Copyright is owned by the Author of the thesis. Permission is given for a copy to be downloaded by an individual for the purpose of research and private study only. The thesis may not be reproduced elsewhere without the permission of the Author. A NON-LINEAR BOUNDARY VALUE PROBLEM ARISING IN THE THEORY OF THERMAL EXPLOSIONS A thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University Michael Rowlinson Carter 1976 ii ABSTRACT When a heat-producing chemical reaction takes place within a confined region, then under certain circumstances a thermal explosion will occur. In investigating from a theoretical viewpoint the conditions under which this happens, it is necessary to study the behaviour of the solution of a certain non-linear parabolic initial-boundary value problem . A frequently used approach is to study the problem indirectly, by investigating whether positive steady-state solutions exist ; the underlying assumption is that positive steady-state solutions exist if and only if a thermal explosion does not occur . The main theme of this t hesis is the development and application of an alternative direct approach to the problem, involving the construction of upper and lower solutions for the parabolic problem and the application of appropriate comparison theorems . The assumption here is that a thermal explosion will not occur if and only if the solution of the parabolic problem remains bounded for all positive time . Following three chapters of introductory material, Chapter 4 contains a survey of some of the important known results concerning the existence of positive steady-state solutions, especially those dealing with the effect on the theory of different assumptions as to the rate at which heat is produced in the reaction . The comparison theorems that are used in the alternative approach, which are modified versions of known results , are proved in Chapter 5 . In Chapter 6, the equivalence of the two criteria mentioned above for the occurrence or non-occurrence of a thermal explosion is established under fairly general conditions . Also in this chapter , a critical value A* is defined for a parameter A appearing in the problem , s uch that a thermal explosion will not occur if the value of A is smaller than A*, but will occur if the value of A is greater than A*? In Chapter 7, upper and lower solutions are constructed for the time-dependent problem under a variety of assumptions as to the rate at which heat is produced in the reaction, and these are used to obtain a number of theorems concerning the behaviour of the solution of the problem, especially as the time variable tends to infinity . The information obtained from these theorems is related to and compared with that known from investigations of the existence of positive steady-st ate solutions . In conclusion , a theorem is proved concerning the effect of reactant consumption on the theory. This is examined in the light of iii some recent research, and an apparent defect which is thereby revealed in the usual criteria for the occurrence'of a thermal explosion is discussed. The theorems of Chapter 7 are employed in Chapter 8 to obtain rigorously derived bounds for the critical parameter A*, for a number of different shapes of the region in which the reaction takes place; these bounds are compared with known estimates for A* obtained using an empirically derived formula . The thesis concludes, in Chapter 9, by using the methods of Chapters 7 and 8 to obtain some results concerning the case where the boundary condition is non-linear . ACKNOWLEm EM ENT I would like to thank Dr . Graeme C. Wake for his ready advice and unfailing encourageme nt and enthusiasm during the preparation of this thesis . iv INDEX TO THEOREMS 1. INTRODUCTION 2. AN EXAMPLE 3. DEFINITIONS AND NOTATION CONTENTS 4. THE STEADY-STATE PROBLEM - A SURVEY 5. COMPARISON THEOREMS FOR THE TIME-DEPENDENT PROBLEM 6. RELATIONS BETWEEN SOLUTIONS OF THE TIME-DEPENDENT AND STEADY-STATE PROBLEMS Monotone Iteration Asymptotic Behaviour of Solutions 7 . CONSTRUCTION OF UPPER AND LOWER SOLUTIONS FOR THE TIME-DEPENDENT PROBLEM Notation Construction of Lower Solutions on V m Construction of Upper Solutions on V m Extension to Other Domains Discussion The Effect of Reactant Consumpt ion 8. BOUNDS FOR THE CRITICAL PARAMETER Preliminaries Bounds for the Critical Value X* Comparison with Known Values 9. NON-LINEAR BOUNDARY CONDITIONS APPENDIX LIST OF REFERENCES vi 1 6 8 14 21 28 29 41 52 52 5 3 6 3 72 77 80 86 86 89 91 102 116 127 INDEX TO THEOREMS Theorem Number Page 1 22 2 24 3 35 4 41 5 41 6 43 7 46 B 47 9 48 10 50 11 53 12 54 12A 74 12B 104 13 55 14 57 15 58 16 60 16 (Corollary) 62 17 63 17A 75 17B 103 18 66 1BB 103 19 68 20 70 20B 104 21 80 1 INTRODUCTION The mathematical problem discussed in this thesis arises from a topic in chemical kinetics , the study of the evolution in time of chemically reacting systems. Suppose we are dealing with a heat? producing reaction taking place in a confined region. For simplicity , we suppose for the time being that there is no consumption of reactant. If the heat produced by the reaction cannot all be removed at the boundary of the region , the temperature of the reactant will rise , leading to an increase in the reaction rate , in turn producing more heat. In practice , one of two things then happens. Either the rate of increase of temperature gradually diminishes and the system approaches a steady state , or the temperature increases rapidly and without limit , and what is usually called a thermal explosion takes place. An elementary discussion of this phenomenon is given by Boudart(S , pp.160-163], and additional information may be found in the book by Bradley(6 , especially pp.2 , 8-15]. The problem with which we shall be concerned is that of determining whether or not a thermal explosion will t ake place in a given situation . Suppose we have a heat-producing reaction taking place within a region V bounded by a surface S. We shall continue to ignore for the time being the e ffect of reactant consumption , which will be commented on in Ch.7 , and we also assume that the thermal parameters of the system are constant in space and time. Then ( see , for example , Ozisik's book[26 , p . 6]) the system is described by a differential equation of the form 2 K( a u + ax2 for ( x ,y;z) in the interior of V , and time t> 0 , together with the initial-boundary conditions K au + Hg( u ) = 0 av for ( x ,y ,z) on S and t > 0 u = 0 for ( x ,y , z) on V US when t = 0. Here u( x ,y ,z ,t ) is the difference between the temperature T at any point and the ambient temperature T , K , p and c are the thermal a conductivity , density and specific heat respectively of the reactant , H is the surface heat transfer coefficient and ?? is the outward normal derivative to S . We shall say that thermal explosion takes 1 place if u( x,y , z ,t ) -m as t - m or if u( x ,y,z , t ) -m as t - T- ( T finite ) . The form o f the heat-generation function f is still a matter for debate. It is proportional to the rate at which the reaction takes place , and the classical form for it is the empirical one due to Arrhenius: c2 = C exp( - ---) 1 T Cc1 , c2 > 0 and independent of u ) . Later theories lead t o the replacement of the constant c1 by an expression of the form c3T n/2 where n is a positive integer which depends on the nature of the reaction and can at present be found only empirically ( see , for example , the books by Glasstone and Lewis(15 , pp.626-638] and Kaufman(17 , pp.198-214 and 233-240]) . In work on the theory of thermal explosions, however , it is usual to use the so-called Frank-Kamenetskii approximation for f , introduced by D.A.Frank-Kamenetskii[1 3] : c5u f( u) = c4e ( C4 , c5 > 0 ) . While this i s indeed an approximation to the Arrhenius expression for f ( u ) when u is small , the theoretical justification for using it in the