d.Q2, z), where 
q(t, J.Q2 , z) = I D 
dc = I Ii(C - c) = IidC-IiI c ,  aDz dn aDz aDz 
and where d is the surface area of J�. 
(5.1 .8) 
(5.1.9) 
Let Ln denote the linear system of equations and couplings (5. 1 . 1 )-(5. 1 .4), and Bn denote the system 
of external boundary conditions, (5. 1 .5)-(5 . 1 .7) and the initial conditions 
c = Co in flxA, C = Co in A at t = O. (5. 1 . 1  0) 
5.2 Mean Action Times, Time Lag Constants and Mean Residence Times 
Let us consider solutions (c, C) of the system Ln, Bn which start at t = 0 with Co and Co zero, and for which 
the given boundary veclor functions CI are independent of time. As time progresses, these solutions lend lo 
steady state solutions (c, C) , and the chemical fluxes q(t, JA3) ,  q(t, J.Q2, z) which start zero, gradually 
approach constant values q(JA3) ,  q(J.Q2 , z) . 
Let Q(t, dA3) , Q(t, d.Q, z) measure the accumulated fluxes across dA3 and d.Q2 respectively, in the 
lime t, so that 
5.2 MEAN ACTION TIMES, TIME LAG CONSTANTS AND 
MEAN RESIDENCE TIMES 
= q(dA3)t - f�fali3 
V3 [C:\Z) - C(s, z)]ds 
- f v3[C(z)t -3(z») . ali3 
for large t, where 
3(z) == f;[C(z) - C(s, z)]ds . 
Likewise, for large t, 
where 
Q(t, df22 , z) - J D�[c(x, z)t - -r(x, z») . 
aQ2 an 
-r(x, z) == f; [c(x, z) - c(s, x, z)]ds . 
143 
(5.2.1) 
(5.2.2) 
(5.2.3) 
(5.2.4) 
These vector functions Q(t, dA3)and Q(t, df22 , z) have l inear asymptotes for each component for 
increasing time, which intersect the time axis at times tddA3)j and tL(z, d.a2)j called flux time lags. The ith 
component of the vector tL(dA3) gives the time lag 
(5.2.5) 
for the chemical flux of component i over dA3, and the time lag 
(5.2.6) 
for the chemical flux of component i over d!h 
The vectors 'r(x, z) and 3(z) are the mean action time vectors for this problem (MCNABB and WAKE 
[ 1 90, 1 93]) and the following systems of equations are obtained for them from Ln , Bn. 
2 
f 
d-r ' -9)V 3+ u · V3-B3+ D-=C in A, 
ail2 dn 
(J3 
-;-= 0  on dAa, for a = 2, 3, una 
(5.2.7) 
(5.2.8) 
(5.2.9) 
(5.2.10) 
(5.2.1 1) 
(5.2.12) 
5.2 MEAN ACTION TIMES, TIME LAG CONSTANTS AND 
MEAN RESIDENCE TIMES 
where (c, C) are the steady slate solutions of Lno Bn, satisfying 
ac -D-= H(C - c) on a.Q2xA,  an 
2 - - - J ac . - 9)V C + u · VC - BC + D- = O  m A, 
iJil2 an 
- ac VIC + 9) an = vlCI on aAI ,  
ac -;--= 0 on aAa, for a = 2, 3. 
una 
144 
(5.2.13) 
(5.2.14) 
(5.2.15) 
(5.2.16) 
(5.2.17) 
(5.2.18) 
Various physical interpretations of these average times are discussed in some deLail for single component 
systems in McNABB and WAKE [ 190, 1 93] .  
The solution c, C above, corresponding to a step function boundary condition CI on aAI and zero 
initial conditions gives rise to another solution g, G for Ln, Bn given by 
G(l, z) = ac in (0, TjxA, 
at 
gU, x, z) = �� in (0, 71x.QxA. 
(5.2.19) 
(5.2.20) 
This solution is generated by a delta function pulse of particles of strength C I released on aAI at I = O. This 
pulse input can be regarded as a tracer solution giving a picture of various transitory aspects of the system. 
There are a number of natural time concepts associated with g, G which like the time lags for c and C, can be 
expressed as functionals of ,g and r. 
For example, the mean first passage time tt(aA3) for the chemical component i leaving A via aA3 is 
given by 
(5.2.21) 
(5.2.22) 
The mean residence time t; (A) for the ith chemical component in A is given by 
(5.2.23) 
Likewise, the mean residence tinle t;(.Q) at a point z in A is given by 
(5.2.24) 
5.3 GEOMETRIC F ACTORISA nON 
and the mean residence time t; (.Q u A)  for the ith component in !.be whole system is given by 
1 45 
(5.2.25) 
The time lags at aA3 for various chemical components coincide with their mean first passage times from the 
system, whereas the time lags for the fluxes to be established in the region D at any point z in A, and the 
mean residence times of the components in the same region D are different functionals of the mean action 
time vector 'f and the steady state vector c .  
5.3 Geometric Factorisation 
The linear system of equations Lno B" in full geometric generality present a computational, dimensional 
crisis, since the solutions are defined in seven dimensions; three in x-space, three in z-space and one in time. 
This is marginally relieved by considering just steady state solutions. Fortunately the equations for (c, C) 
can be geometrically factorized in the following way. If n = n(J) = n(J), and all c are coupled to C by (5 . 1 .3) 
then the vector c (x, z) can be written as the product 
c(x, z) = O(x)SC(z) , (5.3.1) 
where 0 and S are nXn matrices, 0 is a function of x only and S is constant. If 8(x) satisfies the equations, 
-D''V; O - a O = O  in D, 
ao = 0  on aDl an ' 
D �� = H (S- I - 0) on aD2, 
then (c, C) are steady state solution vectors of L", B", provided C satisfies, 
2 
• • - f ao · - 9)\1 C + u · \1C - BC + (  D-)SC = O  in A, an! an 
ac -a = 0  on aAa, [or a =  2, 3.  na 
(5.3.2) 
(5.3.3) 
(5.3.4) 
(5.3.5) 
(5.3.6) 
(5.3.7) 
The linear problems (5.3 .2)-(5.3.4) and (5 .3.5)-(5.3 .7) arc each in three dimensions only, and the first can 
be solved independently of the second for any given nonsingular matrix S. If D is nonsingular and positive 
definite, then equations (5.3 .2)-(5.3 .4) may be uncoupled by linear matrix transformations if a is 
"quasisymmetric" in the sense that 
a=db , (5.3.8) 
where d is a positive definite diagonal matrix and b is symmetric. Furthermore, if H and D are proportional 
in the sense that 
H=yD, (5.3.9) 
for ra positive scalar, then the boundary conditions are uncoupled by the same transfonnation. 
5.3 GEOMETRIC FACTORISATION 146 
In this case, there is a nonsingular matrix T with transpose T' for which T'd-JDT = J the unit matrix, 
and T'bT = j.1., a diagonal matrix. If we let (J = Tcp and S = y-l, then 
-V�Cp-j.1.cp= 0 in .Q, 
dcp 
dn 
= 0  on d.QJ , 
�: = r(J - cp) on d.Q2, 
(5.3.10) 
(5.3.1 1) 
(5.3.12) 
and hence, cp is a diagonal matrix. A similar geometric factorization works for rand 5, by expressing r in the 
form 
rex, z)= (J(x)S5(z)+ '(x)WG(z) . (5.3.13) 
Equations (5.2.7)-(5.2.9) are satisfied for 'l(x, z) if (J is a solution of (5 .3.2)-(5.3 .4) and '(x) is a solution of 
d' - = 0  on d.QJ 
dn ' 
(5.3.14) 
(5.3.15) 
(5.3.16) 
This system may also be uncoupled algebraically  when the earlier conditions (5 .3.8)-(5 .3.9) prevail, by 
choosing 
(J = Tcp, 
S = T- J , 
W = T'd-I , 
and 
'(x) = T�(x) . 
The matrix cp given by (5.3 . 1 0)-(5 .3 . 1 2) is diagonal and �(x) given by the Helmholtz problem, 
d� 
dn 
= 0  on dQI , 
�� + r� = 0 on df2z, 
(5.3.17) 
(5.3.18) 
(5.3.19) 
(5.3.20) 
(5.3.21 )  
(5.3.22) 
(5.3.23) 
is also diagonal. The coupling term f D dr in equation (5.2 .10) for 5(z) can be expressed in the form Jil2 dn 
(5.3.24) 
and O(x) and '(x) are given by (5.3 .2)-(5 .3 .4) and (5.3 . 1 4)-(5.3 . 1 6) ,  so that 5(z) and G(z) are given by 
coupled systems in the z variables. We note that in the algebraically uncoupled system, 
5.3 GEOMETRIC FACTORISATION 147 
(5.3.25) 
(5.3.26) 
The vector C(z) is given by the boundary value problem (5 .3 .5)-(5 .3.7), and in high Peclet number 
situations, the diagonal matrix 9) may be considered a scalar operator since all the elements are equal along 
the diagonal, each being a dispersive mixing parameter due to fluid motion that is almost independent of Dj. 
Since u ·  V is a scalar operator too, these equations are uncoupled by a similarity transformation 9'which puts 
the matrix B* given by 
into Jordan normal form. Let 
C = 9''/J, 
so that 
9'-IB*9'= r, 
where ris the Jordan canonical form for B*. Then if Cj = !AJj , 
(hJ 
-a = 0  on aAa, for a =  2, 3 .  na 
(5.3.27) 
(5.3.28) 
(5.3.29) 
(5.3.30) 
(5.3.31) 
In this form, each component '/Jj of '/J may be solved separately or iteratively. The equations for .J can be 
treated in a similar fashion, since from (5 .2. 10) ,  (5 .3.24), 
a.J 
-
a 
= 0  on aAa, for a = 2, 3 .  na 
Let .J(z)=9'S, so that 
� 
(i1\ as 0 aA vi': + ;;v- = on I , an 
as 
-a = 0  on aAa, for a =  2, 3. na 
(5.3.32) 
(5.3.33) 
(5.3.34) 
(5.3.35) 
(5.3.36) 
(5.3.37) 
5.3 GEOMETRIC FACTORISATION 1 48 
Once again, the components of the vector E are algebraically uncoupled and may be solved separately, or 
iteratively, depending on the nature of r. 
The geometric factorization described here has counterparts in other applied problems (MCNABB 
[ 1 85]), and can be generalised LO apply to some time dependent problems for which the initial value function 
co(x, z) can be factored. Suppose 
co(x, z)=u(x)cut:z), 
and e ,  C denotes the Laplace transforms 
and 
In the transform variable p, the system Ln, Bn gives 
-DV;e + (p - a)"c = co= u(x)cut:z) in ZxilxA, 
ae 
an = 0 
on Zxa.alxA, 
D�� = 1I(C-C)on £X()!22xA,  
(5.3.38) 
(5.3.39) 
(5.3.40) 
(5.3.41) 
(5.3.42) 
(5.3.43) 
where Z is an appropriate strip of the complex p-plane. Once again we see e(p,  x, z) is factorizable in the 
form 
c = Vi(p, x)SC (p, z) + 1E(p, x)WCU(z) , 
provided lji satisfies the boundary value problem, 
a-
a
� = 0 on zxa.a ) ,  
D alji = !-I(S-l - lji) on  Zxafh on 
and It satisfies tlle boundary value problem 
olt � = O  on ZxoD1 , 
un 
D
olt 
- 0 :l 
on 
+ !-Ire = on Zxufh 
(5.3.44) 
(5.3.45) 
(5.3.46) 
(5.3.47) 
(5.3.48) 
(5.3.49) 
(5.3.50) 
Algebraic uncoupling may be pursued in a similar fashion since pl-a is of the form d(Pd-Lb) and matrix 
transformations can diagonalise pd-Lb and reduce d-1D to the unit matrix simultaneously. 
5.3 GEOMETRIC FACTORISATION 149 
If S(P) is of the form pS* then S-I (P) = ls*- I is the Laplace transform of a constant matrix S*. The p 
problems (5.3.45)-(5.3 .47) and (5.3.48)-(5.3.50) can be inverted and we find V/(I, x) is given by: 
Oil' -DV;V'- aV'== O  in (0, oo)x.Q, 
ot 
�� == 0 on (0, 00 )xo.Q1 ,  
D �� == H(S*-I - 11') on (0, 00)XO.Q2, 
11'= 0 in .Q at l = 0, 
and n(1, z) by 
�; -DV;n-a1r == O  in (0, oo)x.Q, 
on 
- == 0 on (0, oo)XO.QI'  on 
on D 
on 
+ H n == 0 on (0, 00 )X().Q2, 
n = W-Iu(x) in .Q at I = O. 
From equation (5.3 .44) and the observation that p Vi is the Laplace transfonn of all' , we find dl 
rl 011'(1 - s, x) e(l, x, z) = Jo 01 S*C(s, z)ds + n(I ,  x)W'U(z) . 
(5.3.51) 
(5.3.52) 
(5.3.53) 
(5.3.54) 
(5.3.55) 
(5.3.56) 
(5.3.57) 
(5.3.58) 
(5.3.59) 
In the transform space, C satisfies a conventional convection-diffusion equation but in I space, the integral 
term is a convolution inLegral in time, so that C(/, z) satisfies the nonhomogeneous partial differential causal 
integral equation, 
oC 2 J il 02 V'(t-s x) J on' . T-9JV C+u·VC-BC+ D ' S*C(s, z)ds==(- D-)W'U(z) 111 (0, TjxA,(5.3.60) ul aQ2 0 dlon aQ2 on 
Fortunately, a broad picture of the transient behaviour of the system is provided by the mean action time 
vecLors r(x, z) and �(z) and as observed earlier, they saLisfy equaLions which can be geometrically facLored in 
a simple way, and algebmicaUy uncoupled under very general circumstances. 
More detail concerning the nature of tracer distributions gj, Gj can be obtained from higher moments. 
The Laplace transforms If(p, x, z) and G(p, z) of the vectors g, G describing the solution of Ln, Bn, created 
by a delta function pulse of strength C) on oA I at 1 =  0, are given by the system, 
-DV;g + (pl - a)g == ° in Zx.QxA, (5.3.61) 
(5.3.62) 
5.3 GEOMETRIC FACTORISATION 
a- -DJ... = H(G -g) on Zxa.a2xA, an 
2- - - J dg -9JV G + u · VG + (pJ -B)G + D-= O in ZxA, aD2 an 
aa -= 0 on ZxaAw for a = 2, 3.  ana. 
The definitions of g and G lead to the following expansions for small p: 
150 
(S.3.63) 
(S.3.64) 
(5.3.65) 
(S.3.66) 
_ 100 p2t2 p3t3 p2 g (p, x, z) = [ l- pt +----+ . . .  ] g(t, x, z)dt = go (x, z)+pg) (x, z)+-g2(X' z)+ . . .  , (S.3.67) o 2! 3! 2! 
_ p2 G (p, z)dt = Go (z) + pG) (z) +-G2(z)+. . .  (5.3.68) 2! 
Equations (5.2. 19) and (5 .2.20) give the following connections between gn, Gn and functions defined 
earlier. The functions go and Go are equal to the steady state solutions (2, C) satisfying conditions (5. 1 .5) for 
CI since 
100 ae A go (X, z) = - - dt = c(x, z) . o at 
The functions 81 and GI are given by --rand -g respectively, since 
roo dc roo g\ (x, z)  = -Jo tal dt = I t (c - c)JO' - Jo (c - c)d/ = -rex, z) , 
and likewise 
(S.3.69) 
(S.3.70) 
(S.3.71) 
Differential equations for these functions 8m, Gm can be readily derived from the equations for g and G by 
differentiating the system (5 .3 .61) and (5 .3 .64) above with respect to p, m times, and putting p equal to 
zero. For example, the equations for 82 and G2 are 
ag2 = 0 on a.alxA, an 
D �: = J-J (G2 - 82 ) on a.f22xA, 
-9JV2G2 +u ·  VG2 - BG2 + J D a82 = 2g in A, aD2 an 
(S.3.72) 
(5.3.73) 
(5.3.74) 
(S.3.7S) 
(S.3.76) 
5.3 GEOMETRIC F ACTORISA TION 
dG2 "l 
-- = 0 on aAa, for a = 2, 3 .  
dna 
151 
(5.3.77) 
These functions 82 and G2 lead to expressions for variances about the various mean first passage times and 
mean residence times discussed earlier. These equations also allow geometric factorization and algebraic 
decoupling. 
We assumed D was nonsingular in equation (5 . 1 . 1 )  and throughout this section. In many 
applications, there wil l  be some chemical components Cj in Q which are immobile and for which Dj = O. 
Such components can only interact with the fluid concentrations C via the reaction tenn and interacting mobile 
ingredients. A more general linear formulation explicitly recognising the existence of immobile components 
in Q can be formulated by expressing (5. 1 . 1 )  in a partitioned matrix form 
(S.1.1b) 
where all the components of C2 are immobile in Q. 
For these ingredients, Hj like Dj is also zero. In reality, the boundary condi tions (5 . 1 .2) , (5 . 1 .3) 
have no relevance for C2. 
The geometric factorization of this section is still valid for singular positive semidefinite matrices D 
and H but the algebraic uncoupling procedure needs re-examination. Is there a nonsingular matrix l' with 
transpose 'J" for which 
Td-1D1' = J  (5.3.78) 
and 
Tb1' = J.l, (5.3.79) 
where J.l is a diagonal matrix and J is diagonal with clements 1 or O? 
We wish to find matrices Tj, components of a partitioned form compatible with (5. 1 . 1  b) and diagonal 
matrices J.lj satisfying, 
and 
(
1" 
J� 1" )( b 1� b� 
If the symmetric matrix b4 is invertible, this is possible if we choose J.l1 and 1'1 such that 
1'1 (bl - b2 bilb2 )1'j = J.lI ' Tj dj"" IDI7) = I ,  
12 = 0 ,  
13 = -bi1b2 1[ 
and J.l2 and 1'4 such that 
(5.3.80) 
(5.3.81) 
(5.3.82) 
(5.3.83) 
(5.3.84) 
(5.3.85) 
5.4 NOTES AND COMMENTS 
The algebraic uncoupling can then be accomplished as before. 
152 
There may also be some chemical components Cj which are immobile in A and for which q)j = O. The 
geometric factorization of this section is still valid for singular semidefinite matrices q) but the algebraic 
uncoupling procedure has to be re-examinied by expressing (5. 1 .4) in a partitioned matrix form similar to 
(5 . 1 . 1  b). The theory can also be extended to include the case when n(l) :f. n(J). 
5.4 Notes and Comments 
This chapter is adapted from McNABB,  PARSI IOTAM and WAKE [ 194] . 
It is very common to uses tracers (a detectable fluid which has similar properties to the flowing fluid) 
in particle reactors. The most commonly used methods for detecting flow maldistribution (such as "dead" 
zones and "stagnant" zones) utilize tracers (usually responses to impulses of tracers) and tracers can also be 
utilized in studying many other physical characteristics of particle reactors. The tracer-determined residence 
time distribution (RTD) has been utilized in the analysis of many kinds of flow systems, including chemical 
reactors, biological systems and underground reservoirs (ROBINSON and TESTER [250] ,  HANRATTY and 
DUDUKOVIC [ 1 14] .  This time distribution for the fluid in a packed bed reactor for example reflects the 
combined history of the fluid flowing external to the porous particles in the reactor bed and the fluid which 
enters the porous structure. DANCKWERTS [79] popularised the tracer method by developing the mathematics 
necessary to interpret an inlet-outlet tracer experiment. A list of internal residence time distribution functions 
within particles and external residence time distribution functions within the bulk fluid in particle reactors 
from the time of Danckwerts to the present is given by ROBINSON and TESTER 1 2501 .  
In th is chapter we have shown how appropriate Ule definitions of mean first pass sage times and lI1ean 
particle residence times corresponding to such tracer pulse imputs of chemicals within such particle reactors 
are in order to achieve some useful results that rely only on solving uncoupled differential equations . Some 
of the other internal and external residence time distribution functions from l i terature may also be treated in a 
simi lar way. 
In (5 .3.78)-(5.3 .79) we look for a nonsingular matrix T with transpose 1" for which T'd-1DT = J 
and T'bT = j1, where j1 is a diagonal matrix and J is diagonal with elements 1 or O. This problem is 
equivalent to the simultaneous reduction of a pair of Hermitian forms one of which is nonnegative definite. A 
necessary and sufficient condition for the existence of such a transformation as well as its method of 
construction is given by RAO and MITRA l239, p . 120- 134J . 
The geometric factorization in this chapter has its analogy in uncoupling systems of linear first order 
differential equations by reducing Ulem to Jordan Canonical form. There is also considerable work on 
factorisation techniques for almost linear systems of first order differential equations. The geometric 
factorization in th is  chapter may in a similar way be genera l ised to almost linear systems of ell iptic and 
parabolic equations. 
6 
Examples 
6.0 Introduction 
This thesis w ith all its theory would not be complete without some suitable examples. In this ChapLer, we 
shall bring some of the theory developed in this thesis and look at some specific examples. The examples we 
shall present will generally be those that have motivated this study. 
Since these sections are written as completed published or publishable papers, there may be a 
repetition in the model development and problem description. 
In section 6.1  we look at an example where a kinetic model is proposed for a surface supported 
biological film (bioparticle) employed in many biological processes. The equations that govern kinetic and 
diffusion controlled substrate uptake by the attached organisms are invariably nonlinear and analytical 
solutions if any are impossible to find. It is therefore desirable to determine approximate analytical solutions, 
failing this, bounds for the exact numerical solution. This example represents an attempt to provide 
increasingly beLter bounds to such equations by a linearisation technique. An iterative scheme relying on the 
maximum principle is presented for obtaining upper and lower bounds to solutions to resulting non-linear 
reaction-diffusion equations. In particular, a spherical bioparticle with Michaelis-Menten type of reaction 
kinetics is considered. Its application to more general equations is also discussed. These bounds are in good 
agreement wilh the numerical solutions obtained by shooting and finite difference procedures. The method, 
which can easily be generalised to other geometries, is relatively simple to use and converges rapidly to a 
very good upper and lower bound. 
In section 6.2, a kinetic model is proposed for substrate conversion in a fluidised bed biofilm reactor 
(FBBR) where substrate conversion obeys Michaelis-Menten type kinetics. This model results in a reaction­
diffusion equation which is coupled in the boundary conditions to outside bulk fluid concentrations. Upper 
and Lower analytical bounds to the solutions are developed which agree with numerial results obtained by 
orthogonal collocation. In particular, i t  is shown that if substrate conversion follows Michaelis-Menten 
kinetics, the bulk fluid concentration never reaches zero concentration in the bulk liquid nor in a bioparticIe. 
In section 6.3 ,  we shall develop and compare some monotone iteration methods. These methods 
prove not only to be powerful tools in constructive existence proofs but are also useful for numerical 
computation of solutions. These iteration schemes may produce either monotone or alternating sequences and 
under special conditions, Newton's method may be applied which accelerates the rate of convergence. The 
objective of this example is to help us to understand the relationship between the properties of the reaction 
functions and the resulting sequences. 
1 53 
6.1 EXAMPLES 154 
In section 6.4, we shall look at a specific example from literature of a modified urea transfer model 
for predicting urea removal in a compact artificial kidney. Uniqueness and Existence theorems are developed 
for the steady state problem. The method of proving uniqueness and existence differs from the methods 
discussed in this  thesis and we shall suggest how it may be useful for proving uniquenss and existence 
theorems for general systems discussed in this thesis. 
In section 6.5, we set out the equations for a tubular fluidised bed biofilm reactor (FBBR) problem of 
applied interest. The b ioparticle reaction kinetics involve three chemical components, a substrate s such as 
phenol or nitrogenous wastes which needs to be converted to harmless byproducts, oxygen 0, an active 
ingredient which helps faci lilate the reaction kinetics, and a product p which linearly inhibits the reaction 
kinetics. We shall look at the question of existence and uniqueness of solutions and see how these solutions 
relate to the steady slate solution. We shall also study the stability of this system. 
In section 6.6, we look at an example of a Continuous Stirred Basket Reactor (CSBR) from literature. 
This involves the study of a reaction inside porous pellets. The enzyme reaction kinetics involve three 
chemical components, a substrate s which is converted to a product p under the action of immobilised 
enzyme, e. The bulk fluid fluid involves only two chemical components, bulk substrate Sb which is 
introduced into the reactor as steady flow and bulk product Pb that is pnxluced in the reactor. This example is 
slightly different from the general model developed in this thesis but we find that this does not cause any 
d ifficulties. We shall study the question of existence and uniqueness to the time dependent problem 
illustrating the usefulness of the concept of coupled upper and lower solutions and using methods developed 
for arbitrary kinetics. 
We have chosen to concentrate on some specific examples throughout this chaptcr such as the 
fluidised bed biofilm reactor (FBBR) since it has motivated our study. Our development is also intended to 
be generally applicable to other particle reactor models. However, cerlain analytical approximations derived 
in this chapter may prove to be very useful to these specific systems but may not be favourable generally. 
6.1 A SIMPLE METHOD FOR OBTAINING GOOD BOUNDS FOR 155 
SOLUTIONS OF A BIOPARTICLE MODEL 
6.1 A Simple Method for Obtaining Good Bounds for Solutions of a Bioparticle Model 
The biological growth attached to surfaces (commonly referred to as biofilms) is extensively employed in 
microbial processes such as fermentation and waste water treatment. Such processes give rise to reaction­
diffusion problems with non-linear kinetics. The solutions to these problems are usually obtained by 
numerical techniques such as the shooting method (PARSHOTAM [223] ,  McELWAIN [ 1 80]) and finite 
difference procedures (KELLER [ 140]) and the orthogonal collocation methods (VILLADSEN and MICHELSEN 
[296]). These approaches all suffer from the drawback that they require extensive computation. Various 
asymptotic techniques such as regular and singular perturbation are suggested for more general equations 
with very low and very high Thiele moduli (MURRAY [203] ,  FINLAYSON [90] and VEGA and LINAN [290] 
of parameter space. FINLAYSON [89] has used construction techniques by trial functions for finding 
bounding solutions. Numerical calculations for examples of such problems have been carried out by 
ANDERSON and ARTHURS [ 12, 1 3 ]  using variational methods based on extremum principles and these 
methods were compared by TOSAKA and MIYAKE [284] using integral equation methods. In all previous 
work, except one isolated case of an unpubl ished work of VILLADSEN reported by ARIS [2 1 ,  Ch.3] ,  the 
maximum principle has not been used to directly generate approximate solutions to such differential 
equations. The method appears however to be adumbrated by VILLADSEN and MICHELSEN [296] . A 
technique was suggested by ANDERSON and ARTHURS [ 14] and VARMA and STRElDER [289] based on this 
method to directly generate approximate solutions and this method was shown to be much simpler than 
variational methods to use and could be applicable to situations when a problem admits multiple solutions and 
where the variational method simp
,
ly docs not apply. This method was also discovered independently by 
P ARSHOTAM, BI-lAMlDIMARRI and WAKE [225] and we shall discribe it in this section. 
It is interesting to note that the method discribed in this section has the disadvantage of producing a 
good lower (upper) bound and a weak upper (lower) bound. This method has recently been improved to find 
optimum lower and upper bounds tly REGALBUTO el al. [245, 2461 .  The method has also been generalised 
to systems of equations exibiting multiple solutions (REGALBUTO el ai. [247]). It is also interesting to note 
that the methods of obtaining lower and upper bounds based on the integral representation of solutions that 
were introduced by TOSAKA and MIYAKE [284] have also been developed further (GARNER [98] and 
ASAITl-lAMBI and GARNER [24]). These methods result in sharp polynomial approximations to the solution, 
are compared to the methods of ANDERSON and ARTHURS [ 1 4] and are shown to yield sharper bounds with 
fewer integration steps. 
In this section we will use an iterative technique to obtain successive better bounding solutions and 
approximations. These approximate analytical solutions and bounding solutions are obtained via a 
Iinearisation technique which yields good upper and lower bounds to the exact solution and which apply to 
all parameter values. It can be shown analytically that this lower bound is always strictly positive which 
implies the solution to these type of reaction-diffusion equations is always strictly positive. The linearisation 
of Michaelis-Menten type kinetics with diffusion has been reported using Taylor's series expansions, 
(MURRA Y [203]) but has often been done at an arbitrary point. This approach invariably results in significant 
errors in solutions. At times, the point of linearisation can even be shown to be outside of the region in which 
the solution lies and even if the point of linearisation is in the region the solution lies, it could still result in a 
negative concentration as an approximation. In this work, a novel method is developed to determine a point 
for linearisation which would give minimal errors in solutions over parameter ranges of significance. It also 
produces good linear approximations to Michaelis-Menten kinetics over all parameter space and using these 
approximations, universal bounds are obtained for the solutions to resulting reaction-diffusion equations. 
6.1 A SIMPLE METHOD FOR OBTAINING GOOD BOUNDS FOR 1 56 
SOLUTIONS OF A BIOPARTICLE MODEL 
6.1 .1 B iotilm Model Formulation 
For a biofilm supported on a spherical inert panicle shown in FIG. 6. l ,  if the mass-transfer within the biofilm 
is governed by Fick's first law and the biochemical reaction follows Michaelis-Menten type of kinetics, the 
substrate transport and reaction within the biofilm at steady state is written as 
where Rmm(S) is the Michaelis-Menten type reaction rate defined as 
kS Rmm(S) = -- . Km + S 
The boundary conditions are 
FIG. 6.1 Schematic of a bioparticle 
The dimensionless mass balance of the substrate in a spherical bioparticle may be expressed as 
with 
F(y) =-y_ , 1 +  fiy 
and the corresponding boundary conditions are 
lea) = 0 and y(1 )  = I ,  
where the variables arc 
S y = Sb ' 
r X =- , rbp 
and the parameters are 
r Sb 2 _ rlpk a = ...E!!. , fi = - and n. K 'I' - DK rbp m m 
(6.1. 1) 
(6.1.2) 
(6.1.3) 
(6.1.4) 
(6.1.5) 
(6.1.6) 
(6.1.7) 
(6.1.8) 
6.1 A SIMPLE METHOD FOR OBTAINING GOOD BOUNDS FOR 157 
SOLUTIONS OF A BIOPARTICLE MODEL 
6.1.2 Upper and Lower solutions and Monotonicity 
Here we use the one dimensional maximum principle (pROrI'ER and WEINBERGER [234]) for solutions to 
y" (x) + H(x, y, i ) =  0, a < x < b 
with boundary conditions 
-i(a)cos () + y(a)sin () = Yl } 
i(b)cos w + y(b)sin w = Y2 '  
(6.1.9) 
(6.1.10) 
where 0 $ () $ n12, 0 $ w $ nl2 and () and w are not both zero. We also have the condition that aHl dy $ O. 
This is given in Theorem 22 in PROTTER and WEINBERGER [234]. In our case, 
, 2 dy 2 y H(x, y, y ) = -- - I/J  -- , x dx 1 +f3y 
(6.1 .11) 
and 
(6. 1.12) 
as required. Also a = a, b = l ,  () = 0, w = n12, Yl = 0 and 12 = 1 as required . 
Direct application of this theorem gives the following : 
Let y be a solution of the boundary value problem 
1 �(x2 dY ) _ 1/>2_y_ = 0 ,  7 <Ix <Ix 1 +fJy 
with boundary conditions y'(a ) = 0 and y(l )  = 1 .  
If � satisfies 
with the boundary conditions 
t(a) = O  and �(l) = l ,  
and if y satisfies the same conditions with the signs reversed then the upper and lower bounds 
�(x) $ y(x) $ Y(x) , (6.1 .13) 
are valid. 
The above result implies that a solution of (6. 1 .4) which satisfies boundary conditions (6. 1 .6) exists and 
must be unique, for if y and y are solutions, we can let � = y = y to find y == y. We note that � (x) == 0 and 
y(x) == 1 are respectively lower and upper solutions, that is 
0 $  y(x) $ 1 .  (6.1.14) 
6.1 A SIMPLE METHOD FOR OBTAINING GOOD BOUNDS FOR 1 58 
SOLUTIONS OF A BIOPARTICLE MODEL 
It also follows direclly from the maximum principle lhal minimum and maximum values of y(x) are al lhe 
boundaries x = a and x = 1 ,  respeclively. 
Monotonicity with parameters 
We now show thal y(x) is monolonically decreasing with t/J. Likewise a similar proof may be obtained lO 
show that y(x) is monolonically increasing with {3. 
Theorem 
y(x) is monotonically decreasing with t/J. 
Proof 
( 1 )  Assume t/Jl > IP2, {3 � O. 
(2) Assume Yl ,  Y2 > 0 (the slrict inequality i s  shown in equation (6. 1 .2 1 » . 
(3) Lel YI be a Solulion of (6. 1 .4) with IPI: 
L[y J = _1 �(x2 
dyl ) = t/,2_Y_l _ I x2 dx dx '1'1 1 + {3YI 
' 
with boundary condilions 
yi (a) = 0 and Yl ( l) = 1 .  
(4) Lel Y2 be a Solulion of (6. 1 .4) wilh 1/J2: 
with boundary conditions 
Y2 (a) = 0 and Y2(1 )  = 1 .  
(5) Consider the equation 
L[Y2 J - IPf 1 
Y
f3
2 = (t/Ji - t/Jf )
1 
Y
f3
2 < O = L[Yd- IPf 1 
Y
f3
1 . 
+ Y2 + Y2 + YI 
Using the theorem in PraHer and Weinberger lhis gives Y2 � YI for a $ x $ 1 .  
(6) Look al the difference z = Y2- YI . The equalion below is obtained 
1 d 2 dz 2 Y2 2 YI L[zJ = 2"-(x -) = L[Y2 - yd = L[Y2 J -L[yd = IP2---IPI --X dx dx 1 + {3Y2 1 + {3YI 
with homogeneous boundary conditions 
z '(a) = z( I) = O. 
6.1 A SIMPLE METHOD FOR OBTAINING GOOD BOUNDS FOR 159 
(7) Simplifying, 
since Y2 (x) � Yl (x) . 
SOLUTIONS OF A BIOPARTICLE MODEL 
(8) Therefore, we have the problem 
L[z]-</>rz <O , 
with boundary conditions 
z '(a) = z(1 )  = 0. 
(6.1.15) 
(6.1.16) 
(9) It follows from the strict form of the positivity lemma (KELLER [ 143 , p.222]) lhat z > ° for a < x < 1 
iff -</>r < J11 ' where J11 is the principal (i.e. , leasl) eigenvalue of the problem 
Llp] + Jip = 0, 
with boundary condilions 
p'(a) = p( l )  = 0. 
This in lurn implies lhal Y2 > Y I for a < x < 1 .  Thus Y is monolonically decreasing with </>, across the 
bioparticle.D 
The physical imporLance of" this is seen by noting that as the Thiele modulus, <f>2 increases (say k increases or 
D decreases), the substrale concentralion decreases. This is physically reasonable. 
Lower bounds 
Any linearisation by a Taylor's series expansion of <f>2F(y) will be strictly greater (for f3 ":f:  0) than <f>2F(y) and 
will lead to a lower solution, since 
F(y) � F(e) + (y-e)F'(e) for Y, e � 0. 
The linearised problem about some e E [0, 1 ]  is 
(6.1. 17) 
where 
(6.1.18) 
and satisfies the boundary conditions 
t(a) = 0 and 2::(1) = 1 .  (6.1.19) 
Solving for �(x), a lower bound is obtained to y(x) where 
6.1 A SIMPLE METHOD FOR OBTAINING GOOD BOUNDS FOR 160 
SOLUTIONS OF A BIOPARTICLE MODEL 
6. cosh ",(1 - x) !J. sinh ",(1 - x) A. 
y(x) = + - -2 ' - x x '" 
and A. and !l. are determined by the boundary conditions (6. 1 . 1 9). 
(6. 1.20) 
Linearising lfJ2F(y) at various e and choosing the maximum lower bound over the whole region produces a 
good lower bound. Alternatively, knowing that �(x) is a lower bound, the region below its minimum, �(a) 
may be eliminated and we limit ourselves to l inearising ¢J2F(y) only in the interval e E [�(a) , 1 ] .  
Linearising lfJ2F(y) in the interval l(a) $ y(x) $ 1 does not exclude the possibility that a negative 
concentration may be obtained as an approximation. This situation would be avoided if the first 
approximation of the lower bound is strictly greater than zero and subsequent iterations produce increasingly 
better lower bounds. 
We do this as follows. The solution of the l inearised problem at e = 0 is taken as a first lower bound: 
( )  alfJ cosh lfJ(x - a) + sinh lfJ(x - a) y x = -1 x[alfJ cosh lfJ(l - a) + sinh lfJ( l - a)] , (6.1.21) 
which is always strictly positive since all the terms in the above expression are positive. This is enough to 
guarantee that y(x) is strictly positi ve. 
A second linearisation is performed at e= l) (a) and a third at e =  l2 (a) 
Subsequently, an nth linearisation performed at e = �n_) (a) would be an improvement on the previous 
iteration and would be positive. 
Upper bounds 
A trivial upper bound to y is Y such that 
and satisfies the boundary conditions 
HI) = 1 and Y'(a) = O . 
This has the solution 
aR
lfJ COSh:l � (X- a) + sinh:l � - (x - a) 
_( ) 1 + f3 1 + f3 1 +  f3 y x = 
x[a b coshb (1 - a) + sinh b (1 - a)] 
-y 1 +f3  -y1 +f3 -y 1+f3 
and is an upper solution to y since for 0 $ Y $ I , 
F(y) � 1; f3 ' 
(6.1.22) 
(6.1.23) 
(6.1.24) 
and therefore y(x) � y (x) for a $ x $ 1 .  Subsequent improved upper bounds are Yn ' where 
l(a) $ y(x) $ Yn (x) $ 1  and where Yn satisfies the equation 
6.1 A SIMPLE METHOD FOR OBTAINING GOOD BOUNDS FOR 161 
where 
SOLUTIONS OF A BIOPARTICLE MODEL 
1 �( 2 dyn ) = -2- ., < th2F( ) -:2 X lI'nYn + lI.n - 'I' Y ,  x dx dx 
-2 
= 
F(l) - F(l..n (a» and I = F(l) _ -2 lI'n l - y (a) n lI'n ' -n 
with the boundary conditions 
Yn(l) = l and y�(a) = 0 . 
Solving for Yn(x) , successive upper bounds are obtained to y(x) which depend on y (aL as -n 
_ ( ) _ A cosh Vln (1 - x) B sinh Vln(1 - x) In Yn X - + - -2 ' X X lI'n 
and where it and B are determined by the boundary conditions (6. 1 .27). 
6.1.3 Results and Discussion 
In general, the following iteration scheme may be used to find lower and upper bounds for y(x) 
wiLh 
�n cosh lI' (1 - x) §.n sinh lI' (1 - x) A y (x) = n + n - -� , n = 1 ,  2, 3, . .  -n x X VI -n 
and where An and ll.n are determined by the boundary conditions as 
Similarly, 
with 
- ( ) _ An Cosh Vln(1 - X) Bn sinh Vln (l - x) In Yn X - + - -2 ' n = 2, 3, 4, . .  x X lI'n 
and where An and En are determined by the boundary conditions as 
A- 1 In B"':""" = -A- Vlna sinh Vln(1 - a) + cosh Vln(1 - a) n = + -2 ' n n . lI'n lI'na cosh lI'n (I - a) + sinh lI'n (1 - a) 
(6.1 .25) 
(6. 1.26) 
(6.1.27) 
(6. 1.28) 
(6. 1 .29) 
(6.1.30) 
(6. 1.31) 
(6.1.32) 
(6.1.33) 
(6. 1.34) 
(6. 1.35) 
6.1 A SIMPLE METHOD FOR OBTAINING GOOD BOUNDS FOR 162 
SOLUTIONS OF A BIOPARTICLE MODEL 
The linearisations of F(y) are shown in FIG. 6.2 where V'� , tji;; , a:.n and In are functions of �n-I (a) . 
y 
¢l2y 
1 +  f3 
F IG .  6.2 Schcma tlc of succcssive IIneur hounds ohtalned for F (y) . 
Since a linearised solution (by a Taylor's series expansion) of equation (6 . 1 .4) would always be a lower 
solution, it is not expected lhal lhis ileralive scheme would converge to the exact solution every time. It does 
however perform very wel l  within numerical l imils as shown in FIG 6.3. FIG . 6.3(a) shows the 
approximations for the variable y itself  for typical parameter values and FIG.  6.3(b) shows it in terms or 
percentage relative errors, that is 
cYn - 1)% or (�n - 1)%. 
y y 
An arithmetic average of �3 and Y3 is also plotted (where Yay = (�3 + Y3 ) / 2 ) .  In this typical example, lhe 
maximum relative error of Yay to the exact solution y(x) was shown to be 0.05%.The method is sensitive to 
an increase in the Thiele modulus, ¢l2 when ¢l2F(y) becomes very nonlinear. This is seen in FIG. 6.4. For 
diffusion limited reactions in bacterial films the Thiele modului are usually higher lhan those for films of 
moulds or yeasts (tjJ2 = 20 is sti l l  considered to be physically ex tremely high) and this technique results in 
increased error between the exact solution and upper and lower bounds for such biofilms. The method is also 
sensitive to a decrease in a as seen in FIG. 6.5 .The linearisation technique may not therefore be as effective 
for thick biofilms with very small support particles or bacterial flocs. It is however, considerably better than 
, 
its first approximation which in the example of FIG. 6.5 where typical parameter values were chosen has a 
maximum relative error lO the exact solution of -64% (not shown complelely in this graph) and an average of 
�.' 
Y2 (a) and �3 (a) produce a maximum relalive error to the exact solution of only 0. 1 %. The iterative method 
produces an increase in relative error for intermediate values of f3 as shown in FIG. 6.6. The approximation 
. of Michaelis-Menten type of kinetics by either zero order or first order has been widely reported (BAILEY and 
OLLIS [3D)). However, such an approximation resulLs in a solution that is given by only the first iteration of 
the scheme presented in this work and may not be the best. On the other hand, this technique leads to a better 
approximation of the solution in a parameter space in addition to obtaining the universal bounds on the exact 
6.1 A SIMPLE METHOD FOR OBTAIN ING GOOD BOUNDS FOR 
SOLUTIONS OF A BlOPARTICLE MODEL 
1 63 
solution. For instance, an average or upper and lower bounds after 3 iterations produces an approximate 
analytical solution of within 3% relative error over the parameter space of interest in biological systems, 
namely , q,2 E [0, 20] ,  f3 E [0, 00) and a E (0, 1 ] .  
a )  
� 
-!:!. • .... 
...:: 
-!:!. 
� 
...:: 
-!:!. 
• 
�I 
b) 
� 
· 
�I � �  � �  I;;� 
� 
0 81� ��  ;0.., ;0.., � 
, 
� �  �� . �  
tOO 
.95 
.90 
.85 
.80 
.76 
LEGEND 
y (x) 
.................. l3(X) 
.-.-.-.-.-.- Yl(x) 
--------- ll(X) 
.-.-.-.- Y1 (x) 
.40 _____ Vex) 
.36'F-=.:-:.....---- --,.------r. -----.-----. ----� 
h � .7 B � to 
6 
0 
-5 
-10 ' 
-15 
-20 
-25 
x 
-- - - - - - ._.-.-._- - - - -_0_-
,......., ...un." ......... aa,rr'-...ft-...n&.-..#'r_ ... _. - - - . : . . .
. 
' 
.....
.....
.....
.... -.
......
......
......
......
.....
 -....
.....
......
... -..
.. -...
.....