study of thermal explosions (where large values of u occur) is unclear. It may well be that for large values of u the Arrhenius expression is no longer valid , and the Frank-Kamenetskii expression is in fact more accurate. Alternatively , it may be that situations in which the use of the Frank-Kamenetskii approximation would lead to significantly inaccurate answers have not yet arisen in practice. We shall be particularly concerned in this thesis with the effect on the theory of different assumptions as to the form of f. 2 Various assumptions may also be made about the form of the function g which appears in the boundary condition. The usual approach is to assume that heat loss at the boundary follows Newton's law of cooling , so that g(u) = u and the boundary condition is linear. Most of the discussion in this thesis is concerned with the linear boundary condition , but in Ch.9 we shall discuss the effect of assuming a non? linear boundary condition. There are two non-linear boundary conditions which arise naturally , corresponding to different cooling processes at the boundary. If cooling at the boundary is by natural convection , then g( u ) and g(u) = u51 4 , " while i f cooling is by thermal radiation , then 4 4 = (u+T ) - T and H = cre whe re a i s the Ste fan-Boltzmann con stant a a e is the emissivity of the surface ( see Ozisik 's book(2 6, pp.7- 9 and 3 48-349]) . The discussion in Ch.9 cove rs more gene ral non-linear boundary conditions as well as these two conditions in particular. The customary met hod of tackling the problem of whether or not a the rmal explosion will take place is to e quate the absence of t hermal explosion with the existence of positive stable steady-state solutions, i . e . solution s of t he time-independent e quation 2 2 2 K( o u + o u + o u) + f( u) = 0 ox 2 gy2 oz2 toget her with the boundary condition for (x ,y , z ) in the interior of V K ? + Hg( u) = 0 for ( x ,y , z ) on S . The unde rlying assumption here is that if positive stable steady-state solutions u( x ,y , z ) exist , then the solution of the original time ? de pendent p roblem will approach one of these steady state s as t - ?, and so explo sion will not take place . We shall show in Ch . 6 how this assumption may be justified mathematically . 3 A discussion of thi s question of the existence of positive steady? state solutions, treated from the chemist 's point of view , is given by Boddington , Gray and Harvey( 4] . The se authors, using the linear boundary condition and the Frank-Kamenet skii approximation for f, with a change to a suitably chosen new variable 9 p roportional to u , obtain th e equation o2 e + o 2 e + o 2 e + ye e = 0 ox 2 gy2 oz 2 for ( x ,y , z ) in the interior of V togethe r with the boundary condition K ?? + H 9 = 0 for ( x ,y , z ) on S ? Here y i s a parameter whose value depends on the physical and chemical prope rtie s of the reactant and on the ambient tempe rature . A positive stable solution of this steady- state problem i s known to exi st if and only if y i s le ss than or e qual to a critical value de noted by y crit ( we shall di scuss this point in more detail in Ch.4) . Thus, if the mathematical formulation of the p ro blem is a reasonably accurate model o f the physical situation , a thermal explosion will occur if y > Yc rit but not if y s y ?t ? The value of y 't depends upon the shape of V . cr1 cr1 (\ The authors are chiefly concerned with methods of determining , or determining approximately , the values of y 't for various shap_es V, cr1. using a combination of analytical and empirical methods . In Ch . S , we shall apply the methods developed in this thesis to the problem of obtaining lower and upper bounds for y 't' for various shapes V, and cr1. compare the bounds so obtained with the estimates given by Boddington , Gray and Harvey . 4 In treating the thermal explosion problem from the mathematician's point of view , we shall work for the most part with equations more general than those discussed so far . Letting x denote the n-dimens ional vector ( x1 ,x2 , ? ? ? xn ) , we consider the equation + n ou ou E b . < x , t > ? + c < x , t > u - at + >..f < x , t , u > = o i=1 l. uXi for x in the interior of an n-dimensional region V and t > 0 , together with the initial-boundary conditions ou d0 ( x , t ) g( u) + d1 ( x ,t ) On = 0 for X on the boundary S of V and t > 0 u = u0 ( x ) for x on V U S when t = 0 where the differential operator in the first equation is uniformly parabolic , ? denotes an arbitrary ( not nece ssarily normal ) outward directional derivative , and appropriate conditions are imposed upon the coe fficients a . . , b . , c , d0 , d1 and the functions f and g . l. J l. The corresponding time-independent ( i . e . steady-state ) problem is des cribed by an equation of the form n ... o2u n ... ou .. ' f ... ( x ,u ) E a . . ( x ) CS 0 + E b . ( x ) ? + c ( x )u + 11. = 0 i ,j= 1 l.J xi xj i = 1 1 uxi for x in the interior of V, together with the boundary condition ... ... ou d0 ( x) g( u) + d1 ( x) ?= 0 for x on S where the coefficients are the limits , as t ?m, of the corresponding time-dependent coefficients , and f( x ,u ) is the limit of f( x ,t ,u) . We shall discuss in Ch . 4 some of the more important results that have been obtained on the existence of posit ive stable steady-state soluti ons . In Chs . 7 , 8 and 9 we shall employ an alternative method of investigating the behaviour of u( x ,t ) as t ? ?, by using the comparison theorems proved in Ch . S to directly attack the original time-dependent equation . We should mention here that there are indications that in certain cases neither of these approaches to the thermal explosion problem is adequate ; some remarks on Lhis point appear at the end of Ch . 7 . l 5 I I \ 2 AN EXAMPLE 6 Before proceeding , we give a simple example of the sort of equation we shall be studying . We shall be using this example from time to t ime for illustrative purposes and as a counter-example . Consider first the equation o2u OU -- - - + ku + A = 0 ( - 1 < x < 1, t > 0 ) ox2 ot where we assume k > 0, A> 0; further , u( x , t ) s atisfies the initial? boundary conditions u( x , O) = 0 for -1 ? x ? 1 u( -1 ,t) = 0 for t ? 0 u(1 ,t) = 0 for t ? 0 . Using Laplace trans form techniques ( see Appendix for details ) it may be ( 2n+1 ) 2TT2 shown that if k 1 4 for n = 0,1,2, . . . . , the above problem has the solution ( -1 )n --------??2 2?----- cos ( 2n+1 )TTX 2 2 ( k _ ( 2n + 1 ) TT ) t 4 u( x , t ) ( 2n +?) TT }( 2n+1) 2 e A cos /Kx A + k cos /T( - k ( 2N+1 ) 2TT2 while if k = for some N = 0,1,2, . . . . , the above problem has 4 the solution u( x , t ) 4' ( -1 ) n = 11. 't' ? --------?? 2 2 ?----- cos TT n#N {k - ( 2n+?) TT } ( 2n+ 1 ) 3A( -1 )N ( 2N+1 )nx + TTk( 2N+1) cos 2 + ( 2n+1 ) TTX 2 e N A( - 1 ) X k A - k sin 2 2 ( k _ ( 2n+1) TT ) t 4 ( 2N+ 1 ) TTX 2 From this we see that , regardless of the value of A > O , we have : TT2 ?cos /Kx If 0 0, ? > 0, and the boundary conditions are u( -1 ) = u( 1 ) = 0 . In this case ( again see Appendix for details ) the situation is as follows : m2n2 If k # --4-- for any m = 1 , 2 , 3, . . ? ? , the solution is If k u( x) = !rcos /Kx _ 1] . j(l cos IT< 2 2 = n n for some n = 1 , 2 , 3 , . . . . , the solution is u( x) = e{k ? cos ? - 1] + B sin /kx cos where B is arbitrary . ( 2n+1 ) 2rr2 If k = for some n = 0 , 1 , 2, ? . . . , no solution exists . 