 
�.�.�
.::::
:::::':
:':::: ::
:'.:.::
.
::.::.:
::.:::::
::
_-/ /' 
--------------------------------------
-
� 
/
/ 
,/,/ .-/' 
.-/'.-/' 
--
f---
/' /' 
/
/ 
/' 
� 
� 
/" LEGEND 
---- y.,(X)  
.. -.. -.. -. Y3(X) 
.................. l3(X) 
.-.-.-.-.-.- Yl(X) 
--------- ll(X) 
'-'-'-'- Y 1 (X) 
--- ll (X) 
-30�----�r_----_y-----�-----r_----_1 
b a .7 x i 9 to 
F!G.  6.3 (a) Successive upper lInd lower bounds 10 Ihe exacl solulion y (x)  wilh a = 0.5, 92 = 16 and P = 1. (b) 
l'erccnlllgc rclulivc crror 10 thc cxucl solliliun y(x)  with a = 0.5, ,2 = 16 und P " 1 .  
, 
6 . 1  A SIMPLE METHOD FOR OBTAINING GOOD BOUNDS FOR 
SOLUTIONS OF A BIOPARTICLE MODEL 
6�
----------------------------------------------�� 
: : --------- -- --
.
- .:-�:�::=::.-
o� ----------���-�--�-�-�-�-��-�-� 
.. 
�
-
m 
.. 
� .. � .. -� .. � .. �:.-�_�==�--�
--�-"------------� 
-2 ' " 
---
............ -.......... -----
-
.. ---
-
-
--
--
-
--
-
----
-
.. -
-4 ' 
''\ 
--
-
--
--------------------- _____ _ 
-6'  " 
-8 ' " 
-10 
" " LEGEND 
1 64 
" " -12 ' _______ y ••  (a )  -14 • '-"
"
'" 
-16 ' 
-18' 
-20 ' � � -22 ' tl tl � � -24 ' c "" 
� -26 
" ........ 
........ 
........ -­---
J3(a) 
�3(a) 
Jl(a) 
�l(a) 
J I (a) 
! I(a)  
� 
, 
t;'1 � � tl c � ,"" "" 
� 
, �  � tl > � "" "" 
� 
� t;'  � >:: 
----
------28 ' --
-���--��--�_r--r__r--r__r--r__r--r__r--�_r--��-�-�-��� o 5 411 10 15 20 
FIG 6.4 Percentage relative error to y(a) with a = 0.5, fJ = 1 and various ¢l 
20�----------------------------------------------------� 
1 /::::�:;�:-;��-=�-::=:;;;;::.::---
--5 �\ ... ��:.::-.. --.......... -.. -......... -......... --.. -.. -------- .. ---.-----
-
-
-10 \ --', 
-15 ......... . --------
--------
--
-
--------
--
-45 
-50 
-55 
-60 
\ �� 
\ 
-�--------------
LEGEND 
\ ------- yn(a)  
'\ ------------ J3(a) 
'\ 
'\ -- .. ----.......... �3 (a) "
'-,-
.---- -- h(a) 
" --------- tl(a) "--65 
-___ 
.. --- -- - J I ( a) 
� ----
-75 -r-----,r------r-----..------r------,-------T---.::..=.::;:..:===r===---,.-----J --
--- ! I (a) 
0 ;  2 3 4 p 5 6 7  B 9 10 
FIG 6.5 Perccntage relatlvc crror to y(a) with fJ = 5, 412 10 and vnrious a 
2�--------------------------------------------,
0 
-2 
-4 
-6 
-8' 
-10' 
--
-
-----==� �=- - - - - -
........................ .............. 
" ,-
,,/ 
, 
, , 
,," 
, 
.-,-, , , 
,/ 
,/" , , , , , , , 
�� 
..,,,.,. ..... 
-
--
-
-
-
-
-
---
/ 
I 
/ 
/ 
/ 
/ 
/ 
lEGEND 
/ --y ..  (a)  
/ ----------------.. �3(a) 
/ 
-
-
------
-
-
-
- Jl(a) 
/ _________ �l(a) 
-12 
/ --- - -- .YI(a) �--.-- -..-- __r-___.r__-r__-,_---r--__.LI-____r---J ----- !I(a) 
.2 .3 .4 a .5 � ] B 0 .9 10 
FIG 6.6 Perccntagc rclntlvc error to y(a) with a = 0.2, 41
2 IS und various fJ 
6.1 A SIMPLE METHOD FOR OBTAINING GOOD BOUNDS FOR 165 
SOLUTIONS OF A BIOPARTICLE MODEL 
6.1.3 Conclusions and remarks 
The Maximum Principle may be applied to finding upper and lower bounds to a wide class of equations 
including reaction-diffusion type with nonlinear kinetics. The method developed in this section is that of 
obtaining a better linearisation of the nonlinear terms by narrowing down the region of linearisation. The 
method of obtaining successive bounds for these source or sink terms relies on the fact that these terms must 
be either concave up or down. The technique is sensitive to the diffusion parameter although the relative error 
of the approximate solution is significantly low. The linearisation technique offers a convenient method for 
obtaining the universal bounds for the exact solution which can be used for obtaining bounds for 
performance indicators such as effectiveness factors. 
Nomenclature 
constant, determined by the boundary conditions that relates to the lower solution � 
constant, determined by the boundary conditions that relates to the lower solution � 
constant, defined in eq. (6. 1 .32), that relates to the lower solution y 
-n 
constant, defined in eq. (6. 1 .32), that relates to Lhe lower solution y 
_n 
x 
constant, determined by Lhe boundary conditions tllat relates to upper solution Y 
constant, determined by Lhe boundary conditions Lhat relates to upper solution y 
constant, defined in eq. (6. 1 .35) that relates to upper solution Yn 
constant, defined in eq. (6. 1 .35) that relates to upper solution Yn 
effective diffusivity 
dimensionless Michaelis-Menten term, derined by eq. (6. 1 .5) 
rate constant 
Michaelis-Menten constant 
radial distance 
bioparticle radius 
support media radius 
Michaelis-Menten type reaction rate, defined by eq. (6. 1 .2) 
substrate concentration 
bulk substrate concentration 
dimensionless distance 
dimensionless concenLration 
a lower solution of y 
an nth lower solution of y 
an upper solution of y 
an nLh upper solution of y 
an average of upper and lower bounds 
Greek letters 
a ratio of support media radius to bioparticle radius 
f3 dimensionless Michaelis-Menten constant 
e point of linearisation 
rf Thiele modulus (ratio of Lhe reaction rale to diffusion rate) 
J. constant, defined in eq. (6. 1 . 1 8) 
J.n constant, defined in eq. (6. 1 .3 1 )  
In constant, defined in eq . (6. 1 . 34) 
It' constant, defined in eq. (6. 1 . 18) 
!!!.n constant, defined in eq. (6. 1 .3 1 )  
Wn constant, defined in eq. (6. 1 .34) 
6.2 ANALYTICAL BOUNDS TO A MODEL OF A FLUIDISED BED BIOFILM 166 
REACTOR (FBBR) 
6.2 Analytical Bounds to a Model of a Fluidised Bed Biofilm Reactor (FBBR) 
A significant step in  the numerical solution of packed bed reactor models was taken with the introduction of 
the method of orthogonal collocation to this class of problems (FINAL YSON [88]). This method was shown 
to be much faster and more accurate than that based on finite differences. 
The orthogonal collocation method for solving partial differential equations developed largely by 
VILLADSEN et al. [293-296] and FINLAYSON [88] has been found to require less computer time than 
standard finite difference methods. The method is now usually applied to and is particularly suited to the 
simulation of fluidised, fixed and packed bed reactors (HANSEN [ 1 1 5] ;  KARANTH and HUGHES [ 135] ;  
RAGHA VAN and RUTHVEN [237]; HASSAN and BEG [ 120]), the simulation of an adsoption column (LIAPIS 
and RIPPIN [ 169]) and fixed bed catalytic reactor simulation with moving boundaries (GARDINI et al. [97]) .  
This method has now been the generally accepted procedure in numerically implementing such models. 
These models all involve reaction-diffusion equations involving interacting macro and microstructures 
where outside concentrations govern the boundary conditions for local behaviour. The equations that govern 
kinetic and d iffusion-controlled substrate uptake by the attached organisms in a fluidised bed biofilm reactor 
are invariably nonlinear and analytical solutions if any are impossible to find. 
This section considers such equations and demonstrates that although such equations may be diffucult 
to solve, it is relatively easy to provide analytical bounds on the solution. These universal bounds agree with 
numerical solutions for such equations with parameters found in chemical engineering literature but apply to 
all parameter values. It also provides us with approximate analytical solutions. 
6.2.1 The Fluidised Bed Biofilm Reactor 
The fluidised bed biofilm reactor (FBBR) is a novel biological process which has been applied to both 
wastewater treatment processes and biochem ical manuracture. The application of lluid ised-bed biorilm 
reactors for biological wastewater treatment has been attempted by several resean:hes in recent years and 
include denitrification, nitrification and organic carbon removal (JERIS and OWENS [ 1 34]) .  The fluidised-bed 
biofilm reactor is a high-energy high-efficiency reactor in which the liquid to be treated is passed upward 
through a bed of small support particles such as activated carbon or other support media at velocities 
sufficient to impart motion to, or lluidise the particles. Each particle offers a large surface area for biological 
growth, resulting in biomass concentration of an order of magnitude greater compared to conventional 
dispersed growth systems.  The bed is seeded with microorganisms which eventually grow to form a 
biological growth known as a biofilm around the core particle. The very high growth support surface 
afforded by these bioparticles results in denitrification of volatile sol id concentrations as high as 30,000 
mg 1-1 and a bed detention times as low as 6 minutes for 99% nitrate removal. 
In this work, predictive models for the lluidised bed biofilm reactor are developed. These models 
have existed for many years in l iterature and often with the axial d ispersion term being neglected (e.g. 
MULCAHY et al. [ 199-202]) The model is similar to the modified urea transfer model presented by LIN [ 1 70] 
in predicting the urea removal in a compact artificial kidney by microencapsulated urease particles. In both 
models, a simpler plug flow equation is usually used instead of a gencral dispersion one is employed to 
describe substrate concentration changes throughout the liquid phase. This is usually justified by the rather 
large Peclet number for the above systems when in operation (MULCAHY el. al. [ 199-202] and LIN [ 170]). 
The example in this section howevcr shall assume a more general dispersion model. 
-------- -----------
6.2 ANAL YTICAL BOUNDS TO A MODEL OF A FLUlDISED BED BIOFlLM 167 
REACTOR (FBBR) 
6.2.2 The Fluidised Bed Biofihn Reactor Model Formulation 
The t1uidised Bed Biofilm Reactor, the typical schematic with bioparticles is shown in FIG . 6.7, consists of a 
column reactor in which granular media with high specilic surface area are Iluidised with nutrient solution. 
The key process components of this system with a uniform biof'ilm are: 
(i) reaction-diffusion within a single bioparticle 
(ii) Solute transport through rcac;tor flow 
cflluent out 
substrate reactant liquid in 
FIG. 6.7 Schcmatic of a FIlIlR with Illoparlicics 
Bioparticle-Reactor Model Development 
media 
llioparticic 
The mathematical model of the FBBR substrate conversion process is divided into two submodels. The 
"Bioparticle Model" is concerned with the intra-biofilm diffusion and substrate conversion by m icro­
organisms auatched to the individual support particles which are in a fluid ised state. The "Reactor Flow 
Model" discusses the hydraulic flow transport of substrate through a FBBR. The two models are coupled by 
biofilm-bulk liquid boundary conditions to yield an overall model for substrate conversion in an FBBR. 
We make the following simplifying assumptions 
Bioparticle Model 
1 .  Homogeneous biofilm of constant and uniform thickness 
2 .  Spherical support media of uniform size 
3 .  Internal diffusion of substrate is governed by Fick's first law 
4 .  Substrate-limited biochemical reaction described by Michaclis-Menten kinetics 
6.2 ANAL YTICAL BOUNDS TO A MODEL OF A FLUIDISED BED BIOFILM 168 
REACTOR (FBBR) 
Reactor Flow Model 
1 .  Liquid phase substrate transport is by plug flow convection and axial dispersion. 
2 .  No macroscopic radial gradients exist 
3 .  Bioparticle characteristics are independent of position w ithin the reactor 
4 .  No  substrate conversion occurs in the liquid phase 
5 .  Cylindrical reactor 
Under these assumptions, the following differential equation is derived from a substrate mass balance across 
an axial reactor element 
where the observed substrate conversion rate per unit lluidised bed volume is given by 
R = NAD as(r) 1 v s a r r=rbp 
(6.2.1) 
(6.2.2) 
The number of particles per unit volume of the reactor is calculated from t11e total initial mass of the bed and 
the expanded bed height for a set of operating conditions. 
N _ Total mass of support material M - Mass of single particle = 41t'r;mPsm . 
Within a bioparticle the substrate mass balance on a differential shell may be written as: 
as -!2f.�(r2 as)+J.1mPbf (_S_) __ () f' 0 Z I .. L. lor r.fm < r < rbp' < J < I, I > 0, al r ar dr YX1S S+Ks 
with initial and boundary conditions 
S(I, r, Z) = 0 at 1 =  0 for rsm < r < rbp' 0 < 2 < h, 
as D s & = 0 at , = 'sm' 0 < Z < h, 
and outside the bioparticle in the external fluid, we have from (6.2. 1)  and (6.2.2), 
aSb a2sb aSb aS
I 
--Ds --+Us -+ NAD - = 0 1 > 0 0 < 2  < h a b az2 b az s a "  , t J r r=rbp 
with initial condi tions 
at I = 0, 0 < 2 < It 
(6.2.3) 
(6.2.4) 
(6.2.5) 
(6.2.6) 
(6.2.7) 
(6.2.8) 
(6.2.9) 
and boundary conditions given by WEHNER and WILHELM [306] and DANCKWERTS [79] , respectively. 
Ds aa
Sb = Us (Sb - Sb ;) at Z = O, t > O, b Z b , (6.2.10) 
6.2 ANAL YTICAL BOUNDS TO A MODEL OF A FLUIDISED BED BIOFILM 1 69 
D aSb = 0 at Z = h t > 0 Sb az J ,  . 
REACTOR (FBBR) 
In dimensionless coordinates, these equations have the form 
_ 
ay + 1 �(x2 
ay ) = tjJ2 _y_ ar ? ax ax 1 + f3y 
-x ay + _1 a2y _ ay = � ay ar � az2 az ax 
1 ay y + !fJ/" av = Yet, z) 
ay =0 av 
1 ay Y ---- = 1 
g'" dZ 
ay =0 az 
y(x, 0)=0 
Y(z, 0)=0 
where the dimensionless variables are 
in a < x <  1 , 0 < z <  1 , 1' > 0, 
in 0 < z < 1 ,  x = 1 ,  l' > 0, 
at x = 1 for 0 < z < 1, l' > 0, 
at x = a for 0 < z < 1 ,  r > 0, 
at z = O, 1' > O, 
at z = 1 ,  l' > 0, 
for a ::; x ::;  1 ,  
for 0 ::; z ::; 1 ,  
X = � y = -.£ y = .!iL Z D I  , , , z = - and l' = -t-' rbp Sb,l Sb,l h rbp 
and the parameters arc 
At steady-state the equations take the following form 
_1 � (x2 
ay) = tjJ2 _y_ 
x2 ax ax 1 +f3y 
_1 d2y _ dY _ � ay 
fi'e dz2 dz - dX 
1 ay y + !fJ/" av = Y(z) 
ay = 0 av 
Y _ _  1 dY = 1 � dz 
dY = 0  
dz 
in a < x < 1 ,  0 < z < 1 ,  
in 0 < z < 1 ,  x = 1 ,  
at x = 1 for 0 < z < 1 ,  
at x = a for O < z < l ,  
al z = 0, 
at z = 1 .  
(6.2. 1 1 )  
(6.2.12) 
(6.2.13) 
(6.2.14) 
(6.2.15) 
(6.2.16) 
(6.2.17) 
(6.2.18) 
(6.2.19) 
(6.2.20) 
(6.2.22) 
(6.2.23) 
(6.2.24) 
(6.2.25) 
(6.2.26) 
(6.2.27) 
6.2 ANALYTICAL BOUNDS TO A MODEL OF A FLUIDISED BED BIOFILM 170 
REACTOR (FBBR) 
3.2.3 The Numerical Solution · the Method of Orthogonal Collocation 
The general form of the model (equations (6.2. 12)-(6.2. 19» is solved numerically by converting the partial 
differential equations (pde's) to a system of ordinary differential equations (ode's). Orthogonal collocation, a 
method of weighted residuals, lends itself well to converting similar types of pde's to systems of ode's 
(RAGHA VAN and RUTHVEN [237], VILLADSEN et al. [293-296]). The resulting set of ode's can then be 
solved by a number of standard techniques. 
Weighted residual methods allow separation of the time and spatial dependency of a pde by 
approximating the exact solution with a series of products of time-varying coefficients and spatial basis or 
trial functions. The collocation method requires that the residual between the numerical approximation of the 
pde and its exact value be orthogonal to the Dirac delta function at specified collocation points. 11lis results in 
the residuals being zero at the collocation points (FINLAYSON [90]). 
Orthogonal collocation uses orthogonal polynomials as basis functions and specifies that the 
collocation points be located at the basis function roots. In this work we chose to usc Jacobi polynomials 
since it has this orthogonality property. The polynomials were constructed orthogonal to each other with 
respect to a weight function. The weight functions used in the construction of the polynomials for the 
different equations were chosen to make the numerical solution stable. 
We shall only be interested in this section in bounds for the numerical solutions at steady state. 
3.2.4 Upper and Lower Bounding Solutions 
The equations (6.2.22)-(6.2.27) are interesting in that the coupl ing is found in the boundary condition 
(6.2.24) of (6.2.22). Also, (6.2.23) depends on the solution to (6.2.22). Despite these features, we are able 
to show that we can obtain comparison results as that found in (PARSHOTAM, BHAMIDIMARRI and WAKE 
[225]) .  
Suppose that we can find functions X, y, r and Y so that the rollowing differential inequalities arc 
satisfied for y and r :  
_I .i!.... (x2 dl.. ) � ¢J2� x2 ax ax 1 + (3"t. 
1 d2 Y dY ay 
---= - -= > ,---= � dz2 dz - ax 
1 ay y + ----= $ Y(z) 
- fiX av -
al.. $ 0 
dV 
y _ _ l dI :S; 1 - � dz 
dK: 
$ 0  
dz 
and the inequalities arc reversed ror 
in a < x < l , O < z < l , (6.2.28) 
in 0 < z < 1 ,  x = 1 ,  (6.2.29) 
at x = 1 for ° < z < 1 ,  (6.2.30) 
at x = a for 0 < z < 1 ,  (6.2.31 )  
a t  z = 0 ,  (6.2.32) 
at z = I . (6.2.33) 
y and Y .  
We see that the functions [.-K, r -K, J+K and Y +K for K positive constants satisfy conditions o f  Theorem 
3 .5.2 and so all solutions or (6.2 . 1 2)-(6.2 . 19) (including (6.2.22)-(6.2.27» are globally stable. This also 
6.2 ANALYTICAL BOUNDS TO A MODEL OF A FLUIDISED BED BIOFILM 171 
REACTOR (FBBR) 
implies uniqueness of solutions to (6.2.22)-(6.2.27) and so (see the end of section 4.3), it follows that the 
upper and lower bounds 
�(x, z) � y(x, z) � Y(x, z) (6.2.34) 
and 
K(z) � Y(z) � fez) (6.2.35) 
are valid. 
One way to obtain upper and lower analytical bounds would therefore be to choose two functions !... and 1 
such that 
(6.2.36) 
For example, we may choose f=4>2y and 1=4>2 Y and let y(x, z) and [(z) be the solution to the - - 1+f3 -following system 
_1 �(x2 
a�) = 4>2y 
x2 ax ax -
1 d2 y dY ay 
---=--= = ,-= � dz2 dz ax 
1 ay y +--= = Y(z) 
- Y;" av -
ay -= = 0  
av 
y _ _ l d[ = 1  - � dz 
dI = 0  
dz 
in a < x < I ,  0 < z < 1 ,  
in 0 < z < I ,  x = 1 ,  
at x = 1 for 0 < z < I ,  
at x = a for 0 < z < 1 ,  
al z = 0, 
al  z = 1 .  
These equalions are solved analytically lO obtain the following equalions for y(x, z )  and nz) : 
and 
where 
( ) _ [1 (z) [ 4>a cosh 4>(x - a) + sinh 4>(x - a) ] y x, z - -- ..:...---'-'----.:.----'---'.-� - c3x 4>a sinh 4>(x - a) + cosh 4>(x - a) 
(6.2.37) 
(6.2.38) 
(6.2.39) 
(6.2.40) 
(6.2.41) 
(6.2.42) 
(6.2.43) 
(6.2.44) 
(6.2.45) 
(6.2.46) 
6.2 ANAL YTiCAL BOUNDS TO A MODEL OF A FLUIDISED BED BIOFILM 1 72 
REACTOR (FBBR) 
C3 = 1 -_1_ + .L[ ¢a sinh ¢(1 - a) + cosh ¢(1 - a) ] 
fJJj. fIJI. ¢a cosh ¢(1 - a) + sinh ¢(1 - a) 
C4 = �1 + 4'cs 
� 
_ 1 m [ ¢a sinh ¢(1 - a) + cosh 1/>(1 - a) ] c3 - - + 'r 
c5 =--------�¢-a-c-o-sh�I/>(�I---a�)-+-s-in-h-I/>�(�I---a�) 
To find Y and Y we solve the following equations 
__ 
1 � (x2 (fy) = Ly 
x2 ax ax 1 + {3 
1 d2f dY _ , ay 
� dz2 - Tz - ax 
. 1 (fy -
Y + fIJI. av  = Y (z) 
(fy = 0 av 
- 1 dY Y --- = 1 
.c¥'e dz 
dY = 0 
dz 
in a < x < 1 , 0 < z < 1 ,  
i n  0 < z < 1 ,  x = 1 ,  
at x = 1 for 0 < z < 1 ,  
at x = a for 0 < z < 1 , 
at z = 0, 
at z = 1 .  
where the solutions for y and Y are given as in (6.2.43)-(6.2.49) but with ¢ replaced by _ � . 
-'J 1 +{3 
To show that constants C t ,  c2 > 0 we are only required to show that Cs > O. We note that 
O [ l/>asinh l/>(l-a)+cosh l/>(1 -a)] 1 � > 
= 
> I/>a cosh 1/>(1-a)+ sinh 1/>(1-a) 
= (I/» e2¢>(I/>-I) > e2a¢>(al/>-l) (al/» g 1/>+1  al/>+1 g 
(6.2.47) 
(6.2.48) 
(6.2.49) 
(6.2.50) 
(6.2.51) 
(6.2.52) 
(6.2.53) 
(6.2.54) 
(6.2.55) 
(6.2.56) 
(6.2.57) 
and this is clearly true since g can be shown to be a monotonically increasing function for 0 S:a < 1 and 
I/> > O. Finally, since Ct ,  C2 > 0, it follows that Y(z) > 0 since Y(z) > K(z) and K(z) is exponentially decaying 
and is strictly bounded below by zero . This in tum implies that y(x, z ) > 0, since J' is similarly strictly 
bounded below by zero. 
For a given set of parameter values found in l iterature, we plot a graph of analytical bounds [(z) and 
Y(z) with Y(z) obtained by orthogonal collocation (see FIG 6.8). 
---------
6.2 ANALYTICAL BOUNDS TO A MODEL OF A FLUlDISED BED BIOFILM 1 73 
REACTOR (FBBR) 
1 .00 r--�--'------'-----"--�--"'---�---' 
0.90 ..... 
. 
......... . .... ......................... 
E I ... 
..: 
� 
� >:'1 
................ 
·············· .. r ................ . ........................ 
: 
O.BO 
- ....................... . 
0.70 L-_� __ -'--_� __ .L-_� __ -'-_��._--l _ _ �_--l 
0.0 0.2 0.4 0.6 O.B 1 .0 
FIG. 6.8 Upper and Lower Bounds to Y (z) with a=0.3, 11=0.5, �=5, �=0. 1 ,  9'" =20 and 9'.= 1 0 
6.2.5 Conclusions and Remarks 
The method demonstrated in this work may be applied to finding upper and lower bounds to a wide class of 
such equations with reaction terms very nonlinear. In the problem demonstrated in this section, since the 
Michaelis-Menten reaction term is concave down, a lower bound is always obtained when this term is 
linearised by a Taylor's series expansion. This lower bOllnd is always strictly greater that zero and gives a 
good indication of concentrations al lhe outlet of llle reactor. For d i ffusion l im ited reactions i n  microbial films 
where the Thiele modulus is very high, these bounds may not be so effective approximate solutions. The 
method is also sensitive to a decrease in a and may not therefore be as effective for biofilms with very small 
support particle sizes. For typical parameter values it is a very convenient method of checking numerical 
results as well as numerical results that involve very small concentrations. It can also be demonstrated 
analytically with the help of the maximum principle that dimensionless substrate concentration within a 
bioparticie is monotonically increasing with the parameters a, {3, .Y'hand monotonically decreasing with the 
parameter ¢fl. Also it can be shown that bulk concentration is monotonically decreasing with all parameters 
g:" .Y'h and ,. These properties can all be interpreted physica\1y. 
Nomenclature 
A cross sectional area, typically 20cm2 
Ds effective diffusivity constant, typica\1y 1 -2xlO -6cm2sec-1 
DSb axial dispersion coefficient 
h dimensional height of the reactor, typical ly 80-90cm 
1-1 s Mass transfer resistance 
Ks Michaelis-Menten saturation constant (kg m-3) 
N number of bioparticles per unit volume 20- 100cm-3 
g:, Peclet number given in (6.2.2 1 )  
USb 
superficial l iquid velocity- typica\1y O. 1 -2cmscc- l 
r radial distance within a bioparticle 
6.2 ANALYTICAL BOUNDS TO A MODEL OF A FLUIDISED BED BIOFILM 174 
REACTOR (FBBR) 
Tsm support media radius 
Tbp bioparticle radius- typically 0.3-0.6cm 
fJJl. Sherwood number given in (6.2.2 1 )  
S substrate concentration within a bioparticle 
Sb bulk substrate concentration which varies up the reactor (kgm-3) 
Sb,i input bulk substrate concentration (kgm-3) 
t actual time 
x dimensionless distance within a bioparticle 
y dimensionlesssubstrate concentration at position x of a bioparticle at height z 
Y dimensionless bulk substrate concentration at given height z 
YXIS growth yield 
Z height up the reactor 
z dimensionless height up the reactor 
Greek letters 
a ratio of support media radius to bioparticle radius 
f3 dimensionless Michaelis Menten constant 
X parameter g iven in (6.2.2 1 )  
lfJ2 Thiele moduli given in (6.2.2 1) ,  typically 0-5 
Jim maximum specific growth rate cOllstant 
Pb! biofilm density 
r dimensionless time 
, parameter given in (6.2.2 1) ,  typically 0- 12  
6.3 MONOTONE ITERATION TECHNIQUES IN THE CONSTRUCTION OF 1 75 
UPPER AND LOWER SOLUTIONS 
6.3 Monotone Iteration Techniques in the construction of Upper and Lower Solutions 
Although upper and lower solutions are useful in the investigation of qualitive properties of solutions and do 
prove to be good as approximate solutions (PARSHOTAM et al. [225, 226]), they can also be improved by 
monotone iteration techniques. The method of combining upper and lower solutions with monotone iterative 
techniques has proved not only to be a powerful tool in proving existence of solutions but has also recently 
been shown to be useful for numerical computation of solutions of boundary value problems for both scalar 
equations and systems of weakly coupled elliptic equations (GROSSMANN [ I l l ] ,  GROSSMANN and Roos 
[ 1 10], PAO [219-222]). The objective of this example is to help us to understand the relationship between the 
properties of the reaction functions and the resulting sequences. We examine some monotone iterative 
techniques and suggest how such techniques may be useful for numerical computation of solutions of the 
general system Sn , Bn. 
Several monotone iteration methods are given by KELLER [ 1 41 ]  for scalar elliptic equations. These 
provide us with either monotone sequences, alternating sequences and in some cases monotone sequences 
with accelerated convergences. These sequences are all obtained with the help of the strong maximum 
principle for the ell iptic operator. In this section, we shall look at the problem developed in section 6.2 but 
for more general arbitrary reaction functions and show that although this problem is non standard in that it 
involves elliptic equations which are coupled in the boundary conditions in a functional way, we can also 
obtain similar monotone sequences. These are also obtained with the help of the strong maximum principle in 
various ways. 
An iteration scheme is set up which generally results in solving either coupled or uncoupled linear 
differential equations. These iteration shemes may produce either monotone or alternating sequences and 
under special conditions, Newton's method may be applied which accelerates the rate of convergence. A 
lower or an upper solution is taken as a good candidate for a first iteration. 
6.3.1 Upper and Lower Solutions 
Consider the boundary value problem 
where 
and 
�y = ¢J2 f(x,y) 
Lz.Y = (�� = (.9Jt.(Y - y) 
1 ay y+ fIJI. av = Y(z) 
ay =0 
av 
y __ l dY = 1  
� dz 
dY =0  
dz 
1 a 2 a � =--(x -) x2 ax ax ' 
in a < x < I ,  0 < z < I ,  (6.3.1) 
in 0 < z < I ,  x = I ,  (6.3.2) 
at x = 1 for 0 � z � I , (6.3.3) 
at x = a for 0 � z � I ,  (6.3.4) 
at z = 0, (6.3.5) 
at z = I ,  (6.3.6) 
(6.3.7) 
(6.3.8) 
6.3 MONOTONE ITERATION TECHNIQUES IN THE CONSTRUCTION OF 176 
UPPER AND LOWER SOLUTIONS 
A lower solution to this problem is a function (�, .D satisfying 
�I ;;:: �Y,4(I - �) 
1 oy �+ y", 0; �I(z) 
o� < 0 
OV -
1 dY Y---=-�l  
- � dz 
dY  -=- � O  
dz 
in a < x < 1 ,  0 < z < 1 ,  
i n  0 < z < 1 ,  x = 1 ,  
at x = 1 for ° � z � 1 ,  
at x = a for 0 � z � 1 ,  
a t  z = 0, 
at z = 1 .  
An  upper solution to this problem is a function (y, y) satisfying 
� y � �y,4(Y - y) 
1 oy -y+ y,4 OV ;;:: Y(z) 
oy > 0 
OV -
- 1 dY Y---� l  � dz 
d Y = 0  
dz 
where (�, .D� (y, Y) . 
6.3.2 Monotone Iteration Methods 
in a < x < I , 0 < z < 1 ,  
in 0 < z < 1 ,  x = 1 ,  
at x = 1 for 0 � z � 1 ,  
at x = a for 0 � z � 1 ,  
at z = 0, 
at z = 1 ,  
(6.3.9) 
(6.3.10) 
(6.3.1 1) 
(6.3.12) 
(6.3.13) 
(6.3.14) 
(6.3.15) 
(6.3.16) 
(6.3.17) 
(6.3.18) 
(6.3.19) 
(6.3.20) 
We shall assume that Yn and Yn+l arc restricted to the set min � � Yn, Yn+ 1 � max y and that Yn and Yn+l arc 
restricted to the set min r � Y n, Y n+ I � max Y . 
In  each of the following methods, we shall define transformations fft and !J2 from (Yn, Yn) to 
(yn+l ' Yn+l )  and show that lhese transformations have some sort of a monotone property. In the general 
case, we shall mean that fft and !J2 are monotone in the sense of COLLATZ [76] ,  i .e., i .e . ,  Yn � Yn+l and 
Yn � Yn+l implies fftYn < fftYn+l and !J2Yn < !J2Yn+l . The monotone property of transformations fft and 
!J2 are shown to usually depend on the properties of the function f. In all these methods BI and B2 are 
relevant boundary operators. 
Assume, for the first three methods thatf is a smooth function on min � � Y � max y and that 
,/,2 of 
'I' oy ' 
is bounded below for a < x < 1 and min :f ::; Y ::; max .y , so that 
Al2 at _ m < 0 
'I' oy , 
for m sufficiently large. 
(6.3.21) 
(6.3.22) 
6.3 MONOTONE ITERATION TECHNIQUES IN THE CONSTRUCTION OF 177 
UPPER AND LOWER SOLUTIONS 
Define 
Then by assumption, 
implies that F is decreasing. The quantity 
is 
F(x, Yn+I ) - F(x, Yn ) , 
so that for Yn � Yn+l , 
Method 1 
We define the (nonlinear) transformations fli and f!li as follows: 
and 
if 
1 dYn+! ( Yn+! + fIJ/" ---av = Yn z) 
dYn+! = 0 
dv 
y _ _ l_dYn+ l _ l n+1 .'Y'e dz -
d Yn+ 1  = 0  
dz 
in ex < x < 1 , 0 < z < 1 ,  
in O < z < l , x = l ,  
at x = 1 for 0 � z � 1 ,  
at x = ex for 0 � z � 1 ,  
at z = 0, 
at z = 1 .  
(6.3.23) 
(6.3.24) 
(6.3.25) 
(6.3.26) 
(6.3.27) 
In this method, the equations for Yn+ 1 and Yn+ 1 are uncoupled and we show that both ffi. and f!li are 
monotone in the sense that Yn � Yn+1 and Yn � Yn+ 1 impl ies fliYn < fliYn+ 1 and f!liY" < f!liYn+ I .  
We have, i f  Yn � Yn+ l and Yn � YII+ I 
(LI - m)fJjYn = (LI - m)YII+I = 1// f(x, Yn ) - mYn in ex < x < 1 , 0 < z < 1 ,  
and 
6.3 MONOTONE ITERATION TECHNIQUES IN THE CONSTRUCTION OF 1 78 
dYn+l = 0  dV 
Y _ _ I_ dYn+1 = 
1 
n+l � dz 
dYn+1 = 0  
dz 
UPPER AND LOWER SOLUTIONS 
at x = a for 0 � z � I , 
at z = 0, 
at z = I ,  
(� - '.Y?A)�Yn+1 = (� - ,.Y?A)Yn+2 = -,.Y?AYn+! in O < z < I , x = I ,  
1 
dYn+2 ( ) Yn+2 +  .Y?Aav
= YII+! z 
dYn+2 = 0  
d V 
y _ _ I_ dYII+2 = 1 11+2 �e dz 
dYn+2 = 0 
dz 
at x = 1 for 0 � z � I ,  
at x = a for 0 � z � I ,  
at z = 0, 
at z = 1 .  
Therefore, 
[ 1 dYII+2 ] l 
I dYII+1 YII+2 + .Y?A av - )'11+1 + .Y?A JVJ = YII+1 (Z) - YII (z) � (} 
d)'I1+2 _ dYII+ 1 = () > () d V d V - , 
[y  2 __ 
I_ dYII+2 ] _ [y 1 -_
I_dYII+ I ] = I - I = O � O 11+ � dz 11+ � dz 
dYII+2 _ dYn+1 = O � O  
dz dz 
and we lherefore have the problem 
in a < x < I ,  () < z < I ,  
in (} < z <  l , x =  I ,  
on x = a, x = I ,  
on z = 0, z = I , 
where, 
in  a<x< l ,  0 < z < I ,  (6.3.28) 
in 0 <z < I ,  x= I ,  (6.3.29) 
at x = 1 for 0 � z � I , (6.3.30) 
at x = a for 0 � z � I ,  (6.3.31) 
at z = 0, (6.3.32) 
al z = 1 .  (6.3.33) 
By  the strong maximum principle, W > () in a < x < 1 and W > () in () < z < I (un less w, W == () in which case 
9Yn+1 = 9Y1I and �YII+ I  = fJ2YII and the right hand sides of (6.3 .28) and (6.3 .29) are identically zero; but 
6.3 MONOTONE ITERATION TECHNIQUES IN THE CONSTRUCTION OF 179 
UPPER AND LOWER SOLUTIONS 
this happens only if Yn == Yn+ l and Yn == Yn+ 1 , since F is strictly monotonic, so !!JjYn < !!JjYn+ l and oo/iYn < 
.o/2Yn+l). It can similarly be shown that Yn � Yn+l and Yn � Yn+l implies !!JjYn > !!JjYn+l and .o/2Yn > .o/2Yn+ l .  
Method 2 
We define the (nonlinear) transformations !!Ii and.0/2 as follows: 
and 
if 
(� - 'Y'h)Yn+1 = -,Y'hYn+! 
Y +_l_OYn+! = y (z) n+l Y'h ov n 
OYn+l = 0 ov 
y __ 1_dYn+l _ 1 n+l �" dz -
dYn+ 1 = 0 
dz 
in a < x < I ,  0 < z < I ,  
i n  0 < z < 1 ,  x = 1 ,  
at x = 1 for ° � z :5 1 ,  
at x = a for 0 :5 z � 1 ,  
at z = 0, 
at z = 1 .  
In this method the equations for Yn+! and Yn+! are coupled and we show that !!Jj and .0/2 are also monotone in 
the sense that Yn � Yn+ l and Yn � Yn+ l implies !!JjYn < !!JjYn+l and .o/2Yn < oo/iYn+ 1 •  
We have, if Yn � Yn+ ! and Yn :5 Yn+ l 
and 
1 oYn+ l  ) Yn+ l + !l'h -;;v = Yn (z 
OYn+l = 0 ov 
y __ l_ dYn+! _ 1 n+l � dz -
dYn+! = 0 
dz 
in a < x < l , O < z < l ,  
in 0 < z < 1 ,  x = 1 ,  
in 0 < z < 1 ,  x = 1 ,  
at x = 1 for 0 � z :5 I ,  
at x = a for 0 :5 z :5 I , 
at z = 0, 
at z = 1 ,  
(0;. - ,!l'h)5;Yn+ 1 = (0;. - ,!l'h)Yn+2 = -,!l'hYn+2 in 0 < z < I ,  x = I ,  
6.3 MONOTONE ITERATION TECHNIQUES IN THE CONSTRUCTION OF 180 
UPPER AND LOWER SOLUTIONS 
aYn+2 = 0  
aV 
at x = a for 0 � z � 1 ,  
y _ _ I_ dYn+2 = 1 n+2 gD" dz 
dYn+2 = 0 
dz 
at z = 0, 
at z = 1 .  
Therefore, 
(� - '.Y?A)(.0/2Yn+1 -.o/2Yn) = -,.Y?A(Yn+2 - Yn+l )  
[ _1 aYn+2 ] _ [ _1 aYn+I ] _ y  ( ) _ Y ( » 0 Yn+2 + .Y?t. av 
Yn+1 + .Y?t. av - n+1  Z n Z -
aYn+2 _ aYn+1 = O � O , 
av av 
[y 2 _ _ I_ dYn+2 ] _ [y 1 __ 
l_dYn+I ] = l _ l = O � O n+ gD" dz n+ gD" dz 
dYn+2 _ dYn+1 
= 0 � 0 
dz dz 
and we therefore have the problem 
in a<x< 1 ,  0 < z < 1 ,  
on x = a, x = 1 ,  
where, 
w = .o/jYn+1 -.o/jYn · 
in a<x<l ,  0 < z < 1 ,  
in 0 <z < 1 ,  x= I , 
at x = 1 for 0 � z � 1 ,  
at x = a for 0 � z � 1 ,  
at z = 0, 
at z = I .  
(6.3.34) 
(6.3.35) 
(6.3.36) 
(6.3.37) 
(6.3.38) 
(6.3.39) 
By the strong maximum principle, w > 0 in a < x < 1 .  This implies that the right hand side of (6.3 .35) is 
negative so that 
L2W $ 0  
B2W �  0 
where, 
in 0 <z < 1 ,  x= l ,  
on z = 0, z = 1 ,  
By  the strong maximum principle, W >  0 i n  0 < z < 1 .  Therefore, Yn $ Yn+ l and Yn � Yn+1 implies .o/jYn < 
.o/iYn+! and §'iYn < §'iYn+! .  It can sim ilarly be shown that, Yn � Yn+ ! and Yn � Yn+! implies .o/iYn > .o/iYn+ ! 
and §'iYn > .o/i.Yn+l .  
Method 3 
We define the (nonlinear) transformations .o/i and §'i as follows: 
and 
if 
6.3 MONOTONE ITER A TION TECHNIQUES IN THE CONSTRUCTION OF 181 
(� - '91�.)Yn+ 1  = -,91I.Yn 
1 JYn+1 y ( )  Yn+ 1  + 
911. av
= n+1 Z 
JYn+1  = 0 
()v 
y __  I_dYn+1 = 1 n+1 � dz 
dYn+1 = 0  
dz 
UPPER AND LOWER SOLUTIONS 
in a<.x<I ,  0 < z < 1 ,  
i n  0 <z < 1 , x= 1 , 
at x = 1 for 0 :5 Z :5 1 ,  
at x = a for 0 :5 z :5 1 ,  
at z = 0,  
at  Z = 1 .  
In this method the equations for Yn+1 and Yn+1 are coupled and we show that !!Ii and [!Ji are mon% ne in the 
sense that Yn :5 Yn+1 implies t11at .o/iYn < .o/iYn+ 1  and this in turn implies that 9jYn < 9jYn+ l . We have, if  
Yn :5 Yn+ I 
and 
1 OYn+1  ( ) Yn+1 + 911. av = Yn+1  z 
()Yn+ 1  = 0 
Jv 
Y __ 1_ 
dYn+1 - 1 n+1  .cI'e dz 
dYn+1 = 0  
dz 
(� - ,91I.)ff2Yn+ 1  = (L-;. - ,91I.)Yn+2 = -,91I.Yn+2 
1 ()Yn+2 Y ( ) Yn+2 + [PI. � = n+2 z 
()Yn+2 = 0 
()v 
y __ I_dYn+2 _ 1 n+2 � dz -
dYn+2 = 0 
dz 
Therefore, 
in a < x <I , O < z <  1 ,  
in 0 < z < 1 ,  x = I ,  
at x = 1 for 0 :5 Z :5 1 ,  
at x = a for 0 :5 z :5 1 ,  
at z = 0, 
at z = 1 ,  
in a < x < 1 ,  0 < z < 1 ,  
in O < z < l , x = l ,  
at x = 1 for 0 :5 z :5 1 ,  
at x = a for 0 :5 z :5 1 ,  
at z = 0, 
at z = 1 .  
i n  0 < z < 1 ,  x = 1 ,  
6.3 MONOTONE ITERATION TECHNIQUES IN THE CONSTRUCTION OF 182 
UPPER AND LOWER SOLUTIONS 
1 dYn+2 [ 1 dyn+1 ] _ ( )  Y ( )  [Yn+2 + YI. a;-]- Yn+l + YI. av - Yn+2 z - n+l z 
dYn+2 _ dYn+l = O � O , dV dV 
[Yn+2 -_
1_ dYn+2 ] _ [Yn+l -_1_ dYn+l ] = 1 - 1 = 0 � 0 � dz £V'e dz 
dYn+2 _ dYn+1 = O � O  
dz dz 
al x = 1 for 0 � z � 1 ,  (6.3.42) 
at x = a for 0 � z � 1 ,  (6.3.43) 
at z = 0, (6.3.44) 
at z = 1 .  (6.3.45) 
Therefore, we have the problem 
L2W � 0  
B2W � 0, 
where, 
in 0 <z < 1 ,  x= l ,  
on z = 0, z = 1 
and by the strong maximum principle, W >  0 in 0 < z < 1 .  This impl ies that the right hand side of (6.3 .42) is 
negative so that 
in a < x < 1 ,  0 < z < 1 ,  
Oil X = a, x = 1 ,  
where, 
w = .o/Yn+l -.o/Yn · 
By the strong maximum pri llciplc, w > 0 in a < x < 1 .  Therefore, Yn � YM I implies f12Y n < f12Y n+ l which 
in turn implies that �Yn < �Yn+ I ' It can similarly be shown that, Yn � Yn+l implies f12Yn > .:12Yn+! which 
in turn implies that �Yn > �Yn+l . 