4 2 In particular , if 0 < k < ? then the steady-state problem has the pos itive solution u( x) = !rcos Jkx - 1] which is also the limit of kL cos JJ< ' the solution u( x,t ) of the time-dependent problem as t - =. For larger values of k , pos itive solutions of the steady-state problem do not exist ; those solutions which do exist can easily be seen to be negative for certain values of x in ( -1 , 1 ) . If we now take k = \, the differential equation becomes o2u - ou + A o}, also regarded as a subset of E 1. n+ denotes closure . We shall next define several important function spaces. We follow, with some modifications, the definitions used by Ladyzenskaja, Solonnikov and Ural ' ceva(21, pp.2-10] . In framing these definitions, we shall write o l .tlu 1.1 1.2 J. '::. -::. -:::.x n ux1 ux2 . ? ? u n where J. = ( .t1 , J.2, . .. .tn ) , the J.i ( i = 1,2, .. . n ) being non-negative integers, and l .t l = .t1 + 1.2 + + .t . n We shall use r to denote l .t l =k summation over all derivatives of a given order k . orv We shall also write, for any non-negative integer r, otr Now let k be a non-negative integer and a a real number with 0 < a < 1. We say a function u :V - R satisfies a Holder condition with exponent a on V if (u)?a) is finite, where sup x,yEV xi-y ju( x ) - u(y) j l x - y jC1 For any function u : V - R which has continuous derivatives up to order k, and whose derivatives of order k satisfy a Holder condition with exponent a on V, we define : SUE_ j u( x ) 1. xEV E ID!ul?o ) l .t l = j for j = 1,2,.. . ? ? k lul?k+a) = I: (o.tu)(a) + I: lulv(j) . l.tl=k X V j =O The Holder space Ck+a(V) is the space of all functions u :V ? R for which lul?k+a) is finite ; with lul?k+a) as norm , the space Ck+a(Y) is a Banach space . Again let k be a non-negative integer and a a real number with 0 o). We shall denote by Lu the expression n + E b . c x , t ) -::..ou + c < x , t ) u i= 1 1 oXi where the coefficients a . . , b . and c are assumed to be cont inuous real 1] 1 functions on DT for all T > O , and a . . = a . . for i , j = 1 , 2 , . . ? n . ?] ]? Stronger assumptions regarding these coefficients will be made from time to time as needed . The differential operator L is assumed to be uniformly elliptic for each T > 0 , i . e . there exists for each T > 0 an A > 0 such that n r i , j = 1 for all real vectors ? = ... a . . (x , t ) ? . ? ? ?] ? J ( ?1 '?2 , . ? . ?n ) ? A and n 2 r ? ? i = 1 ? all ( x , t ) E DT . We shall denote by Lu the expression + n E b . ( X ) ou + c ( X ) u i= 1 ? oxi where the coefficients a . . , b. and c are assumed to be continuous real l. J ? functions on V unless stronger assumptions are needed , a . . = a . . for ... l.J ] ? i , j = 1 , 2 , . . . n , and the differential operator L is assumed ?o be uniformly elliptic on V in a sense similar to that defined for the operator L , but now A does not depend on T . We shall denote by B1 ? u the expression _l.n ou d0( x , t ) u + d1(x ,t ) On and by B u the expression gen where d0 and d1 are assumed to be non-negative , continuous real functions on The function n denotes an on oV of the ST for all g : R - R is outwardly form n(x) T > 0 unless stronger assumptions are needed . assumed t o be strictly increasing . Further , directed , nowhere tangential unit vector field = (n1(x) , n2(x) . , ? . . nn(x) ) where n1 ,n2 , . . . nn 12 are o f class c1+a< oV ) . We shall denote the outward unit normal vector ou to oV by v(x ) ; this is of course a particular case of n(x) . On denotes the directional derivative .. n ou I: n . (x)?. i= 1 1 uXi We shall denote by B1. u the expression ?n A and by B u the expression gen A A OU d0(x) g(u) + d1 ( x) an 13 where d0 and d1 are assumed to be non-negative , continuous real functions on oV unless stronger assumptions are needed , and the notation is in other respects the same as that defined in the previous paragraph . 4 THE STEADY-STATE PROBLEM - A SUR VEY 14 As remarked in Ch . 1 , the que stion of the existence of positive stable steady-state solutions for the heat-generation problem has attracted much attention . In particular , the results of Keller and Cohen[19] , Keener and Keller[18] and Amann[2] give quite a good picture of the relation between the form of the function f ( introduced in Ch . 1 ) and the existence or non-existence of positive stable steady-state solutions . We shall examine this picture , and then later , in Ch . 7 , compare it with the picture obtained by considering the related time? dependent problem . All the above authors restrict themselves to the real self-adj oint problem described by the equation n E ? a . . ( x ) o o u ) i , j = 1 xi ?J xj ao ( x )u + Af( x ,u) = 0 for X E V together with the boundary condition n ... o u -d1 ( x ) I: v . ( x )a . . ( x ) ? - o . . ? ?] uX. ?.]=1 J for X E oV. In the first equation it is required that a . . = a . . E c 1 +a(V) for ?] ]? i , j = 1 , 2 , ? . ? n and some a with 0 0 independent of x , s ince the differential operator in (1) is uniformly elliptic . It follows that n(x ) is an outwardly directed , non-tangential vector for each X e oV. Thus the boundary condition is a special case of the ... condition B1 . u = 0 . Finally , it is assumed that oV = s1 u s2 where s1 ?n has pos itive measure and : d0(x) > 0 , d1 ( x ) = 0 for all x e s1. d0(x ) ... ? 0 , d1(x) > 0 for all x e s2 . 15 The condition that s1 should have positive measure is needed in order to apply a certain uniqueness theorem based on the generalised maximum principle ; one form of this theorem is given by Protter and Weinberger (2 8 , Ch . 2 , Theorem 1 2] . It should be noted that , apart perhaps from this last restriction on the boundary condition , the original steady-state heat-generation problem in the for? studied by Boddington , Gray and Harvey[4) is a special (three-dimens ional ) case of this general self-ad?oint problem in which a . . (x ) = 1(i = :,2,3) and a .. (x ) = O(i 1 j) for all X e V. ?? ?] Following Keller ar.c Cohen[19] , we refer to the set of values of ? for which positive soluTions u(? ; x ) of (1 ) exist as the spectrum of (1 ) , and denote the least upper bound of this spectrum by A*? Keller and Cohen begin by assuming that f satisfies the following hypotheses : Ho: f is continuous for x e V, u ? o . ... H1: f(x , O ) > 0 for x E V . H2: f(x , v ) > f(x ,u ) on V if V > U ? Q, With these hypotheses , Keller and Cohen are able to prove the following : (i ) Only positive ? can be in the spectrum of (1 ) . (ii ) For every ? > 0 in the spectrum of ( 1 ) , there e xists a positive solution u . (? ; x ) of (1 ) which is minimal , Le . which is such that m?n umin(? ; x ) ? u(? ;x ) on V for any positive solution u(A ;x ) of (1 ) . (ii i ) If A1 > 0 is in the spectrum of (1 ) , then all A satisfying 0 < A ? ?' are in the spectrum , and u . (? ; x ) is an increas ing function m?n of A for each X e V and 0 < ? ? A1? (iv) If there exists a positive function F on V such that f(x ,u ) < F(x ) for all u > 0 and all x E V, then all ? > 0 are in the spectrum of (1), i . e . a finite A* does not exist . (v ) If there exist pos itive functions F , p such that for all u > 0 and ? .? 15 all x E V , f( x ,u) < F ( x) + p( x ) u , then the spectrum of ( 1 ) contains all ? such that 0 < ? < ? {p) , where ? {p) denotes the principal eigenvalue of n I: ?a . . ( x ) ":>.ov ) . . 1 oX{ ?] vX]. a0 ( x)v + ?p( x )v = 0 on V ? ,]= .... Thus ?,'; ? ?1 {p) . " n .. ov _ d1 ( x) I: v . ( x )a . . ( x ) ? - 0 . ? ? ?] vX. ?.] =1 J on oV . I f , on the other hand , f( x ,u ) satisfies f( x ,u) > F ( x ) + p( x )u on V , for all u > 0 , then ?* ? ?{p) . .. Assuming that f satisfies H0 ( and possibly H2 ) but not H1 , Keller and Cohen prove also : ( vi ) If there exists a ?ositive p such that f( x ,u) < p( x ) u on V , for all u > O , then no ? such t?at 0 < ? < ?1fp) is in the spectrum of ( 1 ) . An important point that emerges from results ( i ) , ( i i i ) and ( iv ) is that , assuming hypot?eses H0, H 1 , H2 , positive solutions of ( 1 ) , i f they exist at all , exist for A on an interval of one of the forms 0 < A < A??:, 0 < ? ? ? * or A. > 0 ? Keller and Cohen next introduce the strong monotonicity condition f ( x , u ) > 0 and continuous on V for u > 0 . u .. On the assumption that f satis fies H0 , H1 and H2, , and that ( 1 ) has positive solutions for all ? such that 0 < ? < v:, they then prove that each X in this interval satisfies X< ?1 ( ? ) where ?1 ( A. ) is ?1 {p) as de fined above , with p( x ) = f ( x , u . ( ? ; x) ) . Thus ?1( A. ) is the principal u m?n e igenvalue of the linearization of ( 1 ) . Following Keller and Cohen , we say that f is aonaave if it satisfies H2, and in addition H 3a : fu( x ,u) < fu( x ,v) on V i f u > v ? 