If in addition to the assumptions onf, we assume thatf is monotone increasing, the following two iteration 
procedures yield alternating sequences which form two monotone sequences bounding the solution from 
above and below. Therefore, by assumption, for Yn � Yn+ 1 
(6.3.46) 
Method 4 
We define the (nonlinear) transformations � and f12 as follows: 
and 
if 
in a < x < 1 ,  0 < z < 1 ,  
in 0 < z < 1 ,  x = 1 ,  
6.3 MONOTONE ITERATION TECHNIQUES IN THE CONSTRUCTION OF 1 83 
1 dYn+l Y ( ) Yn+l  + fJJl. JV = n Z 
dYn+l =0  
dV 
Y __ I_ dYn+ l _ 1 n+l � dz 
-
dYn+1 = 0  
dz 
UPPER AND LOWER SOLUTIONS 
at x = 1 for 0 :::; z :::; 1 ,  
at x = a for 0 :::; z :::; 1 ,  
at z = 0, 
at z = 1 .  
In this method the equations for Yn+1 and Yn+1 are uncoupled and we show that .o/i and fili are monotone in 
the sense that Yn :::; Yn+ 1 and Yn � Yn+ l impl ies .o/iYn < .o/iYn+1 and filiYn > filiYn+ l . 
We have, i f  Yn :::; Yn+ 1  and Yn :::; Yn+ 1 
and 
1 dYn+1 y ( ) Yn+ l  + fJJl. 
JV = n z 
dYn+1 = 0 
dV 
Y __ I_ dYn+l _ l n+l � dz -
dYn+1  = 0  
dz 
(� - ,fJJl.),o/jYn+ l  = (� - ,Y'I.)Yn+2 = -,fJJI.Yn+l 
1 dYn+2 Y ( ) Yn+2 + fJJl. ---av = n+1  Z 
dYn+2 = 0 
dV 
Y __ 1_dYn+2 - 1 n+2 &'e dz 
dYn+2 = 0 
dz 
Therefore, 
(� - ,fJJl.)(.o/;Yn+1 -S'2Yn)  = -,YI.(Yn+1 - Yn)  $ 0 
in a < x < I , O < z < l ,  
in O < z < l , x =  1 ,  
at x = 1 for 0 :::; z :::; 1 ,  
at x = a for 0 $ z $ I ,  
at z = 0, 
at z = 1 .  
in a < x < 1 ,  0 < z < 1 ,  
in 0 < z < 1 ,  x = 1 ,  
at x = 1 for 0 $ Z $ 1 ,  
at x = a for 0 :::; z $ 1 ,  
at z = 0, 
at z = 1 .  
in a < x <  1 , 0 < z <  1 ,  
in O < z < l , x = l ,  
[ 1 JYn+2 1 I 1 aYn+l ] Yn+2 + 
fJJl. � - Yn+1 +.<h" --a;- =Yn+l (z)-YI1(z)$O  at x = 1 for 0 $ z $ 1  ,
dYn+2 _ dYn+1 = 0 > 0 
dV dV 
- , at x = a for 0 $ z $ 1 ,  
(6.3.47) 
(6.3.48) 
(6.3.49) 
(6.3.50) 
6.3 MONOTONE ITERATION TECHNIQUES IN THE CONSTRUCTION OF 184 
UPPER AND LOWER SOLUTIONS 
[y 2 __  
1_dY,,+2 1 _ [Y J __ 
l_dY"+l l = l - l = O � O n+ ,CVl, dz n+ �e dz 
dYn+2  _ dYn+1 = 0 � 0 
dz dz 
at z = 0, 
at z = 1 .  
(6.3.51) 
(6.3.52) 
Therefore, we have the problem 
in a < x < 1 , 0 < z < 1 ,  
i n  0 < z < 1 ,  x = 1 ,  
on x = a, x = 1 ,  
on z = 0 ,  z = 1 ,  
where, 
w = .o/Y,,+1 -.o/Yn and W = 5Y,,+ 1 -5Y" and B l  and B2 arc boundary operators. 
By Lhe strong maximum principle, w < ° in a < x < 1 and W >  0 in 0 < z < 1 (unless w, W = 0 in which case 
9jYn+l = 9jYn and S"2Yn+ ! = .o/iYn and the right hand sides of (6.3 .47) and (6.3 .48) are identically zero; but 
this happens only if  y" = Y,,+ i and Yn = Yn+ i ,  since f is  strictly monotonic, so .o/jYn > .o/jYn+J and .o/ZYn < 
.o/ZYn+l). Similarly, Yn � y,,+ !  and Y" � Yn+ 1 impl ies fi)Yn > .o/jYn+1 and .o/ZYn < .o/ZYn+ l . This also implies 
that fli2 and S"l are monotone in the sense that y" � Yn+ 1  and Yn � Y,,+i impl ies fli2y "  < fli2Y,,+ 1 and 
.o/i2yn > S",fyn+ l . 
Method 5 
We define the (nonlinear) transformations .'1j and fli as fol lows: 
and 
if 
(Lz - (Y'/")Y,,+I = -(.%Y,,+ 1  
1 JYn+! Yn+ 1 + Y'/" av = Yn(z) 
JYn+1 = 0 
Jv 
Yn+! __ 
l_dYn+1 = 1 
,CVl, dz 
dY,,+1 = ° 
dz 
in a < x < 1 ,  ° < z < 1 ,  
in 0 < z < I ,  x = J ,  
at x = 1 for ° � z 0::::; 1 ,  
at x = a for 0 � z :;:; 1 ,  
at z = 0, 
at z = 1 .  
In Lhis meLhod the equalions for Y,,+ I and y,,+ I are coupled and we show (hat fi) and fJ2 are monotone i n  the 
sense that Yn 0::::; y,,+ I and Y" � Y,,+ I implies fi)y" < fi)Yn+ 1 and .o/ZY n < .o/ZY n+ I ·  
6.3 MONOTONE ITERATION TECHNIQUES IN THE CONSTRUCTION OF 185 
UPPER AND LOWER SOLUTIONS 
We have, if Yn ::; Yn+ l  and Yn � Yn+ l 
and 
LtfJjYn ::: LtYn+l  ::: 1jJ2/(x, Yn) 
(I...-;. - 'Y'h).o/ZYn ::: (I...-;. - 'Y'h)Yn+1 ::: -,Y'hYn+ l  
1 JYn+l Y ( ) Yn+l + Y'h � = n Z 
JYn+l = 0  
Jv 
Y __ I_dYn+l = 1 n+l ,lj'e dz 
dYn+1 = 0  
dz 
1 dYn+2 Yn+2 + Y'h �= Yn+l (z) 
dYn+2 = 0 
Jv 
Y __ 1_dYn+2 _ 1 n+2 � dz -
dYn+2 = 0 
dz 
in a < x < I , O < z < l , 
in O < z < l , x ::: l ,  
at x ::: 1 for 0 ::; z ::; 1 ,  
at x ::: a for 0 ::;  z ::; 1 ,  
at z ::: 0, 
aL z ::: 1 .  
in  a <x < 1 ,  0 < z < 1 ,  
in O < z <  l , x ::: 1 ,  
aL x ::: 1 for () ::; z ::; I ,  
aL x ::: a for 0 ::; z ::; 1 ,  
at z ::: 0, 
aL z ::: I .  
Therefore, 
(� - s9'�)(ff2Yn+l - ff2Yn) ::: -,9'h(Yn+2 -Yn+ l ) in 0 < z < I ,  x ::: 1 ,  
[ 1 JYn+2 I I I  JYn+1 Yn+2 + 9'h �. - Yn+1 + Y'h JY]= Yn+1 Cz)- Yn Cz) ::; O  at x ::: 1 for 0 ::;  z ::;  I ,  d�n:2 _ d�n:l = O � O , aL x ::: a ror 0 ::;  z ::;  1 ,  
[y 2 _ _ 1_ dYn+2 ] _ [y 1 -_I_ dYn+l ] = l _ l = O < O  0 n+ � dz n+ ,Ij'e dz - at z ::: , 
dYn+2 _ dYn+! = 0 ::;  0 at z ::: 1 .  dz dz 
Therefore, we have the problem 
in a<x< I ,  0 < z < 1 ,  
on x = a, x = 1 ,  
where, 
(6.3.53) 
(6.3.54) 
(6.3.55) 
(6.3.56) 
(6.3.57) 
(6.3.58) 
6.3 MONOTONE ITERATION TECHNIQUES IN THE CONSTRUCTION OF 186 
UPPER AND LOWER SOLUTIONS 
w = .o/iYn+1 -.o/iYn 
and by the strong maximum principle, w < ° in a < x < 1 .  This implies that the right hand side of (6.3.54) is 
positive so that 
where, 
L2W � 0  
B2W � 0, 
in ° < z < 1 ,  x = I ,  
on z = 0, Z = 1 
By  the strong maximum principle, W < ° in 0 < Z < 1 .  Therefore, Yn � Yn+ 1 and Yn � Yn+! implies .o/iYn < 
.o/jYn+1 and .o/iYn < .o/iYn+ l .  It can similarly be shown that, Yn � Yn+ 1 and Yn � Yn+ 1  implies .o/jYn > .o/jYn+ 1 
and .o/iYn > .o/iYn+ ! .  
If  the nonlinearitiesflx, y) have continuous y-derivatives, then we  may attempt a fUrlher i teration scheme to 
solve the equations (6.3 . 1)-(6.3.8) .  This is known as Newton's method. Under appropriate conditions we 
wil l  show that the Newton iterates converge monotonically .  This convergence is frequently quadratic 
(KELLER [ 14 1 ]) and hence we expect that the iterates give more accurate approximations to earlier methods 
discussed in this section. However, as we shall sec, the Newton iterates converge either from above (ifjy is 
i ncreasing in y) or from below (if jy is decreasing in y) . We therefore cannot obtain both monotone 
i ncreasing and monotone decreasing sequences and therefore upper and lower approximations for the 
problem (6.3. 1 )-(6.3 .8) as was done in earlier examples. 
Assume thatjsatisfies all previous assumptions and in addition, 
aj >0 
ay -
d 
aj . . .  an 
ay 
IS monotone mcreasmg. 
Defme 
Then it follows that 
Further, if  Yn ::; Yn+l ,  then by the above assumpLions, 
Method 6 (Newton's Method) 
We define the (nonlinear) transformations .o/j and .o/i as follows: 
(6.3.59) 
(6.3.60) 
(6.3.61) 
(6.3.62) 
and 
if 
6.3 MONOTONE ITERATION TECHNIQUES IN THE CONSTRUCTION OF 187 
UPPER AND LOWER SOLUTIONS 
af 2 af (Lt - ay (X,Yn))Yn+1 = ¢J f(x,Yn) - ay (X,Yn)Yn 
(� - '9'h)Yn+1 = -'9'hYn 
1 aYn+l ( ) Yn+l + 9'h av= Yn z 
JYn+l  = 0  
Jv 
y __ l_dYn+ l = 1 n+1 � dz 
dYn+1 = 0  
dz 
in a < x < l , O < z < l ,  
in O < z < l , x = l ,  
al x = 1 for 0 � z � 1 , 
at x = a [or 0 � z � 1 , 
at z = 0, 
at Z = 1 .  
In this method, the equations for Y n+ I and Yn+ I are uncoupled and we show that both !1j and fJi are 
monotone in the sense that Yn � Yn+l and Yn � Yn+ l implies !1jYn < !1jYn+ l and fJiYn < fJiYn+ l . 
We have, if Yn � Yn+ 1 and Yn � Yn+l 
and 
af fiT af 2 af . (L1 - ay (X,Yn)).7IYn = (� - ay (X,Yn))Yn+1 = ¢J f(x,Yn)- Jy (X,Yn)Yn In 
a < x < 1 , 0 < z < 1 ,  
(� - '9'h)fliYn = (� - ,fPh)Yn+1 = -'9'hYn in 0 < z < 1 , x = 1 ,  
1 aYn+1 Yn+1 + 9'h av = Yn (z) 
JYn+1 = 0 
Jv 
y _ _ I_ dYn+ 1 = 1  n+1 � dz 
dYn+1 = 0  
dz 
1 JYn+2 Yn+2 + fPh�= Y"+I (Z) 
aYn+2 = 0  
d V 
y _ _ 1_dYn+2 = 1 n+2 � dz 
dYn+2 = 0 
dz 
at x = 1 for 0 � z � 1 ,  
at x = a for 0 � z � I ,  
at z = 0, 
al z = l .  
in O < z < l , x = l ,  
at x = 1 [or 0 � Z � I , 
at x = a for 0 � Z � 1 ,  
at z = 0, 
al z = 1 .  
. ... Therefore, 
6.3 MONOTONE ITERATION TECHNIQUES IN THE CONSTRUCTION OF 188 
UPPER AND LOWER SOLUTIONS 
Jf !y, 2 2 Jf » (L1 - J/X,Yn+l»(.o/iYn+l - lYn) = <P f(X,Yn+l ) - <P f(x,Yn ) - J/X,Yn)(Yn+l - Yn 
= [<p2F[X;Yn+l ,Yn ]-� (X,Yn+l)](Yn+l -Yn) 
= <p2 F[x;Yn+! ,Yn] -<p2 F(x;Yn+! ,Yn+i 1  � 0 in a < x < I ,  0 < z < I ,  
(Lz - (YA)(.o/;Yn+1 -.o/;Yn) = -(YA(Yn+l - Yn ) � 0 
_1 ayn+2 ] _ [ _1 aYn+! ] _ y ( ) _ y ( » 0 [Yn+2 + YA av Yn+l + flJA Jv - n+! Z n Z -
JYn+2 _ aYn+! = O � O , Jv av 
[y  _ _ I_ dYn+2 J_ [Y _ _ 1_dYn+! ] = 1 _ 1 = O � O  n+2 &'.. dz n+l � dz 
dYn+2 _ dYn+ 1 = 0 � 0 
dz dz 
in 0 < z < I ,  x = I ,  
at x = 1 for 0 � z � I ,  
at x = a for 0 � z � I ,  
at z = 0, 
at z = 1 .  
Therefore, we have the problem 
in a < x < I ,  0 < z < I ,  
in 0 < z < I ,  x = 1 ,  
on x = a, x = 1 ,  
on z = 0, Z = 1 
where, 
(6.3.63) 
(6.3.64) 
(6.3.65) 
(6.3.66) 
(6.3.67) 
(6.3.68) 
By the strong maximum principle, w > 0 in a < x < 1 and W > 0 in 0 < Z < 1 .  (unless w, W == 0 in which 
case .o/jYn+l = .o/jYn and B'2.Yn+! = ff2Yn and the right hand sides of (6.3 .63) and (6.3 .64) are identically zero; 
but this happens only if Yn == Yn+l  and Yn == Yn+ l , since F is strictly monoton ic, so .o/jYn < .o/jYn+l and $iYn 
< $iYn+ ! ). It can similarly be shown that Yn � Yn+ l and Yn � Yn+l implies .o/jYn > .o/jYn+ l and $iYn > 
92Yn+ l . However, to show this we need to assume thatfsatisfies all previous assumptions, but 
df is now 
monotone decreasing instead of increasing as in (6.3 .59). Define, as before 
dy 
Then it foIlows that 
Further, i f  Yn � Yn+l , then by the above assumptions, 
The proof foIlows exactly, with some reversals of inequalities. 
(6.3.69) 
(6.3.70) 
(6.3.71) 
- -- ------------
6.3 MONOTONE ITERATION TECHNIQUES IN THE CONSTRUCTION OF 1 89 
UPPER AND LOWER SOLUTIONS 
The monotone iterative methods discussed in this section have appl ications il l constructive existence proofs. 
Iteration schemes are used and yield monotone convergence to the maximal solution from above, to the 
m inimal solution from below or both for some unique solutions. Newton's method may be shown to 
converge from above or below to a unique solution (the uniqueness is guaranteed from the monotone 
property of f and applying the maximum principle). Newton's method can be shown to converge 
quadratically (KELLER [ 1 4 1 ]) . However some additional continuity properties are needed. In our case h is 
required to be Lipschitz continuous in x. 
To see how these montone iterations are useful in a constructive existence proof let us consider Method 1 and 
define 
where 
We show that the following strict inequalities hold 
[ 1 > [0' r; < Yo, �l > �o and Yl < Yo · 
We have 
So 
(� -M)[, = '.9'h([O -�o ) -M[o 
1 a�l 
J\ + fJJh av = 1:o(z) 
ay, -=-= 0 
av 
(6.3.72) 
(6.3.73) 
(6.3.74) 
in a < x < l , O < z < l , 
in 0 < z < 1 ,  x = 1 ,  
at x = 1 for 0 � z � 1 ,  
at x = a for 0 � z � 1 ,  
at z = 0, 
at z = 1 .  
(� - M)(1:1 -1:0 ) = '.9'h(1:o -�o) -M1:o -� 1:0 + M1:o = '.9'h(r - �)- � r � 0 i n  0 <z < 1 ,  x = 1 ,  
1 a�l . . 1 alo 
[�l + fJlh dV  ] - [ lo + fJJh a;-] = 1:o(z) -r 2 0 
aYI ayo -=---=-- 2 0  
av  dV 
[1:1 - -
1 dII 1 _ I1:o __ 1 dl::O ] 2 0 
� dz fYl, dz 
d[l _ d1:o 2 0  
dz tlz 
at x = J for 0 � Z � 1 ,  
at x = a for 0 � z � 1 ,  
at z = 0, 
at z = 1 .  
6.3 MONOTONE ITERATION TECHNIQUES IN THE CONSTRUCTION OF 190 
UPPER AND LOWER SOLUTIONS 
Therefore, by LIle sLrong maximum principle, �I > �o and I:I > I:o. (Assume that LI �o :t rpz f(x,�o) and 
LzI:o :t '9'�(I:o -�o » '  A similar argument holds jf we have to show LIlat .YI < .Yo and Pi < Yo. 
Now define �z = $i�I , I:Z = .o/'z [z · Then [I > I:o and �I > �o i mplies that �z = $ill > $i�o = �I and 
I:z =.o/'ZI:l >91I:o = I:l ' By induction, LIle sequences defined by �I = .o/i�o' �n =.o/i�n-l and I:I = .o/'zI:o , 
I:n = .o/'ZI:n-l are monotone increasing. Similarly, the sequences defined by .YI = .o/i.Yo ,  Yn = .o/iYn-1 and 
Pi = ,q;Vo, � = ,q;�_1 defines monotone increasing sequences. 
FurLllermore, we have �n < Yn and rn < � for all n: 
�o < �I < �z <" '�n-l < �n <· · ·< .Yn < .Yn-I < . . . < YI < Yo 
ro < I:I < I:z < .. ·I:n-I < L < . . .  < � < �-I < . . . < r; < Yo 
In fact, �o < Yo and I:o < ft> ; suppose that �n-l < Yn-I and rn-I < �-I ' Then �n = .o/iln-I < .o/iYn-1 = Yn and 
I:n = .o/'2I:n- 1 < 91�_1 = �, so the proof follows [rom induction. 
Since LIle sequences (lk l , (I:k l , (Yk l and (�l are monotone and unifonnly bounded, the pointwise limits 
y(x,z) = lim yk (x,z) , Y(x, z) = lim Yk (x, z) - k�oo - k�oo 
rez) = lim rk (z) , y(z) = lim � (z) k�oo k�� 
all exist. 
(6.3.75) 
(6.3.76) 
To show that the sequences defined above converge uniformly and LIlat the l imits above are is in CZ+a, we 
may usc standard continuity arguments described in Chapter 4. A similar proof holds for all other iterative 
methods described in tJl is section. 
6.3.3 Conclusions and Remarks 
The novelty of such problems discussed in th is thesis is that the coupling of differential equations in the 
boundary condi tions of the particle model and the mass transfer resistances in a particle is incorporated into 
the l iquid phase balance equations. These equations pose no problems in applying several monotone iteration 
methods given in l iterature for scalar ell iptic equations. 
Such monotone iterative techniques may also useful in numerical computation. We have seen in 
section 6 . 1 and section 6.2 that upper and lower solutions arc useful in the investigation of qualitive 
properties of solutions and do prove to be good as approximate solutions. These upper and lower solutions 
can be computationally improved by such monotone iteration techn iques. These techniques may also be 
generalised to LIle general system Sn, Bn and its corresponding steady state system Sn ' Bn . 
We have discussed in section 3.7 how the method of upper and lower solutions for the systems Sm 
Bn and its corresponding steady slate system Sn ' Bn may be extended to a nonlinear finite difference system 
which is a discrete version of tJle continuous problems Sn, Bn and Sn ' Bn . It was noted that solving coupled 
systems involves solving fewer elluations. We would thus expect convergellce for the iterative methods 
given in LIlis section involving solving coupled systems to give a faster rate of convergence. 
This section also gives us an idea of conditions that our nonlinear reaction functions have to satisfy in 
order to obtain mOllotone sequences, alternating sequcnces and accelerated monotonc sequences and how 
these results may be general ised to the systems Sn, Bn and Sn ' Bn . We have already studied in section 3 .6 
-------
6.3 MONOTONE ITERATION TECHNIQUES IN THE CONSTRUCTION OF 191 
UPPER AND LOWER SOLUTIONS 
some o[ the properties that our nonlinear functions in the system Sn. Bn have to satisfy in order to get 
monotone and alternating sequences (see Lemma 3.6.6 and Remark 3 .6.6). We have also seen in section 4.3 
that such properties also hold for the system Sn. En (see Lemma 4.3.6 and Remark 4.3.6). It would be 
useful to generalise and to study some monotone sequences which give accelerated convergences to the 
systems Sn. Bn and Sn , Bn using the methods given in this section. Such a method may be a generalisation of 
Newton's method to systems of equations and may prove to be computationally useful. 
6.4 UNIQUENESS AND EXISTENCE THEOREMS FOR A MODEL OF AN 192 
ARTIFICIAL KIDNEY 
6.4 Uniqueness and Existence Theorems for a Model of an Artilicial Kidney 
In this section, we shall look at a specific example found in literature (LIN [ 1 70]) of a modified urea transfer 
model for predicting urea removal in a compact artificial kidney. S ince the transient behaviour of such models 
are determined by the steady-state behaviour of resul ting equations, this section shall only examine the 
uniqueness and existence of solutions to the steady-state problem. This problem involves three elliptic 
equations which arc all coupled together in their boundary conditions. The approach taken in proving these 
theorems differs from the approach discussed in the early part of this thesis and the problem differs slightly 
as well. 
The method used in this section may be generalised to proving existence and uniqueness to the 
general system Sn, Bn and its corresponding steady state system Sn ' Bn . 
6.4.1 The Artificial Kidney 
An artificial kidney is  a compact device that is that is worn by a patient and uses a replacement unit for 
removal of niLrogeneous waste products from the blood. Within this device are micro-encapsulated enzyme 
particles which consists of a layer of membrane and an inner urease solution. This particle membrane pcnnits 
the urea to diffuse through and into the urease solution but retains thc urease because of thc larger urease 
molecules. Ammonia is generated by the enzymatic conversion of urea and in turn reacts with the ion 
exchange resins which are suspended in the urease solution. In addi tion, the urease solution may also contain 
a small amount of activated carbon for removal of uric acid and creatinine. 
We shall develop a generalised model for the artificial kidney. The model applies as well to removal 
of other toxic uremic molecules, which is also of main concern of the artificial kidney . Many early models 
(CHANG and POZNANSKY [68] , VIETH el al. [292]) are mainly concerned with reaction and diffusion only 
within the m icro-encapsulated enzyme particles and are not concerned with what happens outside of these 
particles. However a few models such as that of LEVINE and LACOURSE [ 1 68] examine the performance of 
the whole artificial k idney. These people also show that an artificial kidney of l Ocm long is sufficient to 
remove 90% of the urea from the bloodstream. 
In this work, predictive models for the artificial kidney are developed. This model is similar to that 
proposed for a Ouidised bed biofilm reactor (SHI El l el al. [263] , PARSI IOTAM el al. [226]). However, for 
the present model, the boundary condition for the particles has a correction factor to account for external 
diffusion within the layer of membrane. This diffusion wi thin the membrane could be quite significant 
(MCELWAIN [ 1 80]). Despite this  feature we are sti l l  able to obtain uniquness and existence theorems. 
However, in this section, we shall develop melhods for proving un iqueness and exislence theorems that 
differs from earlier melhods and appears to be more suitable to this range of problems. 
The model is developed by assuming M ichaelis-Menten kinetics but for the sake of uniqueness and 
ex istence there is no additional difficulty in assuming the general case where micro-encapsulated enzyme 
particle kinetics are monotone increasing and nonnegative. 
6.4.2 The Artificial Kidney Model Formulation 
The artificial kidney, the typical schematic is shown in FIG. 6.9, consists of a compact device in which 
micro-encapsulated enzyme particles (see FIG . 6. 1 0) are in a Iluidised state. 
The key processes of this system with uniform membrane thickness is: 
(i) Urea transport through the anilkial kidney. 
(ii) diffusion within \lIe microencapsulated urease particle lIlembrane 
(iii) reaction-dirrusion wi\llin a single microencapsulated urease particle 
6.4 UNIQUENESS AND EXISTENCE THEOREMS FOR A MODEL OF AN 1 93 
ARTIFICIAL KIDNEY 
Microencapsulated urease particle-Artificial K idney Model Development 
The mathematical model of the artificial kidney urea conversion process is divided into three submodels. The 
"Microencapsulated urease particle Model" is concerned with diffusion within the urease solution and urea 
conversion by the microencapsulated urease particles. The "Microencapsulated urease particle membrane 
Model" is concerned with diffusion within the urease particle membrane. The "Artificial Kidney Plug Flow 
Model" is concerned with the hydraulic plug flow transport of urea through the artificial kidney and is 
external to a microencapsulated urease particle. The three models are coupled by urease particle-bulk liquid 
boundary conditions to yield an overall model for urea conversion in an the artificial kidney. 
i 
F IG .  6.9 Artificial Kidney F IG. 6. 1 0  Microencapsulated particle with urease solution 
The one dimensional dispersion equation for describing the urea concentration in the blood stream passing 
through the artificial kidney can be represented by 
d2Sb dSb I D-.,.--u--nAkL(S/" -sm ) = 0, dZ" dZ / r=rmp (6.4. 1 )  
where Sb  i s  the urea concentration in  the blood stream passing through the artificial kidney, Sm i s  the urea 
concentration in the membrane, D is the dispersion coefficient, u is the velocity of blood flow, n is the 
number of microencapsulated particles per unit volume of fluid, A is the microencapsulated particle surface 
area, h is the mass transfer coefficient and Z the axial coordinate. 
The Reynolds number for the present system is given by 
rile = dpup 
J1 ' 
(6.4.2) 
where dp is the diameter, p is the density and J1 is the viscoscity of the microencapsulated particle and the 
Peclct number for this system is given by 
� = ul 
D ' 
(6.4.3) 
6.4 UNIQUENESS AND EXISTENCE THEOREMS FOR A MODEL OF AN 1 94 
ARTIFICIAL KIDNEY 
with I being the length of the artificial kidney. It is common to neglect the second order differential term in 
(6.4. 1 )  in the present system since the Peclet number for this system is often quite high (LEVEN SPIEL [ 166, 
p .275] ,  LIN [ 1 70] ,  SHIEH et al. [263]). However, we shall include this term in our model equations as there 
is no additional difficulty for the purposes of uniqueness and existence theorems. 
The urea distribution profiles in both membrane and solution phases are expected to be different because of 
different urea diffusion coefficients in these phases. Therefore, the urea balance in both phases can be 
expressed as 
D ( a
2Sm +�aSm )= 0  m ar2 r ar ' (6.4.4) 
D ( a
2s +�  as ) _ Vms 
S ar2 r ar - s+ km ' (6.4.
5) 
where Sm is the urea concentration in the membrane and s is the urea concentration in the urease solution, Dm 
and D s are the corresponding urea diffusion coefficients, V m is the maximum reaction rate, km i s  the 
Michaelis constant and r is the radial coordinates. 
The boundary conditions are given by 
dSb =0 
dZ 
DdSb = u(Sb -Sb ) 
dZ ,I 
D aSm D as II mTr= of ar ' Sm = S 
at Z = I, 
at Z = 0, 
at r = rus, 
at r = rmp, 
where H is the partition coefficient between the membrane and the urease solution. 
In d imensionless form, the equations (6.4. 1 )  and (6.4.4)-(6.4.9) take the following form 
_1 �(x2 aYm ) = 0 x2 ax ax 
_1 �(x2 ay) = ¢2_Y_ x2 ax ax 1 +(3y 
1 d2y dY -----= �.Y?h(Y -Ym) fl'e dz2 dz 
aYm ay 
ax 
= r ax '  Ym 
= lIy 
1 ay Y + -� =  Y(z) m 
.Y?h av 
ay = 0 av 
i n  a < x < 1 ,  0 < z < I ,  
in 0 < x < a, 0 < z < I ,  
in 0 < z < I ,  x = I ,  
in 0 < z < I ,  x = a, 
at x = 1 for 0 < z < I ,  
at x = 0 for 0 < z < 1 ,  
(6.4.6) 
(6.4.7) 
(6.4.8) 
(6.4.9) 
(6.4.10) 
(6.4.1 1) 
(6.4.12) 
(6.4.13) 
(6.4.14) 
(6.4.15) 
6.4 UNIQUENESS AND EXISTENCE THEOREMS FOR A MODEL OF AN 195 
Y __ 1 dY = 1 
� dz 
dY = 0  
dz 
ARTIFICIAL KIDNEY 
at z = 0, 
at z = 1 � 
at steady-state the equations, where the dimensionless variables are 
and the parameters are 
Equation (6.4. 10) is l inear and can be integrated to give 
which is further integrated to give 
Ym =_£L+C2 , x 
(6.4.16) 
(6.4.17) 
(6.4.18) 
(6.4.19) 
(6.4.20) 
(6.4.21) 
where Cl and C2 are constants of integration. Substitution of equations (6.4 . 10) and (6.4.21 )  into equation 
(6.4 . 14) g ives 
(6.4.22) 
The integration constant C2 can be eliminated by inserting equations (6.4. 10) and (6.4.22) into equation 
(6.4. 13)  so that 
where, 
ay 
ax 
= [JJl.m(Y -Hy) 
fA - [JJI. � - ya([JJI.+  a( I -.%)) 
at x = a, 
From equations (6.4. 13) and (6.4.20), we can show that 
(6.4.23) 
(6.4.24) 
(6.4.25) 
Therefore equation (6.4.25) can be substituted into (6.4. 14) which is substituted into equation (6.4. 13) so 
that the original set of equations is reduced to the following set of equations 
in 0 < x < a, () < z < 1 ,  (6.4.26) 
6.4 UNIQUENESS AND EXISTENCE THEOREMS FOR A MODEL OF AN 196 
ARTIFICIAL KIDNEY 
1 d2y dY 2 aY
I 
--
2 
-- = a y-
� dz dz ax x=a 
in 0 < z < 1, x = a, (6.4.27) 
ay = .9'hm(Y - /-ly) ax at x = a for 0 � z � 1 , (6.4.28) 
ay = 0  av at x = 0 for 0 � z � 1 ,  (6.4.29) 
Y __ 1 dY = I at z = 0, (6.4.30) 
� dz 
dY = 0  at z = 1 .  (6.4.31) 
dz 
We see that the boundary condition (6.4.28) has a correction factor to account for the external diffusion 
within the microencapsu lated urease particle membrane. We note that this problem reduces to the case 
discussed in sections 6.2 and 6.3 in the limit as Il tcnds to unity and [Phm tcnds to [Ph. 
For purposes of un iqueness and existence, we shal l substi tute equation (6.4.26) for the more general 
equation 
in 0 < x < a, 0 < z < I ,  (6.4.32) 
where the Michaelis-Menten term in (6.4.26) is replaced by an arbitrary function f(y) which is monotone 
increasing, nonnegative and Lipschitz continuous with respect to y. 
6.4.3 Uniqueness and Existence the()rems 
We have reduced three coupled ell iptic equations to two coupled ell iptic equations by solving (6.4. 10) and 
have obtained a system similar to that in section 6.2 and section 6.3.  The system (6.4.26)-(6.4.32) is 
nonstandard in the way that (6.4.32) is coupled to (6.4.27) via the boundary condition (6.4.28). 
Although i t  may be observed from well known results (e.g.  HILTMAN AND LORY [ 1 26]) that for a 
given a unique Y(z) there exisL<; a unique y(x, z) , il cannol be assumed because of the coupled nature of lhe 
equations for y(x, z) and Y(z) that Y(z) is in fact unique for a given z. We therefore consider the one­
parameter family of solutions, Yjl(x) where Yjl(x) is a solution to the microencapsulated urease problem and 
where J1 has captured the z. We shall establish a one-to-one correspondence between Yjl(x) and y(x, z) . In 
summary, we have the fol lowing 
Vz E [0, 1 ] ,  :3 ! J1(z) : y(x, z) = Yjl(x), 
Consider the following o.d.e. that is paramelriseel by height z 
1 d ely 2-(x2�)= lP2 f(yJ.l ) in ° < x < a, x dx dx 
with the boundary conditions 
at x = 0, 
(6.4.33) 
(6.4.34) 
(6.4.35) 
6.4 UNIQUENESS AND EXISTENCE THEOREMS FOR A MODEL OF AN 197 
1 dy --�+Hy =J1 al x = 1 ,  f/J;., dx j.l m 
ARTIFICIAL KIDNEY 
(6.4.36) 
and where yj.l(x) may be thought of as the family of solutions y(x) wilh given const.anl J1 andf is monolone 
increasing, nonnegalive and Lipschitz with respecl lo yj.l with Lipschitz const.ant K. We have the following 
comparison results for yj.l' 
Lemma 6.4.1 (Comparison Results) 
Let Yj.l(x) denote the solution of (6.4.34)-(6.4.36) and suppose that two functions �j.l and Yj.l can be 
found so that the differential inequalities 
d--A-�(x2 yp. ) � lP2 f(Yp. ) in 0 < x < a, (6.4.37) x dx dx 
hold with the boundary conditions 
and the differential inequalities 
1 d dy 
2-(x
2 -po ) ? lP2 fey ) in 0 < x < a, x dx dx -po 
hold with the boundary conditiolls 
Then 
Proof 
dy 
-j.l � O  
dx 
at x = 0, 
I dy 
--�+J-1y �J1 at x = a. f/J;"m dx -po 
From (6.4.34) and (6.4.37), we have 
1 d 2 d 2 2 
x2 dx
(x 
dx
(�j.l -yp.»? lP  f(�)-lP f(Yj.l ) ?m(�j.l -yj.l ) 
[or m ? 0, since f is monolone increasing. Hence, 
1 d 2 d --(x -(y -y »-m(y -y ) ? O x2 dx dx -j.l p. -j.l p. 
From (6.4.35)-(6.4.36) and (6.4.38)-(6.4.39), we have 
(6.4.38) 
(6.4.39) 
(6.4.40) 
(6.4.41) 
(6.4.42) 
(6.4.43) 
(6.4.44) 
(6.4.45) 
6.4 UNIQUENESS AND EXISTENCE THEOREMS FOR A MODEL OF AN 198 
implying by the maximum principle Illat 
y -y" � O  or y � y" .  -JI. r -JI. r 
ARTIFICIAL KIDNEY 
(6.4.46) 
(6.4.47) 
(6.4.48) 
I n  the same way from (6.4.34)-(6.4.36) and (6.4.40)-(6.4.4a.), we find that yJl. � Yw Combining these 
results, we therefore have shown that the upper and lower bounds 
(6.4.49) 
are valid.O 
The equation (6.4.49) is a special case of theorems established for ell iptic equations by AMANN [5] .  The 
functions y and y" where y � y" are lower and upper functions of (6.4.34)-(6.4.36) respectively and so -JI. r -JI. r 
there exists a solution yJl. of (6.4.34)-(6.4.36) where y � yJl. � y" . 
-JI. r 
Lemma 6.4. 1 implies a solution of (6.4.34) which satisfies boundary conditions (6.4.35)-(6.4.36) 
must be unique for if u and v are solutions, we can let Y..JI. = u and yJl. = v ,  to rind that u == v. 
Thus given J1 is unique for a given z, then yJl. is unique. However, we cannot assume that J1 is unique 
for a given z and for that we shall need the fol lowing lemma: 
Lemma 6.4.2 
YJl.(x) is monotonically increasing in J.1 and is Lipschitz in II. 
Proof 
Consider the equations YJl.l and y� with J11 � J12 and where YJl.l satisfies the equations 
1 d dy 
2-(x
2 ---.&..)- ip2 f(y" ) = 0 in 0 < x < a, x dx dx rl 
with the boundary conditions 
dYJl. l 
=0  
dx 
1 dy 
-----.&..+Hy =}1 fIJI. dx Jl.l 1 m 
and y� satisfies Ille equation 
at x = 0, 
at x = 1 ,  
1 d dy 
2-(x
2
-.!!:l..) - ip
2f(YII ) = 0 in O < x < a, x dx dx r2 
with Ihe boundary conditions 
(6.4.50) 
(6.4.51)  
(6.4.52) 
(6.4.53) 
6.4 UNIQUENESS AND EXISTENCE THEOREMS FOR A MODEL OF AN 
ARTIFICIAL KIDNEY 
dyJlz = 0 
dx 
_
1
_
dYJl2 +1-1  -f)J/" dx YJl2 P2 m 
at x = 0 , 
at x = 1 .  
These equations satisfy the differential inequalities 
1 d dy 
""3-(x2--..!:l.)- ¢J2f(YII )= 0$ 0 in O < x < a, x dx dx ,-1 
with the inequalities in the boundary conditions 
dYJll = O � O  
dx 
1 dYJl 
r;;-__ 1 + I-IYII = PI � P2 .Y'J"", dx ,-1 
at x = 0 , 
at x = 1 ,  
and YJlz satisfies the differential inequalities 
with the inequalities in the boundary conditions 
at x = 0, 
at x = 1 .  
and from Lemma 6.4. l ,  this implies that 
Hence YJl is monotonically increasing in p. 
To show that YJl(x) is Lipschitz in Il, we need to only consider the case whcn ILl � P2. 
199 
(6.4.54) 
(6.4.55) 
(6.4.56) 
(6.4.57) 
(6.4.58) 
(6.4.59) 
(6.4.60) 
(6.4.61) 
(6.4.62) 
Let us consider the equations (6.4.50)-(6.4.55) and look at the difference. We then obtain the 
inequalities 
with the inequalities in the boundary conditions 
d 
d./YJl1 - YJl2 ) = ° at x = 0, 
1 d 
f)J/" dx 
(YJl1 - YJl2 ) + H(YJll - YJl2 ) = PI -P2 � 0 at x = 1 ,  m 
(6.4.63) 
(6.4.64) 
(6.4.65) 
6.4 UNIQUENESS AND EXISTENCE THEOREMS FOR A MODEL OF AN 200 
ARTIFICIAL KIDNEY 
.- ._-_._._ ._--_ ..... _------_. --.---- -----_. 
From the maximum principle we see that the max imum value for Ill is  problcm is at thc boundary x =  I and 
Jil - Ji2 ;::: 0 implies Y/-l1 - Y/-l2 ;::: 0 so that 
(6.4.66) 
Hence Y/-l(x) is Lipschitz in pwith Lipschitz constanl k = I .U 
Let us define the following function 
(6.4.67) 
which is obtained from integrating (6.4.32) and applying the boundary conditions (6.4.29). 
Lemma 6.4.3 
F(J1) is monotonically increasing in p and Lipschitz in p. 
Proof 
To show that F(p) is monotonically incrcasing in p, wc notc that Y/-l is monotonically increasing in Ji, 
f(Y/-l(x» is monotonically increasing in Y/-l and Illcrcfore FClt) is monotonically increas ing in Jt. 
To show that F(J1) is Lipschitz in Ji, we note from Lemma 6.4.2 that Y/-l(x) is Lipschitz in Ji . Therefore 
rlX 
I I F(p)- F(J1+ )1 I = cp2 11 J o x
2 f(Y/-l )-x2 f(Y/-l. )dxll 
= cp2 11J: x2 (f(Y/-l )-f(Y/-l. » dxll 
� cp2 J;"x2(f(Y/-l )-f(Y/-l. » lIdx 
� cp2 IoIX 1I(f(Y/-l )- fey /-l. » l l dx 
� cp2 K JolX l Iu -p + l ldx 
� cp2aKIlJi -Ji+ 1 I  
and F(J1) i s  Lipschitz in p with LipschilZ constant k = fK {l-a).O 
Consider the following differential equation 
in 0 < z < I ,  x = a, (6.4.68) 
6.4 UNIQUENESS AND EXISTENCE THEOREMS FOR A MODEL OF AN 201 
with boundary condiLions 
1 dJ1 
J1---= 1  
� dz 
dJ1 = 0  
dz 
F{J1) is  given by (6.4.67). 
Lemma 6.4.4 
ARTIFICIAL KIDNEY 
at z = 0, 
at z = 1. 
The solution of (6.4 .68)-(6.4.70) exists and is unique. 
Proof 
(6.4.69) 
(6.4.70) 
The existence and uniqueness of solutions fol lows from applying Theorem 2.6 of AMANN [8] given that 
F(J1) is monotonically increasing in J1 and is Lipsch itz in J1 from Lemma 6.4.3. This theorem will also 
require that upper and lower solutions to (6.4.68)-(6.4.70) ex ist and these can be constructed from y and -J1. 
YJ1. in (6.4.37)-(6.4.42) and using the monotonicity of F{J1) in (6.4.67).0 
The equations (6.4.68)-(6.4.70) are obtained from (6.4.27), (6.4.30)-(6.4.3 1 )  and we may no longer treat J1 
as a parameter. We are now justi fied in stating that Y(z) is unique for a given z E [0, I ]  and that there exists a 
unique solution y(x, z) given that 't.1l and YJ1. arc lower and upper solutions respectively  and therefore Yix) 
in (6.4.34)-(6.4.36) exists and is unique. We have shown that there exists a solution of (6.4.27)-(6.4.32) if 
there exists a solution of (6.4.34)-(6.4.36) and that there is a one-to-one correspondence between the one­
parameter fami ly  of solu tions, Yjl(x) and y(x, z) . Furthermore standard results (HILTMAN and LORY [ 1 26]) 
I l IaY  hc used in t.h is case to show u l l iqucl less alld cx istcl lcc 10 t he SYSICI l I  «().4.2(» -«().4.l l ) .  
Conclusions and Remarks 
The problem discussed in this section differs somewhat from the problems discussed in section 6.2 and in 
section 6.3. In this problem there is diffusion within the urease particle membrane. Literature suggests that 
this diffusion within the particle membrane could be quite significant. This problem involves three elliptic 
equations which are coupled in their boundary conditions and these equations are reduced to two equations 
which are coupled in their boundary conditions by solving a diffusion cquation w i thin the m icroencapsulated 
urease particle membrane. 
The method of proving uniquness and ex istence IJleorell1� di ffers from earlier chapters. In this section 
we prove uniqueness and existence theorems by establishing a one to one correspondence between VI rca 
transport through the arLificial k idney and reaction-diffusion wi thin a single microencapsu lated urease 
particle. 