0 .. and we say that f is aonvex if it satisfies H2, and in addition f ( x ,u) > f ( x , v) on V if u > v ? 0 . u u Keller and Cohen then obtain the following results : ( '' ) If f.. ? f' H d ? {concave ) d 'f ( 1 ) has the v?? sat?s ?es 0 , H 1 an ?s , an ? convex is an {increas?ng)? spectrum 0 < ? < A* or 0 < A ? ?* , then ?1(?) decreas1ng ... function of ? on this interval . Furthermore , i f f is concave . then ? ( A ) < ?* for 0 < ? < ??':, and if f is convex then ?( A ) > V: for 0 < ? < ?''c ? ... ( viii ) I f f satisfies H0, H1 and is concave , then lim ?1CA) = A''c and A fA* ?* is not a point of the spectrum . Thus the spectrum must take one of the forms 0 < ? < A* or ? > 0 . Furthermore , there is exactly one positive solution of (1) for each A in the spectrum. ( ix) If f satisfies H0 , H1 and is concave , and if iri addition lim f ( x ,u ) = p ( x) on V, then A* = ? {p) where we adopt the convention u-- u 1 that ?1{p) = m if p( x) = 0 . Thus Keller and Cohen obtain a reasonab ly comple?e picture of the situation in the case of concave f, but rather less information in the ... ... case of convex f . In the case of convex f , note that it is known in certain special cases that the positive solutions for all A in the interior of the spectrum are non-un ique ( see the paper by Laetsch[23 ] ) . Keller and Cohen conclude by discussing the question of stability . For any ? in the spectrum of (1), they define a steady-state solution u( A ; x ) to be stable if , roughly , any solution of the time-dependent problem which satisfies an initial condition of the form u0 ( x) = u( ? ; x ) + ev (x ) decays exponentially in t to u( A ;x ) , t o first order in e . If one of two stable steady-state solut ions is such that this exponential decay described above is more rapid than in the case of the other steady? state solution , then the first solution is said to be re latively more stab le than the second . Keller and Cohen then prove the following : 17 ( x ) Suppose f satis fies H0, H1 and H2, , and is such that (1) has a non? empty spectrum . Then , for 0 < A. < ?* , the minimal positive solut ion ... of (1) is stable . I f , in addition , f is convex , the minimal positive solution for a given ? is relatively more stable than any other positive solution for the same A ( if f is concave , we already know by ( vi i i ) that the minimal positive solution is in fact the only pos itive solution for a given A ) . F. 11 ' f f.. . {concave ) h 1 t' 1na y , 1 1s , t e re a 1ve convex ab '1' f h ? ? . . 1 . { increases ) , . st 1 1ty o t e m1n1mal pos1t1ve so ut1ons d as A 1ncreases , ecreases on 0 < ? < ??? . The case where f is convex is studied in more detail by Keener ?nd Keller[18 ] . They use the following strong convexity condition : H( 3b ) ' : fuu( x ,u ) > 0 and cont inuous on V for u > 0. 18 A solution u( A ; x ) of (1) is said by Keener and Ke ller to be non-isolated if the linearization of (1) about that solution , i . e . the problem n 't" 0 1,. OV ? a-'a . . ( x ) -'::l. -) i,j=1 xi lJ uxj Af ( x , u( A ;x ) )v = 0 u n ci 1 ( x ) r: v. ( x ) a . . ( x ) 0 ?v = o i , j = 1 l lJ xj on oV on V (2) has a non-trivial solu?ion . A solution u( A ;x ) of (1) is said to be a principal non-isolated so lution i f it is a non-isolated solution for which ( 2 ) has a pos i?ive solution . Keener and Keller then prove the following fundamental result : Let H0 , H1, H2, a?d H ( 3b ) ' hold , and for A = A? > 0, let ( 1 ) have a positive principal nc?-isolated solution , u( A0 ;x ) > 0 on V. Then : ( a ) A = ( b ) A0 = A* , and u( A0 ; x ) is the unique positive solu?ion of ( 1 ) for A:?, ; A minimal pos itive solution of ( 1 ) exists for all A E ( O , A* ) , and no positive solutions exist for A > A* ; ( c ) For some sufficiently small 6. > 0 , a pair of pos i?ive solut ions of ( 1 ) exists for each A E (A*-6. , A* ) . After this , the major question remain ing for Keener and Keller to deal with is that of the existence of a positive principal non-isolated solution . They require for this the following hypothesis of asymptotic linearity : H4.. ll' m { f( x ,u ) - [F u ( x) + uG( x )] } - -- 0 - - on V , where G ( x ) > 0 on V . u-co To complete their pr9of , they also require an hypothesis H5 of a rather technical nature , which need not be given here , since Amann[2) has shown that it can be dispensed with . On the assumption that f satisfies H0 , H1 , H2, , H( 3b ) ' ? H4 and H5 , Keener and Keller prove that a positive principal non-isolated solution of ( 1 ) does in fact exist for some posit ive Ao? This shows that for such f the spectrum of ( 1 ) is an interval of the form 0 < A ? V:. Subsequently, P.mann(2] has sho1m that the crucial hypothesis in the work of Keener and Keller is that of asymptotic linearity. Amann ... assumes only that f satisfies the following conditions : 1 ) f( x ,u ) > 0 for x E V, u > 0 . 2 ) lim f ( x ,u) = f ( x) exists uniformly for x E V, and f ( x ) > 0 for u ? ? u- x E V ( this is the crucial assumption of asymptotic linearity ) . Amann denotes by ?m the principal eigenvalue ?(f?) ( in the notation of Keller and Cohen introduced earlier) . He is then able to prove the following comprehensive theorem : There exists ?;': > 0 such that for every A E ( 0 ,?*) , ( 1 ) has a minimal pos itive solution u . ( ? ; x ) , and ( 1 ) has no solution for m1n A > ?*. ( 1 ) has a solution for A = ?*, in fact a minimal positive solution umin ( A1': ; x ) , i f and only if { llumin (A ; x ) IIC ( V) : 0 < ? < ???: ) is bounded . This is the case if and only if u* ( x ) = lim u . ( ? ; x ) A-A;?:_ m1n exists in c2+a(V), in which case u* ( x ) = u . (A*? x) . m1n ' Further , we have 0 < ? ? ? ;': . ? I f lim JJu . (A ; x ) JJC ( V ) = ?, then A-??L m1n A? = ???: ( compare this with result ( ix ) of Keller and Cohen for concave f). On the other hand , if ? < ?;': , then ( 1 ) has a minimal pos itive ? solut ion for A = A;':, and for every A E ( ? ,Af:), (1 ) has at least two ? distinct positive solutions . Finally , suppose there exists a pos itive function y , continuous on V , and a constant p > 0 , such that for all x E V and all u ? p, f ( x ,u) - f ( x ,u )u ? -y ( x) . u 19 Then ?? < A*? Note that this condition is related to convexity , s in ce it implies that for every x E V and for u ? p, the tangent to the graph of f( x , u) against u intersects the negative y-axis . Amann ' s theorem therefore gives a rather complete picture of the s ituation for the case where f is asymptotically linear , thus filling in the picture drawn by Keller and Cohen[ 19 ] ?and by Keener and Keller[1 8 ] . The example discussed in Ch . 2 s erves to illustrate one of the cases considered by Amann . In this example ( the steady-state problem with k = A) we have f( x ,u) = u + 1 , which satisfies the conditions for Amann's theorem with f ( x) = 1 . Ours is an example of the case dealt ? with by Amann in which lim u . ( ? ; x ) does not exist ; in our example A_A,L m1n cos /Ax n2 urn in 0 .. ; x ) is cos Jt - 1 and A;': = 4 In this case Amann 's theorem tells us that A* = A? = ?1{fc}. Now ?1{fc} in our example is the principal e igenvalue of the linear problem d2u - t AU = 0, u( 1) = u( -1) = 0 dx2 2 2 h . h h ? 1 ' = m n ( 1 2 3 ) th t ' t ? ? 1 w ?c as e ?genva ues I\ -4- m = , , , ? ? ? , so a ? s pr?nc?pa 2 20 ? 