This method of proving un iqueness and existence also makes good use of the derivative tenn in the 
right hand side of (6.4.27). This term may not be expressed in a l inear form by making use of boundary 
conditions as was done in section 6.2 and in section 6.3 and therefore our definitions or upper and lower 
solutions in section 6.2 and section 6.3 would not be the same for th is problem. However, this method of 
proving uniquness and ex istence demonstrated in this section can still be used in proving uniquness and 
existence for the problems in section 6.2 and in section 6.3 and may be generalised LO proving existence and 
uniqueness to the general system Sn, Bn and its corresponding steady state system Sn ' Bn ' It also provides us 
with bounding solutions. 
6.4 UNIQUENESS AND EXISTENCE THEOREMS FOR A MODEL OF AN 202 
ARTIFICIAL KIDNEY 
Nomenclature 
A The microencapsulated particle surface area - typically 4OOcm2 
D The dispersion coefficient, u is the velocity of blood flow -typically 1O-5cm2sec-1 
Dm Urea diffusion coefficients in the membrane 
dp The diameter of the microencapsulated particle - typically 5xlQ-5m 
D s Urea diffusion coefficients in the urease solution 
kL The mass transfer coefficient 
km The Michaelis constant and r is the radial coordinates. 
The length of the artificial kidney 
n The number of microencapsulated particles per unit volume of fluid - typically l OS cm-3 
� Peclet number - typically 450 for I = l Ocm 
r mp microencapsulated particle radius 
r u.s microencapsulated particle radius with urease solution 
flle Reynolds number - typically 0.05 
S The urea concentration in \lIe blood strcam passing through thc artificial kidney, 
s The urea concentration in the urease solution 
u The velocity of blood flow - typically 0. 1 cm sec- I 
V m The maximum reaction rate 
Z The axial coordinatc. 
Greek letters 
a ratio of microencapsulated particl6 radius to microcncapsulated particle radius with urease solution 
f3 dimensionless Michaelis constant given in (6.4. 1 9) 
l/J2 Thiele modulus givcn in (6.4. 1 9) 
r parameter given in (6.4. 1 9) 
f1 The viscoscity of the microencapsulated particle - typical ly O.01 2g cm scc-t 
p The density of the microencapsulated particle - typically 1 .2g cm-
3 
� parameter given in (6.4. 19)  
6.5 A FLUIDISED BED BIOFILM REACTOR (FBBR) MODEL INVOLVING 203 
MUL TICOMPON ENTS 
6.5 A Fluidised Bed Biotilm Reactor (FBBR) Model Involving Multicomponents 
In this section, we bring theory into focus by setting out the equations [or a tubular fluidised bed biofilm 
reactor (FBBR) problem of applied interest. The bioparLicle reaction kinetics involve three chemical 
components, a substrate s such as phenol or n itrogenous wastes which needs to be converted to harmless 
byproducts, oxygen 0, an active ingredient which helps facilitate the reaction kinetics and a product p which 
linearly inhibits the reaction kinetics. The first two components s and 0 have reaction functions of Michaelis­
Menten character. The linear product inhibition term is discussed by LEVENSPlEL [ 1 67] . 
6.5 .1  Model Formulation 
The equations governing bioparticle kinetics are similar to those given in HOSSAIN [ 1 32] where they are 
expressed in dimensionless terms as follows: 
where 
ao 2 2 --;;t - d2V"o = - (hf(s, 0, p) 
ap 2 2 ' --;;t - d3V"p = tPd(s, 0, p) 111 (0, l1xili.t1, 
{_S __ O_(I _ p) if s, o ,� o, p � 1 f(s, 0, p) = 1 + filS 1 + fi20 . ° If s, o < O, p > l , 
with boundary conditions 
as = ao = ap = ° on (0, Tjxa.Qlx.t1 ,  an an an 
� � � . 
an 
= f1'1o(S-s) , an 
= fJ'I.(O-o) , an =f1'Io(P-p) on (0, 1 ]x
a.Q2x.t1, 
and initial conditions 
s(t, x, z) = o(t, x, z) = 1 ,  p(t, x, z) = ° in Q x.t1 at t = 0, 
(6.5.1) 
(6.5.2) 
(6.5.3) 
(6.5.4) 
(6.5.5) 
and where .t1 is the il1lerval (0, 1) .  The external or bulk fluid concentrations S, 0 and P are g iven by the 
following equations 
ap 1 a2 P ap J ap ' .  X----- +- + �3 - = 0 111 (0 1 ]x.t1 at [Ie a/ az (Jilz an " 
willI boundary conditions 
(6.5.6) 
6.5 A FLU IDI5ED BED BIOFI LM REACTOR (FBBR) MODEL INVOLVING 204 
MULTICOMPONENTS 
-_._--_ .. _-_._- _ . . _-- _._._-
1 as 1 dO 1 ap S---= I , 0---= 1 ,  P---=o  at z = 0, I > 0, 
!jO" (}z � (}z !f'" (}z 
(}S ao ap -= - = -= 0 at z = 1 , I > 0, az az dZ 
and initial conditions 
S(I, z) = 0(1, z) = 1 ,  P(I, z) = 0 in A at I = O. 
(6.5.7) 
(6.5.8) 
(6.5.9) 
We see thatfls, o, p) is monotone increasing in s and ° and monotone decreasing in p. Therefore, in (6.5. 1 ) ,  
-¢l f(s, 0, p) and -¢If(s, 0,  p) are monotone decreasing in s and ° and monotone increasing in p and 
¢i f(s, 0, p) is monotone increasing in s and ° and monotone decreasing in p. 
By letting C l = S,  Cl = S, C2 = 0, C2 = 0, C3 = I-p and C3 = I-P,  the associated concentrations c ) ,  
C2, C3 ,  C l ,  C2, C 3  infi and Fj are given by 
_¢2 CtC2C3 , (1 + f3] ct )(1 + f32C2 ) 
o 
_¢2 CIC2 , (1 + f3t Ct )(1 + ,l32c2 ) 
o 
for c) , c2 � 0 , c3 ::; I 
for c3 � I ,  where i "#  3 
for c) < 0, where j = 3, 
where fi is monotone decreasing with respect to c ] ,  C2 and C3. Also, we see that 
In dimensionless terms, the bioreactor model equations for I > 0 are 
aCj - d/v;cj = /; (c) in (0, Tjx.QxA, al 
aCj =0  on (0, ' f IXd!l] xA, 
an 
�� = SP/.( Cj - Cj ) on (0, Tj xa.Q2xA, 
dCj 1 a2cj aCj J: J dCj - O ·  (0 'J'J A X---�+-+ ':> ' -- I II X/l (}I � az az ' iJf22 an " 
1 ac C ---' = 1  at z = ° I >  0 , � az " 
ac -' = 0 at z = 1 I > 0 
dZ " 
Cj(t, X, z) =  1 in !l xA at l = O, 
Cj(l, z) = 1 in A at I = O. 
(6.5.10) 
(6.5. 1 1 )  
(6.5.12) 
(6.5. 13) 
(6.5.14) 
(6.5. 15) 
(6.5.16) 
(6.5.17) 
(6.5.18) 
(6.5.19) 
6.5 A FLUIDISED BED BIOFILM REACTOR (FBBR) MODEL INVOLVING 205 
MUL TlCOMPONENTS 
6.5.2 Uniqueness, Existence and Stabil ity 
We study the stabil ity,  uniqueness and existence of solutions of this system by following the path outlined in 
Chapters 3-4. The comparison functions for el , e2 � 0, e3 � 1 are: 
with the following inequalities in the boundary conditions 
ae · ac 
d� � 0, d� 
� ° on (0, 1'lXd!2lxA, 
de ·  dC· -
d� � !fJ1I.(�j - f.j ), d� 
� !fJ1I.( Cj - Cj ) on (0, TJxdQ2xA, 
and the following inequalities in the initial condi t iolJs 
f.j (t, x, z) � 1, Cj (t, x, z) � I in .f2 xA at t= O. 
Note thaL fj are uncoupled from Cj for each j = 1 , 2, 3. The comparison [uncLions for CI , C2, C3 are: 
x a�I __ l a
2
�1 + a�1 + �I f afl � O  dt � dz2 dz a� dn 
- 2- -
X 
dC1 __ l d C1 + dCI + ;d dCI � O  dt � JT dz aDz an 
a�2 1 a2 �2 a�2 1= J df.2 a XTt- � azr
+
Tz
+
':>2 aDz an � 
- 2- -
X 
aC2 __ 1 a C2 + aC2 + �zf aC2 � o  dt � az2 az aDz an 
dC 1 a2 C ac J ae X -=.1 __ -=.1 + -=.1 + ;3 -=:l � () at � az2 az aDz an 
- 2- -
x aC3 __ 1 d C3 + aC3 + �3J aC3 � o  at � az2 az aDz an 
with the following i nequalit ies in the boundary cond i tions 
6.5 A FLUIDISED BED BIOFILM REACTOR (FBBR) MODEL INVOLVING 206 
MUL TICOMPONENTS 
1 ae. - 1 aE C · _ _ -=l.. $ 1 C. _ __ I � 1 at z = 0 t > 0 
-I � az ' I g'e az " 
ae . ae 
-=l.. $ 0, _I � 0 at z = 1 ,  t > 0, dz dz 
and the fol lowing inequalities in the initial conditions 
�j (t , z) $ 1, C;(t, z) � 1 in A at t = 0, 
for all i = 1 , 2, 3 .  Note that these equations are all uncoupled. 
From Theorem 3 .2. 1 2  (Generalised Strong Comparison Theorem) we see that for i = 1 ,2,3, 
fj = C = 0 are lower solutions and Cj = C; = Kj � 1 are upper solutions for constants Kj so that fj = C= 0 
and Cj = C; = Kj � 1 on aA) provide bounds for Cj and Cj since the functions j; are negative. 
Moreover, fj = �j = 0 and Cj = Cj = Kj � 1 are also lower and upper bounds respectively,  of the steady 
state problem and so by Theorem 4.4. 1 ,  if (; 1 ,  (;2 and (;3 arc soill tiolls of thc systcm (6.5. 1 2)-(6.5. 1 9) ,  then 
or 
and 
dC) aC2 aC3 aCI aC2 aC3 < 0 at ' at ' at ' at ' at ' at - , 
as ao as ao 
Jt' Jt' at' at $ 0 
ap ap > 0 at ' at - . 
The solutions s, 0, S and 0 therefore decrease monotonically with time and the solutions p and P therefore 
increase monotonically with time towards a positive l imit which must be a steady state solution of (6.5 . 1 2)­
(6.5 . 1 9) by Theorem 4.4.2. 
The system S3, B3 is imbedded in the system S6, B6 where inequalities in the above comparison equations 
are replaced by equalities. 
Letting Vj =Cj ,  V j = C; and v3+j = -fj , V 3+j = -�j for j = 1 , 2, 3 and setting j;* = fl ' r;* = � , 
N+j = -lj and F3:j = -Ej [or j = 1 , 2, 3 ,  we obtain the following monotone system SJ, B; : 
6.5 A FLUIDISED BED BIOFILM REACTOR (FBBR) MODEL INVOLVING 207 
MULTICOMPONENTS 
--- - -- ----------- --
()V6 -d V2v = f'(V v )= _n.2 vl v2 v6 ()t 3 x 6 J6 I " "  6 '1'3 (1+,8I VI)(l+,82V2) 
X dY) __ 1 d2Y) + dVI + �I f  dVI = 0 dt � dZ2 dZ aDz dn 
X dV2 __ 1 a2V2 + aV2 + �2f dV2 = 0 dt � az2 az a!J2 an 
X ()V3 __ 1 a2V3 + ()V3 + �3J aV3 = 0 at �" az2 az . r7!l2 an 
X ()V4 __ 1 ()2V4 + aV4 + ';I f aV4 =0 at � az2 az r]!JZ an 
X avs __ 1 �+ avs + �2J ;Jvs = 0 ()t � az az aD2 an 
X aV6 __ 1 a2V6 + JV6 +�3f  JV6 = 0 at � ---;;;r- az r7!l2 an 
where V I is uncoupled from V4, V2 is uncoupled from vs , and V3 is uncoupled from V6. These have the 
appropriate boundary uno in i Lial conditions. The matrix at;* is given below rJv J 
-l/JfVSV6 2 2 0 0 0 -l/J\ V\ V6 -l/J\ v\ vS ( 1 +,8\ VI )2 (1-,82 vs ) (1 +,8\ VI )(1-,82 vS)2 (1+,81 VI )(1-,82 vs ) 
-l/J'iV4V6 2 -l/JtV4V2 0 0 -f/J2 v2v6 0 (1-,81 V 4)(1 +,82 V2)2 (1 -,81 V 4 )2 (1 +,82 V2) (1-,81 V 4 )(1 +,82 V2 ) 
0 0 -l/J�V4VS -l/J�VSV3 -l/J�V4V3 0 (1-,8\ v4)(1 -,82 vS ) (1-,8\ V 4 )2 (1-,82 vS ) (1-,8\ V 4 )(1-,82 vS )2 
0 -l/Jfv4v3 -l/Jfv4v2 -l/Jfv2v3 0 0 
0-,81 V 4)(1 +,82 V2)2 (1-,8\ v 4 )(1 +,82 V2 ) (1-,81 V 4 )2 (1+,82 v2 ) 
-l/J5:VSV3 0 -�ivl vs 0 -�iV\ V3 0 (1 +,81 VI )2 (1 -132 VS) (1 +131 V\ )( 1 -,82 vs ) ( 1 +13\ VI )( 1 -,82 vS)2 
-l/JjV2V6 2 -l/JjVI V2 -l/J3 vI V6 0 () 0 (1+,81 v1l(1+fi2V2 ) (l+fil VI )( 1+fj2V2)2 (1 +,81 Vl )(1 +132 v2 ) 
with matrix aij of the linear equations P(c) of the form: 
or 
6.5 A FLUIDISED BED BIOFILM REACTOR (FBBR) MODEL INVOLVING 208 
MULTICOMPONENTS 
0 0 0 0 .-lL <Pf 
(1+131 ) ( 1  +131 )(1 +132 ) 
0 0 0 � 
( 1+132 ) 
0 fIJi 
(1  +/31 )(1 +/32 ) 
0 0 0 � � 0 
(1+/32 ) (1+/31 ) 
0 � <Pf 
(1+/31 ) ( 1  +/31 )(1 + /32 ) 
0 0 0 
� 0 fIJi 0 0 0 
(1+/32 ) (1  +/31 )(1 + /32) 
� � 0 0 0 0 
(1  +132 ) (1+131 ) 
0 � ¢If 
(1+131 ) ( 1  +131 )(1 + 132 ) (� g) , where B= <pi 0 <pi (6.5.20) 
( 1+/32 ) (I +/31 )(1 +/32) 
-.PL � 0 
(l +flz ) (1+f:J1 ) 
and Vj are all uncoupled for i = ) , . .  , 6  and 
A jj = O. 
IL is clear that!;(cj) satisfies the Lipschitz and H{;lder properties required in assumptions (HI)  and (H2) for 
C l ,  C2 � 0 and C3 $ 1 .  By Remark 3/) .7, these continuity properties also hold by extending the function in 
(6.5 . 1 0) LO all defined values of Cl , C2 and q. 
We may apply Theorem 3 .3 . 1  (Generalised Uniqueness Theorem), Theorem 3 .4. 1 and Theorem 
3 .6. 1 (Generalised Existence Theorem) to show that there ex ists a un ique solution to the system (6.5 . 12)­
(6.5 . 1 9) .  
We now study the slability of  the system S;, B; . I f  we seek functions (w, W) satisfying the system P(e) for 
which all the Wj components are equal to W,  a positive scalar, then the inequalilies (3 .5. 1 2)-(3 .5 . 14) are 
satisfied if we can find w > 0 satisfying 
where 
and 
aw  a;;� O on a.alxA, 
Jw -
- � y(W- w) on J!22xA, 
an 
W � Wj 1'01' all j 
(6.5.21) 
j 
(6.5.22) 
(6.5.23) 
(6.5.24) 
6.5 A FLU IDISED BED BIOFILM REACTOR (FBBR) MODEL INVOLVING 209 
H r=sup-l . 
j dj 
MUL TlCOMPONENTS 
(6.5.25) 
The equations for (3 .5 . 1 5)-(3 . 5 . 17) are likewise satisfied by Wj = W , if we can find W > 0, satisfying 
where 
2 - -_1 d � _ d W  + (t: - �S?t)W+ �f w � O in A, fY'e dz dz af12 
- 1 d W  
W - - - � 1 at z=O, 
[i'" dz 
d W  
- � ° at z= l ,  
dz 
These scalar equations for w and W uncouple if we set 
so that 
w = e(x) W (z) 
ae 
an � 0 on aD! ,  
ae � y( I .- 0 )  Oil  rHh, 
(}n 
and W satisfics the scalar equation 
- 1 d W  
W - -- � l  at z = 0, 
[i'" dz 
d W 
- � O  at z =  1 .  
dz 
(6.5.26) 
(6.5.27) 
(6.5.28) 
(6.5.29) 
(6.5.30) 
(6.5.31) 
(6.5.32) 
(6.5.33) 
(6.5.34) 
(6.5.35) 
(6.5.36) 
Now A, given by sup (LAj)ldj ,  depends on the square or the biopanicle size and positive e solutions [or 
I . 
equations (6.5 .3 1 )-(6.5.33) exist for small enough A. 
In the case of large Pcclet number £f'e, where the macroscopic system is convection dominated, the W 
equation has a solution 
(6.5.37) 
for some constant W, if the fol lowing conditions are met 
1 2 
-K - K +J1 � O , 
� 
(6.5.38) 
where 
6.5 A FLUIDISED BED BIOFILM REACTOR (FBBR) MODEL INVOLVING 210 
MULTICOMPONENTS 
J1 ? E - �.9I + � J e .  
aQ2 
Note, K real and positive exists satisfying (6.5.38) if.cY'e ? 4J1. At z = 0, W must also satisfy 
- 1 dW  K W - --- = re [ 1 - -] ? 1 .  .cY'e dz @Ie 
(6.5.39) 
(6.5.40) 
This requires .cY'e >  K and W large enough. These conditions are met with K=2J1, where � ?  4J1. We 
conclude that all solutions of Lhe system (6.5 . 1 )-(6.5.5) are stable and there is no more Lhan one steady state 
solution when Lhe fol lowing two conditions hold: 
For the given geometry of Q, the boundary value problem (6.5 .3 1 )-(6.5.33) for e has a positive 
solution. This will be the case if A < Al where Al is the smallest eigenvalue of problem 
The first requirement is thell that 
If A < Al and Lhe function e satisfies (6.5 .3 1 )-(6.5.33), and J.1o is given by
, 
J1o = �f ( e - 1) = -.ff de = �A J e ,  aQ2 r aQ2 dn r Q 
the second requirement is that 
� >4J10 =4 �A J e .  r Q 
(6.5.41) 
(6.5.42) 
(6.5.43) 
(6.5.44) 
(6.5.45) 
(6.5.46) 
We therefore have stability and uniqueness for tile system (6.5 . 1 )-(6.5.5) for small enough particles and high 
enough Peelet number, @Ie. 
6.6 A CONTINUOUS STIRRED BASKET REACTOR (CSTR) MODEL 21 1 
INVOLVING M U L T1COMPON ENTS 
6.6 A Continuous Stirred Basket Reactor (CSBR) Model involving Multicomponents 
In recent years there has been considerable interest in the possibility of using immobilised enzymes as 
industrial catalysts. These enzymes are usually trapped in or aLLatched to water-insoluble supports by a 
variety of methods. In the fol lowing example, a general model is developed for the study of an enzyme 
reaction inside such porous support pellets contained in a Continuous Stirred Basket Reactor (CSBR). The 
enzyme reaction kinetics we shall look at in this section involves three chemical components, a substrate s 
which is converted to a product p under the action of immobilised enzyme, e. The porous support pellet 
permits the substrate s and product p to move in and out of these pellets but retains the enzyme e because of 
the larger enzyme molecules. The bulk fluid therefore involves only two chemical components, bulk 
substrate Sb which is introduced into the reactor as a steady flow and bulk product, Pb which is produced in 
the reactor. 
This example is found in literature (HOSSAIN [ 1 32]) and we shall study the question of uniqueness 
and existence to the time dependent problem using methods developed in this thesis and adapting the theory 
to reactors such as the CSBR. The objective is to look at the following arbitrary kinetics: 
(i) Michaelis-Menten type reaction kinetics, 
(ii) substrate inhibition (non competitive or anticompetitive) type reaction kinetics, 
(iii) product inhibition (competitive) type reaction kinetics, 
(iv) product inhibition (non competitive or anticompetitive) type reaction kinetics, 
(v) zero order kinetics. 
6.6.1 The Continuous Stirred Basket Reactor (CSBR) 
The Continuous Stirred Basket Reactor (CSBR) consists of a cyl indrical vessel in which it is essential to 
have perfect mixing of the contents. The effect of good mixing is that all ClclllCJJlS of the fluid in the vesscl 
have virtually the same composition. The liquid motion is induced by mcchanical sLirring. The result is LhaL 
porous support pcllets contained in the CSBR circulate in a somewhat haphazard manner that characterises 
the CSBR. A schematic of a CSBR is given in FIG. 6. 1 1 .  
., 
., 
., 0 
.. 0 
.. OJ .. 0 ., 
FIG.  6. 1 1  Schematic of a Continuous Stirred lIasket Reactor (CSIIR) 
6.6 A CONTINUOUS STIRRED BASKET REACTOR (CSTR) MODEL 212 
INVOLVING MULTICOMPONENTS 
6.6.2 The Continuous Stirred Basket Reactor (CSBR) Model Formulation 
The mathematical model of the CSBR is divided into two submodels. The "Support pel let Model" is 
concerned with reaction and diffusion within a support pellet and the "CSBR Model" which is concerned 
with bulk fluid concenttations within the reactor. We make the following simplifying assumptions 
Model Assumptions 
1 .  The porous pellet is  spherical and uniform in size 
2 .  The effective diffusivities of the substrate and product within the porous pellet are constant 
3 .  The volume and the density of the reacting medium within the porous pellet are constant 
4 .  The reaction kinetics within the porous pellet of the main reaction is arbitrary 
5 .  The process within the porous pellet is diffusion conttolled 
6 .  The bulk fluid i s  perfectly mixed 
The equations describing the concentrations of intrasupport substrate and product, active enzyme 
concentration are 
dP 1 d 2 dP cp - = Dep 2-(r -) + kppf(.I', p)e(r, I )  dl r dr dr 
de Tt = -kdf(s, p)e(r, t) 
with boundary conditions 
dS = dP = 0 at r = 0 dr dr ' 
dS Des dr = kms(Sb -.I') at r = rp, 
dP Dep dr = kmpCPb -p) at r = rp' 
and initial conditions 
.1'(0, r) = So, 
pea, r) = 0, 
e(O, r) = eo(r). 
The bulk concentrations of substate and product in (he CSBR are given by 
dSb mp 3 as l V--=F,(SO -Sb)---(Des- ) ,  dl Pp rp ar r=rp 
dPb mp 3 dP
I V- = -F,Pb ---(Dep- )
, 
dt Pp rp dr r=rp 
and initial conditions 
(6.6.1) 
(6.6.2) 
(6.6.3) 
(6.6.4) 
(6.6.5) 
(6.6.6) 
(6.6.7) 
(6.6.8) 
(6.6.9) 
(6.6.10) 
(6.6.1 1) 
6.6 A CONTINUOUS STIRRED BASKET REACTOR (CSTR) MODEL 213 
INVOLVING MUL TICOMPONENTS 
(6.6.12) 
Pb = 0 at 1 =  O. (6.6.13) 
For the five kinetic expressions mentioned earlier,j(s, p) has the following form: 
f(s,p) = 
for s, p ;::: 0 and 
f(s, p) = 0, 
for s, p < O. 
s 
s 
s 
(s+ Km)(l '+ pi  Kj )] 
1 
Michael is-Menten type kinetics 
substrate inhibition (non competitive) type kinetics 
product inhibition (competitive) type kinetics 
product inhibition (noncompetitive) type kinetics 
zero order kinetics 
It may be observed that in the case of Michael is-Menlen type kinelics f(s, p) is monotone increasing with s 
and independent of p. In lhe case of substrate inhibition (non competitive or anticompetitive) type kinetics 
f(s, p) is concave down, has a maximum positive value, is independent of p and is zero when s is zero and 
when s is very large. In the case of product inhibition (competitive) type kinetics, f(s, p) is monotone 
increasing with s and monotone decreasing in p. In the case of product type inhibition (non competitive or 
anticompetitive type) kinetics, f(s, p) is monotone increasing with s and monotone decreasing in p. Zero 
order kinetics is of course independent of both s and p. 
It may also be observed that this model with constant flow rate Fr, is a limiting case of the more 
general model developed in Chapter 2 which involves the convection operator u ·  V .  Consider, for example 
the foHowing equation: 
aSb 
-=u ,VSb 
at 
' 
where the spatial average is defined as 
(6.6.14) 
(6.6.15) 
Differentiating (6.6. 1 5) with respect to t and applying a result from Gauss's Theorem (RUTHERFORD [253,  
p.77] , we obtain 
(6.6.16) 
6.6 A CONTINUOUS STIRRED BASKET REACTOR (CSTR) MODEL 214 
INVOLVING MUL TICOMPONENTS 
In only one variable, Lhercfore 
dS _b = uA(Sj - So),  
dt 
(6.6.17) 
where Sj is  the inlet concentration and So is the outlet concentration and A is the reactor inlet surface area. 
Here uA = Fr with the units m3 sec-t o  
The methods developed i n  this thesis for proving uniqueness, existence and stability theorems for 
more general type equations with a convection term may therefore be adapted to problems where there is a 
constant flow rate of substrate into the reactor such as that found in our model of the CSBR. For the 
purposes of mathematical convenience, the terms Fr(SO-Sb) and -FrPb may be considered to be reaction 
terms in the bulk fluid as there is no convection term involved. 
Nondimensionalisation of the above governing equations leads to the following equations 
where, 
aCl 1 a 2 aCt 2 . -;h- x2 ax (x -;;;) = -¢ h(cl ' c2)c3(x, -r) for 0 < x  < 1 ,  -r >  0, 
aC2 8 a 2 aC2 2 -a -z-(x -) = ¢ h(Cl '  C2 )C3(x, -r) for 0 < x < 1 ,  -r >  0, -r x ax ax 
dC3 2 
_ . d-r 
= -¢ h(Ct , (2)c3 (X, r) lor 0 < x < 1 ,  r >  0, 
dCl aCt 
I 
_ - = a(1-Ct )-N-- for r >  0, 
dr ax x=l 
dC2 N rJC2 1 . -= -aC2 --- lor r >  0 
dr 8 ax x=l ' 
aCt aC2 
-a;= dx = 0  at x = 0, -r >  0, 
aC2 -B '  (C ) , - 1 0 ax - Ip 2 - c2 at x - , r >  , 
C l (O, X) = C t .o f"or O < x <  1 ,  
C2(0, x) = 0 for 0 < x < 1 ,  
C3(0, x) = C3,O(X) for 0 < x < 1 ,  
(6.6.18) 
(6.6.19) 
(6.6.20) 
(6.6.21) 
(6.6.22) 
(6.6.23) 
(6.6.24) 
(6.6.25) 
(6.6.26) 
(6.6.27) 
(6.6.28) 
(6.6.29) 
(6.6.30) 
6.6 A CONTINUOUS STIRRED BASKET REACTOR (CSTR) MODEL 215 
INVOLVING MULTICOMPONENTS 
_C_I _ 
1 + f3cI 
Michaeli s- Menten type kinetics 
substrate inhibition (non competitive) type kinetics 
product inhibition (competitive) type kinetics 
product inhibition (noncompctitive) type kinetics 
zero order kinetics 
for CI ,  C2 � 0 and 
The parameters are 
(6.6.31) 
(6.6.32) 
Also, 
(6.6.33) 
for zero order kinetics and 
(6.6.34) 
[or Michaelis-Menten, substrate inhibition (non competitive or anticompetitive) and product inhibition 
(competitive and non competit ive) type kinetics. 
The variables are 
r Derl X = -, r = -2-'- ' rp rp cp 
s p e � Sb Pb So co (r) . Sb i cl = -' c2 = -' c3 = ::-' C1 = - , C2 = -, cI O = -' c3 0 = -_-, CI O = -' , 
So So eo ' So So ' So ' eo ' So 
(6.6.35) 
(6.6.36) 
where the non<l imensional enzyme concentration is scaled with respect \0 the mean initial concelHration, 
which is defined as 
- 3 J'p 2 eo =3 r eo (r)dr r 0 p 
and in nondimensional form it becomes 
11 2 3 x c3 0 (x)dx= 1 .  o ' 
(6.6.37) 
(6.6.38) 
6.6 A CONTINUOUS STIRRED BASKET REACTOR (CSTR) MODEL 216 
INVOLVING MULTICOMPONENTS 
6.6.3 Uniqueness, Existence and Comparison Results 
We study the uniqueness and existence of the system (6.6. 1 8) -(6.6.30) by following the palh oUllined in 
chapters 3-4. The comparison functions for C I and C2 are given by: 
d�1 � a(l-�\ ) -N Ofl l • d'l" ox x=\ 
dCI � a( l -E;)-N dCI I 
• 
d-r ox x=1 
d�2 $ _aC2
_ N df2 1 • 
d'l" - (j ox x=1 
dC2 �_aC2
_ N OC2 1 • d'l" (j dx x=1 
(6.6.39) 
(6.6.40) 
(6.6.41) 
(6.6.42) 
and the comparison functions for C I .  C2 and C3 depend on the arbitrary k inelics considered. In the case of 
Michaelis-Menten type reaction kinetics the comparison functions for ct . C2 and C3 are given by: 
Ofl _ 2-�( 2 dfl ) < _Al2 ---.fL - ( ) :l 2 :l  X :l - 'I' f3 C3 x. -r , u'l" x dX dX 1 + f\ 
dC3 2 CI _ -�-¢J -- C3 (X' 'l") . 
d'l" 1 +f3fl 
In the case of substrale inhibilion (anlicompetilive or non compelilive) lype reacLion kinetics the comparison 
funcLions for q,  C2 and C3 are given by: 
6.6 A CONTINUOUS STIRRED BASKET REACTOR (CSTR) MODEL 217 
INVOLVING MUL TICOMPONENTS 
In the case of product inhibition (compet.itive) type react ion kinetics the comparison fllnctjons for Cl , C2 and 
C3 are given by: 
In the case of product inh ibition (anticompetitive or noncompetj tive) type reaction kinetics the comparison 
functions for C l ,  C2 amI C3 are given by: 
In Lhe case of zero order kineLics the comparison funcLions for C I ,  C2 and (;3 arc givcn by: 
d�l __ I �( 2 d�1 ) < _,/,2- ( ) d 2 d x d - 'I' C3 x, r , r x x x 
dC2 _�� ( 2 d(2 ) > ,/,2- ( ) d 2 ... x d - 'I' C3 x, r , r x ox x 
6.6 A CONTINUOUS STIRRED BASKET REACTOR (CSTR) MODEL 218 
INVOLVING MULTICOMPONENTS 
The inequalities in the boundary and initial conditions are 
(6.6.43) 
(6.6.44) 
(6.6.45) 
(6.6.46) 
(6.6.47) 
(6.6.48) 
(6.6.49) 
(6.6.50) 
(6.6.51) 
In the case of Michaelis-Menten, substrate inhibi lion (non competitive or anticompetitive) and product 
inhibition (competitive and non competitive) type kinetics, we see that the comparison functions for c ) ,  C2 
and (;3 arc all coupled in SOIl lC  way. 
In all cases we also see that the nonlinear reaction functions are monotone increasing in some of the 
variables and monotone decreasing in all other variables. From section 3 .4 wc see that all these systems may 
be made into quasimonotone nondecreasing systems (with inequalities replaced by equalities) by the 
substitution Vj =Cj '  v3+j = -fj for i = 1 , 2, 3  and Vj = C; ,  V2+j = -kj for i = 1 , 2. 
However, rather than defining upper and lower solutions for this new monotone system we may stilI 
define coupled upper and lower solutions of our original system. These as we have seen in section 4.3 for 
the time independent problem can also give us uniqueness and existence results. 
In the case of Michaelis-Menten, substrate inhibition (non competitive or anticompetitive) and 
product inhibition (competitive and non competitive) type kinetics, we shall take fJ = f2 = f3 = kJ = k2 = 0 as 
lower solutions and look for upper solutions in terms of these lower solutions. 
In the case of Michaelis-Menten type reaction k inetics the only difficulty is to find functions c) , C2 and c3 that 
satisfy the inequalities 
6.6 A CONTINUOUS STIRRED BASKET REACTOR (CSTR) MODEL 219 
INVOLVING MULTICOMPONENTS 
In this case we may choose 
ci = � = K] ,  c2 = C2 = l/J2 K1 K3t ,  c3 = C3 = K3 ,  ( 1 + {3K1 ) 
where K) and K3 are positive conslants satisfying 
In the case of subSlrate inhibition (non competitive or anlicompetitive) type reaction kinetics the only 
difficulty is to find functions ci ,  C2 and C3 that satisfy the inequalities 
1 dC2 8 d ( 2 d(2 ) > Al2 1JJr _ ( ) --�- X - ,/, C3 X .. d.. x dx dx - (1 + �)(1 + k:) " 
In this case we may choose 
'IfJr 'IfJr 
In the case of product inhibition (competitive) type reaction kinetics the only difficulty is to find functions 
c) , C2 and c3 that satisfy the inequalities 
dc] __ 1 �(x2 dCl ) � O , d.. x2 dx dx 
de2 8 d ( 2 de2 ) > Al2 el _ ( ) ---- x - _ '/' C3 x, .. , a.. x2 dx ax {3e] + (1 + )'f:2) . 
aC3 > 0  a .. - . 
In this case we may choose 
where Kl and K3 are positive constants satisfy ing 
and UlC only difficulty now is 10 find (;2 satisfying 
6.6 A CONTINUOUS STIRRED BASKET REACTOR (CSTR) MODEL 220 
INVOLVING MULTICOMPONENTS 
However, since 
and 
are unifonnly bounded, we are assured that upper solutions e2 exist and may be constructed (PAO [222]). 
For example, we may take C2 = C2 = AeR1 , so that 
AeR1 � ¢2 KI K3, 13K! + ( 1  + rAeR1 ) 
where A is determined from Ule boundary condLions and R can be chosen large enough. 
In the case of product inhibition (non competitive or anticompeLiLive) type reaction kinetics the only difficulty 
is to find functions c! , C2 and C3 that satisfy Ule inequalities 
de2 _ 8 �(x2 d(2 )" �  ifJ2 el c3 (x, 1") , d1" 7" dX dX (I + /3e) )(1 + Jf"2) 
OC3 > 0  o-r - . 
In this case we may choose 
where KI and K3 are positive constants satisfying 
and the only difficulty now is to find (;2 satisfying 
However, since 
and 
are unifonnly bounded, we are assured that upper solutions (;2 ex ist and may be constructed (PAO [222]). 
For example, we may take c2 = C2 = AeR1 , so that 
Aeu1 � ¢2 K) K3, (1 + /3K) ( 1 + rAe/(l ) 
where A is determined from me boundary condtions and R can be chosen large enough. 
6.6 A CONTINUOUS STIRRED BASKET REACTOR (CSTR) MODEL 221 
INVOLVING MULTICOMPONENTS 
In the case of zero order kinetics we may take f2 = f3 = �2 = 0 as lower solutions. The only d i lTiculty is Lo 
find functions fl ' £:1 '  £:2 and £:3 that satisfy the inequalities 
In this case we may choose 
where Kl and K3 are positive constants satisfying 
and the only difficulty now is to find fl and c2 satisfying 
dfl __  l .i.(x2 df1 ) 5, _l/)2K3 . d'r x2 dX dX 
dC2 _�.i.(x2 d(2 ) ? l/)2K3 • d'r x2 ax ax . 
Our comparison functions provide us with valid coupled lower and upper solutions fl ' f2 ' f3 ' �l ' �2 and 
el '  e2 ' e3 '  C; .  C2 • respecti vely for the time dependent problem (6.6. 1 S)-(6.6.30) with fl 5, Cj .  £2 5, e2 ' 
f3 5, e3 '  �I 5, C; and �2 5, C2 . Since our HOlder and Lipschitz continuity properties arc satisfied by the 
nonlinear terms there exists a unique solution of the problem (6.6. 1 8)-(6.6.30) by Theorem 3.3 . 1  and results 
from section 3.6. 
6.6.4 Conclusions and Remarks 
Our coupled upper and lower solutions also provide us with bounds on the solution. C I . C2. q. C 1 and C2. 
These bounds can all be interpreted physical ly .  The existence of solutions to the steady slate problem is 
treated similary by following the methods in this section and using Theorem 4.3.3 (Generalised Existence 
Theorem) for nonmonotone systems. The study of stability and uniqueness of the steady state problem 
follows along the lines set forward in section 6.5. 
Nole lhal fl = �I = 0 is nOl a universal lower bound on (;1 in the case or zero order kinetics and c )  
may possess a negative solution. This o f  course i s  physically unrealistic and the problem would have to 
redefined to avoid this. I t  can be shown with the strong maximum principle that f2 ' f3 and �2 are strictly 
bounded below by zero and so therefore is (;2. C) and C2. However we cannot say this of zero order kinetics, 
(in facl of fractional order kinetics) which may al low for incoJllplete pcnctration of substrate (PARSI IOTAM 
6.6 A CONTINUOUS STIRRED BASKET REACTOR (CSTR) MODEL 222 
INVOLVING MULTICOMPONENTS 
(223]) in our model . The physical impl ications of this is that substrate concentrations may approach and 
reach zero concentration somewhere in a panicle and reactor but product and enzyme concentrations will 
never reach zero. 
Nomenclature 
A reactor inlet surface area 
B is B iot number for substrate mass transfer 
B ip B iot number for product mass transfer 
CI nondimensional intracatalyst substrate concentration 
C2 nondimensional product concentration 
C3 nondimensional immobilised enzyme concentration 
C1 nondimensional reactor substrate concentration 
C2 nondimensional reactor product concentration 
dp diameter of particle 
Des effective diffusivity of substrate in the pellet 
D p effective diff usivity of product in the pellet 
e active imrnobilised enzyme concentration in the pellet 
eo initial distribution of active immobilised enzyme concentration 
iii) mean active enzyme concentration defined in (6.6.37) 
f arbitrary kinetic exprcssion 
F, Volumetric flow rate of substrate to the reactor 
h arbitrary kinetic expression 
k enzymatic reaction rate constant 
kd deactivation rate constant of immobiliscd eni',Ylllc conccntration 
Ki Inhibition constant for the supported catalyst 
Km Michaelis-Menten constant for the supported catalyst 
kmp mass transfer coefficients of the product 
kms mass transfer coefficients of the substrate 
mp total mass of the enzyme pellet in the reactor 
N ratio of support volume to that or reaclOr - defined in (6,6.3 1 )  
p intrasupport product concen\Iation 
P b Bulk concentration of the product in the reactor 
r radial position in thc pellct 
rp pellet radius 
s intracatalyst substrate concentration 
So Inlet concentration of substrate 
Sb Bulk concentration of substrate in the reactor 
S b, i Initial bulk concentration of substrate in the reactor 
real time 
u variable flow rate into the reactor 
V volume of the reacting solutions or of the CSBR 
x nondimensional spatial variable 
6.6 A CONTINUOUS STIRRED BASKET REACTOR (CSTR) MODEL 223 
INVOLVING MU LTICOMPONENTS 
Greek letters 
a parameter defined in (6.6.3 1 )  
fJ parameter defined in (6.6.3 1 )  
o parameter defined in (6.6.3 1 )  
Ep partical voidage, i.e., initial porosity of support pellet 
rp2 Thiele moduli defined in (6.6.33)-(6.6.34) 
r parameter defined in (6.6.3 1 )  
Pp pellet density 
ljI parameter defined in (6.6.3 1 )  
l' dimensionless time 
6.7 Notes and Comments 
6.7 NOTES AND COMMENTS 224 
Section 6.1 is adapted from PARSHOTAM et al. [225], section 6.2 is adapted from PARSI IOTAM et al. [226J, 
section 6.3 is adapted from PARSHOTAM [227], section 6.4 is adapted from PARSI IOTAM [229] and section 
6.5 is adapted from PARSHOTAM [228]. 
Note that the analytical bounds demonstrated in section 6.2 can be improved by using techniques 
developed in section 6. 1 .  These techniques however are not so straightforward and may also be applied in 
some cases to systems of such equations. 
An exact analytical solution to the systcm Sn, Bn with porous spherical particles and a cylindrical 
isothermal adsorption column without reaction in either particles nor the reactor and for a single chemical 
component has been derived by RASMUSON and NERETNIEKS [243] and is improved by RASMUSON [244]. 
The fast Fourier transform (FFT) has also been used in these problems (CI IEN  and HSU) to predict 
breakthrough curves for this problem and this is compared with the orthogonal collocation results of 
RAGl-1A V AN and RUTHVEN [237] .  
In section 6.5 the function fi(c) in equation (6.5 . 1 0) is not only monotonically decreasing with 
respect to Cj but it is also a negative function which is concave up and 1;(0) = O. Therefore a linearisation of 
this function by a Taylor's series expansion, uncoupling the resulting linear equations by using methods 
developed in Chapter 5 and solving these linear equations should give a solution which is a lower bound of 
the original system. Similarly, an upper bound may be constructed along the same lines by using methods 
developed example 6.2 and these bounds can also be improved by using the methods developed in example 
6. 1 .  In general tJlis tJleory may bc applicd to the systems Sn ' Bn and Sn ' Bn if aij, A ij � 0 and Jij or F ij arc 
either concave up or concave down for all i , }. Note that this conclusion would not be reached if the 
'" 
substitution C3 = 1 - P and C3 = 1 - P had not been made. In this case we may stil l have got monotone 
A. 
system but it is not true that aij � 0 for i = j. 
In section 6.5, we have stability and uniqueness lor small enough particles and high Peclet number. 
This is consistent with what is well documented in l i terature from numerical and analytical studies for 
reactors and particles (McGUIRE and LAPIDUS [ 1 8 1 ], GAVALAS [104], HLAVACEK [ 1 27-131], HELLlNCKX 
et al. [ 122], Luss [ 1 74, 175J, Luss and AMUNDSON [173J and WEISZ and HICKS [307]). 
In section 6.6 we see that substrate concentrations may decrease and approach zero somewhere in lhe 
interior of the reactor or particle. From a physical standpoint, this concept of partial penetration or incomplete 
penetration of substrate through a particle is very important to operating conditions. It can be demonstrated 
that certain kinetics cannot allow for parlial or incomplete penetration (PARSI IOTAM [223,225,226]) in a 
particle i n  a reactor. This problem has i ts mathematical analogy in demonstrating the existence or 
nonex istence of a dead core or finding nonnegative solutions with interior zeros (BANDLE and STAKGOLD 
[ 3 1 ],  BANDLE, et al. 1 .3 1 ,  32/, BOBISUD [39-4 1 ] , FRIEDMAN and Pl l lLlPS / 95 /) and thesc problems do not 
exist in positone problems (CASTRO and Sl llVAJI [ 5 1 j) .  It can be shown by the max imum principle thaI for 
some kinetics, this dead core is empty and this behaviour of solutions usually occurs with zero order and 
fractional order kinetics (GRAHAM-EAGLE and STAKGOLD [108]). These results may also be generalised to 
systems of equations by the methods developed in tJlis thesis. 
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