1 ? ? d d n ? d e?genva ue ?s ?n ee ? as requ?re ? Note that , as required by Amann ' s 2 theorem in this case , the spectrum is 2 solution exists for A = n4 . the open interval (O,n4 ) ; no 5 COMPARISON THEOREMS FOR THE TIME -DEPENDENT PROBLEM Comparison theorems , based on various versions of the maximum principle , are a standard tool in the study of differential equations . They are discussed in the books by Protter and Weinberger(28] and Friedman[14] , and used by many authors , such as Chan[9] , McNabb [2 5 ] , Sattinger[3 3] and Wake[35]. The comparison theorems which will play a fundamental role in the rest of this thesis are based on those proved by McNabb , but they differ in certain important details , and so the proofs are given in full here . We require first a lemma due to Fej er[12] , the proof of which we include for the sake of completeness . n LEMMA : If g (x ) = L gikxi? and h ( x ) = i ,k=1 n r hikxi? are two non?i ,k= 1 negative quadratia forms, with gik = gk i and hik = hk i for all n i ,k = 1 , 2 , . . . n, then r gikhik ? 0. i ,k = 1 Proof: The result is obvious if e ither g or h i s identically zero ; as sume therefore that neither g nor h is identically zero. We shall n first show that there are n linear forms z ( x ) = L p x ( where r rs s s = 1 r = 1 , 2 , . . . n ) with real coefficients prs ' such that n 2 g ( x ) = L z ( x ) = r= 1 r ? ? ? ? ? ? ? ? ( 3 ) We know that for all i = 1 , 2 , . . ? n , g . . ? 0, s ince g . . i s the 11 1 1 21 value of g ( x ) when x . = O ( j # i ) and x . = 1 . Further , the coefficients J 1 g . . cannot all be zero , for if they were , then since g( x ) ? O , there 1 1 must be at leas t one g . . # O ( i # j ) . 1] g ( x) could be obtained by choosing x . 1 s ign of g . . ) and ? = 0 for k# i , j . 1] K i = 1 , 2 , ? . ? n , we must have g . . > 0. 11 assume g1 1 > 0. Now write : ? p 1 1 = ( g1 1 ) . In that case a negative value of = 1 , x . = ?1 ( depending on tbe J Thus , for at least one Without loss of generality , P 1 1P 1s = g1s ( s = 2 ' 3'''' n ) . n r p 1sxs . s=1 2 2 g( 1 ) ( x ) = g ( x ) - z? ( x ) . It is easily seen that the quadrat ic? form g( 1 ) ( x ) is independent of x1 . Furthermore , it is non -negative , for suppose we could obtain a negative value for g ( 1 ) ( x ) by tak ing x . = a . ( i = 2 , 3 , . . . n ) . Put ? ? -1 n a = ? I: p 1sas ) . 1 P11 s=2 Then , writing a = ( a1 , a2 , . . ? an ) , we have z1 ( a ) = 0 and so we obtain the contradiction g( a ) = g( 1 ) ( a ) < 0 . I f g ( 1 ) ( x ) is identically zero , then ( 3 ) is proved already , s ince 2 we may take zr( x ) = 0 for r = 2 , 3 , . . . n , and we have shown g ( x ) = z1 ( x ) . I f not , we can carry out for g ( 1 ) ( x ) a construct ion similar to that n carried out for g ( x ) , obtaining a linear form z2 ( x) = I: p 2 x such s=2 s s that g( 2 ) ( x ) = g ( l ) ( x ) - z; ( x ) = g ( x) - [z? ( x) + z; ( x ) ) is a non? negative quadratic form independent of both x1 and x2? Continuing thus , after n steps we will obtain g ( n ) ( x ) = 0 and so g( x) = i z2 ( x) , which r= l r proves ( 3 ) . n It then follows from ( 3 ) that gik = I: Pr?Prk ( i ,k = r= 1 n Similarly , we obtain h . = ?k r q q c i ,k = s i sk Hence n I: gikhik = i ,k= l s= 1 n n 1 , 2 , . ? . = I: { I: Priqs iprkqsk } i ,k= 1 r , s= 1 = n n 2 r c r P .q . ) ? o . ? k 1 J. --1 r] s J ?, = n ) . 1 ' 2 ' . . . n). We prove now the first comparison theorem we require ; the method is similar to that used by McNabb[2 5 , Theorem 1] . THEOREM 1 : Suppose that ( a ) The funations u1 and u2 are defined and aontinuoUB in DT ' their first-order x . -derivatives exist in DT ' their seaond-order x . -derivatives ? ? exist and are aontinuous in DT , and their firs t-order t-derivatives exist in DT . ( b ) For al l ( c ) u1 ( x , O ) ou1 ( x ,t ) E DT ' Lu1 - ot + f ( x ,t ,u1 ) < u2 ( x , O ) for aZZ x E V. ( d ) For al l ( x ,t ) E ST , B u1 < B u2. gen gen Then u1 ( x , t ) < u2 ( x , t ) for all ( x , t ) E ? ? Proof: Suppose , on the contrary , that there is a point P in DT where u1 ? u2. Then (by the continuity of u1 and u2 ) there is a point ( x1 , T1 ) E DT such that u1 :s: u2 in DT' , while u1 = u2 at ( x1, T ' ) , T1 > 0. So v( x , t ) = u1 ( x , t ) - u2 ( x , t ) has a maximum of zero in DT' at the point ( x1, T1 ). Suppose first that we may choose ( x1, T1 ) E DT' ' i . e . such that x1 is not on the boundary oV . Then the quadratic form n I: D D v ( x1, T1 ) x . x . . . 1 X . X . ? ] ? , ] = ? J n is non-positive . Since the quadratic form I: a . . ( x1 , T 1 ) x . x . is non- i , j = 1 ?] ? J negative by definition of L , it follows by the lemma that : Further , v( x1, T1 ) = 0 . Dtv( x1, T ' ) ? n I: i , j = 1 n i . e. I: i , j = 1 for each i = a . . ( x' , T ' ) D D v( x1, T1 ) ?] x . x . ? J a . . ( x' , T ' ) D D v( x', T ' ) ?] x . x . ? J 1 ' 2 ' ? . . n , D v( x1, T' ) = X . ? ? 0 :s: 0. o , and also Thus Lv :s: 0 at the point ( x', T 1 ) . Since also 0 ' it follows that Lv - ov dt :s: 0 at the point ( x' , T1 ) . ou1 ou2 Lu1 - at :S: Lu2 - Tt at the point ( x', T1 ) ? Finally , since u1 ( x', T 1 ) = u2 ( x', T1 ) , we have that at the point 2 3 ou1 ou2 ( x1, T' ) , Lu1 - -at + f( x ,t , u1 ) ? :s: Lu2 - Tt + f ( x ,t ,u2 ) , contradicting hypothes is ( b ). If x1 cannot be chosen away from oV , then we must have u1 = u2 at ( x' , T' ) with x1 E oV , and u1 < u2 in DT'' 0 Thus ?u1 - u2 ) :s: 0 at ( x', T1 ). ? B B h ? ( I T1 ) ? t ( I T 1 ) ? . e. genu2 :s: genul at t e po?nt x , , s?nce u1 = u2 a x , . This contradicts hypothesis ( c ) , and so the proof of the theorem is complete . 2 4 In the case where the function f( x ,t ,u ) satisfies a uniform Lipschitz condition in u on any finite interval , a stronger comparison theorem can be proved . The method of proof is an extension of that used by McNabb(2 5 , Theorem 2 ) . THEOREM 2 : Suppose that ( a ) The functions u1 and u2 are de fined and continuous in DT , their first-order x . -derivatives exist in DT , their second-order x . -derivatives ? ? exist and are continuous in DT , and their first-order t-derivatives ( d ) The coeffiaient d0 ( x , t ) of g (u ) in B u is strictly positive for a l l gen ( x ,t ) E ( e ) For ST . aU ( x ,t ) E ST ' B u1 ? B u2 ? gen gen ( f ) On any finite interval [a ,b] , the function f( x , t ,u ) satisfies a uniform Lipschitz condition in u , ? . e . there exists a constant M[a ,b] > 0 (depending on the interval (a ,b] J such that j f( x , t , u1 ) - f( x ,t ,u2 ) I ? M(a ,b] lu1 - u2 l for a l l u1 ,u2 E (a ,b] and al l ( x , t ) E DT . Then u1 ( x , t ) ? u2 ( x ,t ) for a l l ( x , t ) E DT . in ? ' it is bounded there . Let Proof: Since u2 is cont inuous m1 = inf _ u2 ( x ,t ) and m2 ( x ,t )EDT = sup _ u2 ( x ,t ) . Choose M' > 0 such that ( x ,t ) EDT M' > sup {M ) + c( x , t ) } ( re call that c ( x , t ) is the coefficient ( x , t )EDT [m1 ,m2+1 of u in Lu) . For UA( x , t ) = u2 ( x ,t ) all ( x , t ) E DT A M' ( t -T ) + e ? and all A E [ 0 , 1) , define Then for all ( x ,t ) E DT and all A E m1 ? u2 ( x ,t ) < UA ( x ,t ) ? m2 + 1 . Thus , for all ( x ,t ) E DT ?and all A E ( 0 , 1 ) , we have : oUA [LUA - ? + f( x ,t ,UA) ) = f ( x , t ,UA ) - f( x ,t ,u2 ) + AC ( x ,t ) eM' ( t-T ) - AM' eM' ( t -T ) ? M AeM' ( t -T ) + Ac ( x , t ) eM' ( t -T ) - AM' eM' ( t -T ) by ( f ) [m1 ,m2 +1 ] = AeM' ( t -T ) {M ( J + c ( x ,t ) - M' } < 0 . m1 ,m2 +1 ( 0 , 1 ) , Thus , using hypothesis ( b ) , we have that for all ( x ,t ) E DT and all A E ( 0 , 1] , ou1 oUA Lu1 - 3't + f( x ,t ,u1 ) > LUA - ? + f (x, t ,UA ) . Also , for all ( x , t ) E ST and all A > 0 , BgenUA - Bgenu2 = do ( x ,t ) {g ( UA ) - g( u2 ) } > 0 by hypothesis ( d ) and the fact that g is strictly increasing by definition of B gen Thus , using hypothesis ( e ) , we have : B u < B U gen 1 gen A for all ( x ,t ) E ST and all A > 0 . Further , we have that u1 ( x , O ) ? u2 ( x ,O ) < UA ( x , O ) for all x E V and all A > 0 . I t follows by Theorem 1 that , for all A E ( 0 , 1] , u1 C x ,t ) < UA( x , t ) for all ( x , t ) E DT . Since UA( x , t ) - u2 ( x ,t ) as A - 0+ for each ( x ,t ) E DT , it follows that u1 C x ,t ) ? u2 C x , t ) for all ( x , t ) E DT , as required . Notes : ( i ) I f , in the statement of Theorem 2 , we omit hypothesis ( f ) , that f( x , t ,u) should satis?; a uniform Lipschit z condition in u on any finite interval , then the thecrem fails to hold , as the following counter-example demonstrates . o2u Take Lu to be -- V to be 2 , ox {x : -1 < x < 1 } , ? ou ou d f( ) b 576e lu l B u to be gen u + 0\1 = u + x ox an x , t ,u to e Consider first the function y (x ) 1 2 2 = ?x - 1 ) ( 5 -x ) 5 + 1 . Here y' (x ) 1 2 = 5{ C x - 1 ) ( -2x ) + ( 5 -x2 ) 2x} y * ( x) For - 1 ? x ? 1 , the = ?< 3 -x2 ) . 5 4 2 = ? 3-x ) + = ?< 1-x2 ) . graph of y (x ) point of inflection /: ? I w?th non-horizontal tangent ?-2x) 5 is as follows : y 1 - - - - - - - - :""' . . f . 1 po?nt of ?n lect ?on : with non-horizontal tangent 26 Then : ( a ) u1 ( x , O ) = 0 = u2 ( x , O ) , so certainly u1 ( x ,O ) ? u2 ( x , O ) for - 1 ? x ? 1 . ou1 ( b ) Lu1 - dt + ou1 [Lu1 - ? + f( x , t , u1 ) ] 2t + 5 76e.ff'. = 2t {y (x ) - 1 } - t2y ' ( x ) + say . We have that A( x , O ) = 0 for -1 ? x ? 1 . Assume now that t > 0 and - 1 < x < 1 . Then : a? A ( x , t ) = 2 (y ( x) -1 } - 2ty' ( x) + 5 76?? -(??%/' e/t'(y( xll''} ( 4 ) J,... For -1 < x < 1 , we have 0 ? y (x ) < 1 , so 0 ? {y( x) } ? < 1 . Put {y ( x ) }? = 1 - 6 (x ) where 0 < 6 ( x) ? 1 . Then y ( x ) = {1-6 ( x) }4 = 1 4 6 ( x ) + 662 ( x) - 4 o 3 ( x ) + o4 ( x ) 1 2 2 = 1 "?-< 1-x ) ( 5-x ) . 6 C x ) {4 - 6 0 C x ) + 462 ( x) - 6 3 ( x) } = ? 1-/ ) ( 5 -x2 ) . 6 ( x) = ( 1-x 2 ) ( 5 -x2 ) 5 {4 - 6 6 ( x ) + 4 62 ( x) - 63 ( x) } ( 1-x2 ) ( 5-x2 ) > 40 for -1 < x < 1 , s ince then 0 < 6 ( x) ? 1 . Thus ( 4 ) becomes : c! A( x ,t ) = ?/-1 ) ( 5 -x2 ) - 2?t( 1 -x2 ) + Tf?e/t - {1-6C x ) }e/t{1-6 ( x ) } } 21 2 1 ) ( 5 2 ) _ 24t( 1-x2 ) = 5' x - -x 5 /t 2 2 2, 2 1 ) ( 5 2 ) _ 24t( 1-x2 ) 2 88e ( ( 1 -x ) ( 5 -x ) } > S'x - -x 5 + If 40 If t tit Further , for t > 0 , e = 1 + .If + 2T + V + > /t + t? . from above . 0 2 2 2 24t 2 72 t 2 2 ot A ( x , t ) > px - 1 ) ( 5 -x ) - y-< 1-x ) + fo-<1+6) ( 1-x ) ( 5-x ) 2 2 2 24t 2 1 2 } = p 1-x ) ( 5 -x ) {- 1+ 1 8 } + y-<1-x ) { - 1 + "fr-< 5-x ) . 2 7 The first tei'l!l i s ?obviously pos itive for - 1 < x < 1 ; the second is also . . . 5 2 f pos 1t1ve , s1nce -x > 4 or -1 < x < 1 . Thus , for - 1 < x < 1 , 0 A( x , O ) = 0 and dt A ( x , t ) > 0 for t > 0. I t follows that A (x , t ) > 0 for t > 0 and -1 < x < 1 . ou1 ou2 Lu1 - (ft: t f( x ,t ,u1 ) > Lu2 - (ft: t f( x , t ,u2 ) for t > 0, - 1 < x < 1 . ( c ) For x = ?1 , B ou1 u = u1 + x ax- -gen 1 B ou2 u = u2 t X ax- = gen 2 = = t2 . 2 t y ( x ) t xt2y' ( x) 2 t y ( x ) 4 2 2 2 + - t x ( 3 -x ) 5 t2 Bt 2 s ince 2 + -- X = 5 B u < B u2 for x = ?1 , t > 0 . gen 1 gen 1 , y ( x ) = 1 . Thus all the hypotheses o= Theorem 2 except hypothesis ( f ) are satisfied for any T > 0 ( indeed , we have rather more than is required , s ince the inequali ties in ( b ) and ( c ) of the counter-example are strict > and < rather than ? and ? as in the theorem) . Hypothes is ( f ) u? is not satisfied , s ince the funct ion e does not satisfy a Lipschitz condition on any interval [O , a] ?ith a > 0 , as its derivative 1 - 3 / 4 u? . 4 u e 1s unbounded on any such interval . And the conclus ion that u1 ( x , t ) ? u2 ( x , t ) for all t ? 0, - 1 ? x ? 1 is false , s ince 0 ? y ( x ) < 1 if - 1 < x < 1 , and so u1 ( x , t ) > u2 ( x ,t ) for all t > 0 and - 1 < x < 1 . ( ii ) Hypothesis ( f ) o f Theorem 2 can , however , b e rep laced by a condition that f( x ,t ,u ) be monotone decreasing in u ; this allows the proof to be slightly s imp lified . This version of the theorem is of little or no interest for the present discussion , s ince in this thesis the funct ion f( x , t ,u) is generally assumed to be monotone increas ing in u , this being the case in the heat-generation problem which motivates the whole discussion . Note that a comparison theorem involving monotone decreas ing f is proved by Chan[9 , Theorem 1] and used ? to derive some interesting existence and uniqueness theorems . 6 RELATIONS BETWEEN SOLUTIONS OF THE TIME-DEPENDENT AND STEADY-STATE PROBLEMS 2 8 It will b e recalled that in Ch . 1 we discussed two poss ible approaches to the problem of determining whether or not a thermal explosion will take place in a given s ituation . The usual approach is to argue that a thermal explosion will take place if the equation describing the system has no positive steady-state solutions . The approach used in this thesis is to argue that a thermal explosion will take place if the solution of the t ime-dependent equation is unbounded as t - m or as t tends to some finite value . We wish to show in this chapter that , under fairly wide conditions , these two approaches are mathematically equivalent . Accordingly , we wish to investigate the relation between the boundedness over all ti?e of the solution of the time-dependent problem ou Lu - dt + f( x ,t ,u) = 0 for ( x ,t ) E D B 1 . u = 0 ?n for ( x ,t ) E S u( x ,O ) = u0 ( x ) for x E V and the existence of pos itive solut ions of the corresponding steady? state problem Lu + f( x ,u ) = 0 for x E V Bl . u = 0 for X E oV ?n where the coefficients are the limits , as t - m, of the corresponding time-dependent coefficients , and f ( x ,u) is the limit as t - ? of f( x , t ,u) . We restrict the discussion to the linear boundary condition because the fundamental theory on which this chapter is based is not available for the non-linear boundary condition . The parameter A which appeared in the equations mentioned in Ch . 1 is , for the purposes A of the present discussion , absorbed into the functions f and f . It will reappear later . In the book by Friedman [14 , Ch . 6) there are some important theorems concerning the case where f and f are independent of u . Friedman uses less general boundary conditions than ours , but his methods are easily adapted to our boundary condit ions , as we shall show later . With certain restrictions on the coe ffic ients , Friedman proves that , if it is known that the steady-state problem has a unique solution , then the solution of the t ime-dependent problem will tend to this steady-state solution as t - ..., , Reynolds( 29] extends Friedman ' s ... method t o prove a s imilar theorem for the case where f and f are 2 9 dependent on u ( Reynolds ' theorem is rather general , s ince it allows for non-linearity of the differential operators as well as the functions f ... and f ) . However , Reynolds ' result is not quite what we want , s ince we are frequently concerned with s ituations where the steady-state problem is known to have multiple solutions , as we have seen in Ch . 4 . It should also be mentioned that Liapunov methods have been used to study problems of this type , for example by Chafee and Infante(B] in the case of a special one-dimens ional problem . ' The technique which was found to be appropriate for our purposes was that of monotone iteration , introduced by Courant(1 1 , pp . 370 , 371] and developed further by Cohen(10] and others . Using this technique , Sattinger(3 1 , 3 2] has proved two existence theorems , for parabolic and elliptic problems , which are of great value in our present study , and which we will now discuss in some detail . Monotone I teration : We consider first the parabolic initial-boundary value problem ou Lu - at + f( x ,t ,u) = 0 for ( x ,t ) E DT Blinu = 0 for ( x ,t ) E ST u( x , O ) = u0 ( x) for x E V . ? ? ? ? ( 5 ) where L and B1 . are as defined in Ch . 3 , with the addit ional assumptions ?n a - a - a -that a . . = a . . E H ( DT ) ' b . E H ( DT ) and c E H ( DT ) for all T > 0 ; also ? ] ] ? . ? 1+a -d0 and d1 are of class H ( ST ) for all T > 0 . We assume also that u0 E c 2+a(V) and f is continuous for ( x ,t ) E DT and at leqst some u- interval . We call ? x ,t ) an upper soLution for ( 5 ) if ? is continuous in DT , I has continuous first-order xi-derivatives in DT ' continuous second-order xi-derivatives in DT and continuous first -order t-derivatives in DT ' and s at i sfies : Lcp - ? + f( x ,t ,?) ? 0 for ( x ,t ) E DT Blincp ? 0 for ( x ,t ) E ST Cj)(x ,O ) ?? uo (x ) for X E V . We call ? x ,t ) a strict upper solution for ( 5 ) if ? is cont inuous in DT , has? cont inuous first-order xi-derivat ives in DT , continuous second-order xi-derivatives in DT and continuous first-order t? derivatives in DT , and satisfies : L? - ? + f( x ,t ,?) < 0 for ( x ,t ) E DT Blin? > 0 for ( x , t ) E ST ?x ,O ) > uo ( x) for X E V. The terms lower solution and strict lower solution are defined analogously by reversing the inequalities in the above definitions . 30 By a solution of ( 5 ) we shall understand a classical solution u( x ,t ) of ( 5 ) which is cont inuous in DT , has continuous first-order xi? derivatives in DT , continuous second-order xi-derivatives in DT and cont inuous first-order t-derivatives in DT . It follows from Theorem 1 that if ? is a strict upper solution for ( 5 ) and u a solut ion of ( 5 ) , then ? x ,t ) > u( x , t ) for all ( x ,t ) E DT . I f f( x ,t ,u) satisfies a uniform Lipschitz condition in u on any finite interval , and the coefficient d0 ( x , t ) of u in Blinu is strictly positive for all ( x ,t ) E ST , and if ? is an upper solution for ( 5 ) and u a solution of ( 5 ) , then Theorem 2 shows that ? x , t ) ? u ( x ,t ) for all ( x , t ) E DT . Analogous results with reversed inequalit ies hold for lower solutions . Note that , in the case where f and d0 satisfy the hypotheses of Theorem 2 , we can use Theorem 2 to prove that the solution of ( 5 ) , if it exists , is unique ; if u1 and u2 are both solutions of ( 5 ) , then by Theorem 2 , u1 ( x ,t ) ? u2 ( x , t ) and u2 ( x ,t ) ? u1 ( x , t ) for all ( x ,t ) E DT , whence u1 ( x , t ) = u2 ( x ,t ) for all ( x ,t ) E DT . We shall now give a detailed proof of Sattinger ' s existence theorem for parabolic problems [ 32 , Theorem 2 . 3 . 2 ] to illustrate the method of monotone iteration , and also for the sake of completeness , s ince the proof is not given in detail by Sattinger , who proves in detail the corresponding theorem for elliptic problems . We require first a number of lemmas . LEMMA 1 : If u0 satisfies the boundary condition, i . e . if BlinuO = 0 for t = 0 and a l l X E oV , then for any g E Hn(DT ) , the prob lem ou Lu - ot = g ( x ,t ) for ( x , t ) E DT Blinu = 0 for ( x ,t ) E ST u( x , O ) = u0 ( x) for x E V 2+a -has a unique so lution u E H ( DT ) with llu ll?2+a) s: c1C I Ig ll? a) + l u0 1?2+a) ) T T where c1 does not depend en g or u0 ? Proof: This is a special case of Theorem 5 . 3 on p . 320 of Ladyzenskaj a , Solonnikov and Ural ' ceva(2 1] . LEMMA 2 : Suppose q > 1 . :or any h E Lq( DT ) , the prob lem ou Lu - d-t = h ( x ,t ) for ( x ,t ) E DT B ; _ u = 0 for ( x ,t ) E ST u( x , O ) = O for x E V has a unique (not necessarily classical) solution u E w2 ,q( DT ) with ' . ( L ,q ) 11 1 1 J ,t: I IJT s; c2 h q , DT where c2 does not deper.? en h . Proof: This is a speci?? case of a theorem analogous to Theorem 9 . 1 on p . 341 of Ladyzenskaj a , ? o lonnikov and Ural ' ceva[21] , but with a different boundary cor.ci? ion ; see p . 35 1 of the same reference . 3 1 Definition : We say that V satisfies the cone condition if there e xists a fixed finite cone K such that , no matter at what point of V its vertex is p laced , the cone can be swung so that all of it is contained in V. LEMMA 3 : If V satisfies the cone condition, if u E Ha(DT ) and also h n+2 d n+2 u E W 2 , q ( DT ) for some q such t at q > -2- an o < a < 2 - q' then l !u l l?a) s: c3 llu ll? 2 ,q ) T T where c3 does not depend on u . ( . ) Proof: If we write ((u)) J , q = I: I lD? Dsu ll D for j a non- . DT 2r+ I s I = j x q ' T negative integer , and also write r (nr 04-2r- l s I< 2 t sup j u( x , t ) j ( x , t ) EDT for "- > 0 32 then by the second part of Lemma 3 . 3 on p . 80 of Ladyzenskaj a , Solonn"ikov and Ural ' ceva(21) we have that for any u E w2 ,q( DT ) : (u} 0.) ? c O.) ((u)) ( 2 ,q) + c 0 . ) l !u l l DT 4 DT S . q ,DT ? ? ? ? ? ? ? ? ? ? ? ? ( 6 ) i f 0 ? X < 2 - n?2 , where c?X) and c?X ) depend on n , q , T , X and the dimensions of the cone K , but not on u . Now l !u l l?a) = (u} ?a) + sup _ lu( x , t ) I T T ( x ,t ) EDT ? c(a.) ((u)) ( 2 , q) + c (a) llu l l + (u) ( O ) by ( 6 ) with X=a 4 DT S q , DT DT ? c?a) ((u))?2 ,q) + c?a) !lu l l D + c? O ) ((u))?2 ,q) T q , T T + c( O ) !lu l l by (6 ) with X=O S q , DT { ( a) ( O ) (a) where c3 = max c4 + c4 , cS ( 0 ) + cs } does not depend on u . lemma follows since = ? ((u)) ( j ,q ) j = O 0T Proof: l lfg ll?a) = T = !lu l l + ((u)) ( l ,q) + ((u)) ( 2 ,q ) . q , DT DT DT sup l f( x ,t ) g( x ,t ) I ( x , t ) EDT The + sup (x , t ) , (y ,t )EDT x"#y . lf( x , t )g( x , t ) - f(y,t )g(y ,t ) I I X - y 1(1 ? sup l f( x , t ) I ( x , t ) EDT + + sup ( x , t ) , (y , t ) EDT xt-y sup (X' t ) ' (X ' T) EDT tf. T + lfC x ,t )g(x ,t ) - f( x ,T)g( x ,T) I sup - I la/2 ( x ,t ) , ( x , T)EDT t - T t #T sup j g( x ,t ) I ( x ,t )EDT lfC x,t )g( x,t ) -f (x,t )g(y , t ) +f(x ,t )g(y , t ) -f(y, t )g(y, t ) j l x - Y la jf( x ,t )g( x ,t ) -f( x , t )g( x , T) +f( x ,t )g( x , T) -f( x , T )g( x , T) I l t - T la/2 ? sup l f( x , t ) I sup l g ( x , t ) I + ( x ,t )EDT ( x , t )EDT sup ( X ' t ) , ( y 't ) EDT x?y lg( x ,t ) - g(y,t ) l l x -. Y la + sup lg ( x ,t ) I sup ( x ,t ) EDT ( x ,t ) , (y , t )EDT xt-y + + sup ( X , t ) , ( X , T ) EDT t?T lf( x ,t ) - f(y,t ) l lx y ja sup lg( x ,t ) - g( x , T ) I j t - T la/ 2 ( X , t ) , ( X , T) EDT ti-T jf( x ,t ) - f( x , T) I l t - T la/ 2 Definition : Given two functions f :DT x [a ,b] - R and u :DT - [a ,b] , we define the function f[u] :DT - R by f[u] ( x ,t ) = f( x ,t ,u( x ,t ) ) for all ( x , t ) E DT . LEMMA 5 : ( a ) If the function f( x , t ,u) is uniformly Lipschitz in ( x , t ) and in u for a s u s b and ( x , t ) E DT ' and if the function u( x ,t ) is such that u E Ha(DT ) and a s u( x , t ) s b for a l l ( x , t ) E DT ' then f[u] E Ha(DT ) . ( b ) If, in addition to the hypotheses of ( a ) , we have that u( x , t ) = u ( x ,t ) = A ( x , t ) + TB ( x ,t ) is a linear function of the T parameter T, where 0 s T s 1 and A ,B E Ha(DT ) , then for all T E [0 , 1 ] we have l lf[uT] 11?? ) ? M1 + M2 ( I lA 1 1??) + I lB 11??) ) where M1 and M2 are independent of T and u ? T ( c ) If the first partial derivative f of f is uniformly u Lipschitz in ( x ,t ) and in u for a ? u ? b and ( x , t ) E DT ' and if u ,v E Ha(?) and a ? u( x , t ) ,v( x ,t ) s b for al l ( x ,t ) E DT , then l lf[u] - f[v) ll?a) s ( K1 + K2 1 lu l l?a\ K3 l lv l l?a ) ) l lu-v l l?a ) T T T T where K1 , K2 , K3 are independent of u and v . Proof: ( a ) l lf[uJ I I ( a) = sup l f[u] ( x ,t ) I DT ( x ,t ) EDT 33 + sup jf[u]( x , t ) -f[u](y, t ) I + ( x , t ) , ( y , t ) EDT I x - y I a sup ( x , t ) , ( x , T ) EDT lf[u]( x ,t ) -f[u]( x , T ) I i t - T la/ 2 x#y t i- ?\ ::s: sup tf( x ,t ,u( x ,t ) ) I ( x , t ) EDT 3 4 + sup lf( x ,t ,u( x ,t ) ) -f( y ,t ,u( x ,t ) ) l+ lf(y ,t ,u( x ,t ) ) ?f(y,t ,u(y,t ) ) j ( x , t ) , (y ,t )EDT l x - y ja + x#y up jf( x,t ,u( x ,t ) ) -f( x , T ,U( x ,t ) ) j+ lf( x,T ,U( X,t ) ) -f( x , T ,U( X , T) ) j s a/2 ? ( x ,t ) , ( x , T)EDT l t - T l t #T Because of the uniform Lipschitz conditions sat isfied by f , we have that there exist const ants K1 and K2 such that : llf[uJ II?a) ::s: sup _ l f( x ,t , u( x , t ) ) I T ( x , t )EDT + + sup ( x , t ) , ( y , t )EDT x#y sup ( x , t ) , ( x , T)EDT tiT K1 1 < x ,t ) - (y ,t ) I + K2 1 uC x , t ) - u(y , t ) I l x - Y la K1 1 ( X , t ) - ( x , T ) I + K2 1 uC x ,t ) - u( x , T ) I ::s: sup I f( x , t , u( x , t ) ) I < x , t )ED.,.. I 1 1 -a + sup_ K1 x - y x ,yEV J. x#y a 1- - + sup K1 1 t - T I 2 O::s:t , T? t#T S ince we are assuming throughout that 0 < a < 1 , it follows that llf[u] ???a) is finite if llu ll?a) is finite , which proves ( a ) . T T (b ) This follows from the above argument , since ( i ) sup _ l fC x ,t , uT( x ,t ) ) l ::s: sup_ l f( x , t ,u) j , independent of T ( x , t ) EDT ( x , t )EDT and u . T ( ii ) llu u and ? for ( 5 ) , with Hx , t ) :s: cp( x , t ) for all ( x ,t ) E DT , and ? ,q> E Ha(DT ) ; ( v ) f satisfies a uniform Lipschitz condition in u on any finite u? interval, for ( x ,t ) E DT , and in ( x ,t ) on DT , for inf ? ( x , t ) :s: u s ( x , t ) EDT sup cp( x , t ) ; (X , t )EDT ( vi ) the partial derivative f is uniformly Lipschitz in ( x ,t ) and in u u for inf vc x ,t ) :s: u :s: ( x , t )EDT sup q>( x ,t ) and ( x , t ) E DT . ( x ,t ) ED".r Then there exists a unique so lution u E H 2+CI(DT ) of ( 5 ) such that for all ( x ,t ) E DT , V ( x ,t ) :s: u( x , t ) :s: cp(x ,t ) . Proof: By hypothesis ( vi ) , fu is bounded for ( x ,t ) E DT and inf V ( x , t ) :s: u :s: sup cp( x ,t ) . Fix 0 such that f +c( x ,t ) +O > 0 ( x ,t )EDT ( x , t )EDT u for all ( x , t ) E DT and inf v c x ,t ) :s: u :s: sup q>( x ,t ) . ( x ,t )EDT ( x ,t )EDT 35 36 For any u E Ha(DT ) such that inf tjr ( x , t ) ? u( x ,t ) ? sup cp( x ,t ) ( x , t ) EDT ( x , t ) EDT for all ( x , t ) ( L - c( x , t ) - e ? , we define Tu by saying that V = Tu if and only if : ov O)v - ot = - {f( x ,t ,u ) + c ( x ,t )u + nu} for ( x ,t ) E DT Blinv = 0 for ( x , t ) E ST v( x , O ) = uo ( x) for X e V. Using hypothesis ( v) , it follows by Lemmas 4 , S ( a ) and 1 that Tu is uniquely defined for each u as specified above , and Tu E H2+a( DT ) . ( a ) We show first that T is monotone , in the sense that if u( x ,t ) ? v( x , t ) for all ( x ,t ) E DT ' and Tu and ( Tu ) ( x , t ) ? ( Tv ) ( x ,t ) for all ( x ,t ) E DT . Tv exist , then Suppose then that inf tjr( x , t ) ? u( x , t ) ( x , t )EDT ? v( x ,t ) ? sup ?( x ,t ) ( x ,t )EDT for all ( x , t ) E DT , and u ,v E Ha(?) . Then : ( L c( x ,t ) O)Tu 0 if< Tu) - - (f( x ,t ,u ) + c( x , t )u + Ou} for ( X , t ) E DT ; ( L - c ( x ,t ) O)Tv - ? Tv ) = - {f( x ,t ,v) + c ( x ,t )v + Ov} for ( x , t ) E DT ; B 1 . ( Tu ) = B1 . ( Tv ) = 0 for ( x , t ) E ST ; 1n 1n ( Tu ) ( x , O ) = ( Tv) ( x , O ) = uo ( x) for X E V. Put w = Tv - Tu . Then : Ow ( L - c( x ,t ) - O)w - ot = - [f( x ,t ,v ) + c ( x , t ) v + Ov - {f( x ,t ,u) + c ( x , t )u + nu} ] ? 0 for ( x , t ) E DT ' s ince , by the choice o f 0 , f( x ,t ,u) + c ( x ,t ) u + nu i s an increasing function o f u for ( x ,t ) E DT and inf tjf ( x , t ) ? u ? ( x ,t )EDT Also , sup c:p( x ,t ) . ( x , t )EDT B1 . w = 0 1n w( x , O ) = 0 for x E V. It follows by Theorem 2 , taking u1 ( x ,t ) ? 0 , u2 C x ,t ) = w( x ,t ) in the notation of that theorem , that w ( x , t ) ? 0 for all ( x , t ) E DT . i . e . ( Tv ) ( x ,t ) ? ( Tu) ( x ,t ) for all ( x ,t ) E DT ? as required . ( b ) Now put u1 = T?. We prove first that u1 ( x ,t ) ? cp(x ,t ) for all ( x ,t ) e DT . We have : ou1 ( L - c( x , t ) - O)u1 - at = - {f( x ,t ,?) + c ( x ,t )q> + Oq> } for ( x ,t ) E DT ; B1i?u1 = 0 for ( x , t ) E - ST ; ul ( x , O ) = uo ( x) for X E V. ( L c ( x , t ) O) ( u1 - cp) - ft< u1 - cp) oul ? = ( L - c ( x ,t ) O) u1 - ? - { C L - c ( x , t ) - O)cp - ) - - { f( x ,t ,cp) + c ( x ,t )cp + O cp } - {Lcp - t?J + c ( x ,t )cp + O cp ? 0 Also : {Lcp + f( x ,t ,cp) - trJ for ( x ,t ) E DT since cp is an upper solution for ( 5 ). -B1 . cp ? 0 l.n ul ( x , O ) - cp(x , O ) = uo ( x ) - cp( x , O ) ? 0 for X E V. Applying Theorem 2 , we obtain u1 ( x , t ) - cp( x ,t ) ? 0 for all ( x , t ) E DT , as required . S imilarly , i f we put v1 = T ? , then v1 ( x ,t ) ? $ ( x ,t ) for all ( x , t ) E DT . Furthermore , since w c x , t ) ? cp( x ,t ) for all ( x ,t ) E DT ' it follows by the monotone property of T that T $ s Tcp on DT , i . e . v1 ( x ,t ) ? u1 ( x ,t ) for all ( x ,t ) E DT . So we have : $ s v1 s u1 ? cp on DT . 37 2 +a - a -Since u1 E H ( DT ) c H ( DT ) ' we may de fine u2 = Tu1 . S ince u1 s cp on DT ' we have Tu1 ? Tcp on DT ' i . e . u2 ? u1 on DT . S imilarly , if we define v2 = Tv1 , then v2 ? v1 on DT . S ince v1 s u1 on DT , it follows also that Tv1 ? Tu1 on DT ' i . e . v2 ? u2 on DT . So we have : W ? v1 ? v2 ? u2 ? u1 ? cp on DT . Continuing thus , we obtain two sequences {u } , {v ) , with n n 2+a -un ,vn E H ( DT ) for each n , and such that W s v1 ? v2 ? ? . . . ? u2 ? u1 s cp on . DT . ( c ) S ince the sequences {u ) and {v ) are monotone and n n converge pointwise . In particular , {u ) does so . Let n bounded , u( x , t ) = lim u ( x , t ) for ( x , t ) E DT . S ince , by hypothesis ( v ) , n n-- is continuous in u over the relevant u-interval , the sequence b oth f ( x ,t ,u ) {f( x ,t ,u ) + c ( x ,t )u + Ou ) converges pointwise to f( x ,t , u ) +c ( x , t ) u+ oU, n n n for each ( x ,t ) E DT . Since $ s un ,u ? cp on DT , it follows by Lebesgue ' s dominated convergence theorem that , for any q > 1 , i . e . - } 1/q j f( x ,t , u ) +c ( x ,t ) u +Ou - {f( x , t , u ) +c ( x , t ) u+oU) Iq dx dt - o n n n as n - CD, 3 8 Thus the sequence {f[u ) + cu + Ou ) converges in the Lq( DT ) norm , n n n this sequence is a Cauchy sequence in Lq( DT ) . Hence , i f we write h ( x ,t ) = f( x , t , u ( x , t ) ) + c ( x ,t ) u ( x ,t ) + Ou ( x ,t ) m ,n m m m - {f( x , t , u ( x ,t ) ) + c ( x , t ) u ( x ,t ) + Ou ( x ,t ) ) for all n n n (X, t ) E DT then given any e > 0 , there exists a positive integer Ne such that m ,n : --2- an n+2 0 < a < 2 - --- ( where n here is the dimension of V ) , it then follows by q Lemma 3 that llw l l ( a ) m , n DT :s: c3c2 ljh 11 D m ,n q , T where c3 does not depend on h m ,n m ,n : - 2+a. -It follows that u E H ( DT ) . Also , Tu sat isfies 0 -( L - c ( x , t ) - O)Tu - ?Tu) = - {f( x ,t ,u ) + c ( x , t ) u + oU} for all ( x , t ) E DT . for ( x , t ) E DT Blin ( Tu ) = o for ( x , t ) E sT (Tu) ( x , O ) = u0 ( x ) for x E V. 39 Since Tu = u, we have at once that u is a solution of ( 5 ) , which proves the theorem ; similarly one can show that lim vn = v E H 2+ 0 for all f:u1 f( x ,u1 ) .. f ( x ,u2 ) Thus + > Lu2 + for all X E V . Also , when X .. ,. i . e . X = ?1 , then B l . u1 = u1 = 7 , B linu2 = u2 = 8 . Thus ?n - x . E ov , Blinu1 < Blinu2 for all X E oV. I f theorems analogous to Theorems 1 and 2 held , we would expect at least that u1 ( x ) ? u2 ( x) for all x E V. However , when x = 0 , u1 ( = 3 ) > u2 ( =0 ) , so theorems analogous to Theorems 1 and 2 do not hold in this case . Hence , if there exist upper and lower solutions ? and ? for ( 7 ) , then in contrast to the case of the parabolic problem ( 5 ) , we cannot assert that every solution u of ( 7 ) must lie between ? and ? on V. However , Sattinger ' s method of monotone iteration is still applicable , and shows the existence of at least one solut ion u of ( 7 ) lying between ? and ? on V. In this connection , though , it is interesting to note that it has been shown ( e . g . by Parter[27) ) that in cert ain special cases there exist solutions of ( 7 ) whi ch cannot be obtained by monotone iteration procedures . Such solutions are unstable in the sense of Keller and Cohen[ 19 ) ( see Ch . 4 ) . A full discussion of these points is given by Amann[3) . We give now Sattinger ' s existence theorem[32 , Theorem 2 . 3 . 1 ) for elliptic problems . The proof is s imilar to that of Theorem 3 and so is not given in detail , but an outline of the procedure is given , as 41 this will be re ferred to in the sequel . THEOREM 4 : For the boundary value prob lem ( 7 ) , we suppose, in addition to the assumptions already made, that : ( i ) there exist upper and lower so lutions ? and ? for ( 7 ) , with ?( x) s <$( x) for all X e V and ? ,c$ e c2 + 0 such that f + c ( x ) + 0 > o for inf $ C x ) s: u s: sup c$( x ) and u xEV xEV x E V . Then define Tu by say ing that v = Tu if and only if : ( L - c (x ) - O) v = - {f ( x ,u) + c( x ) u + Ou} for X E V ; ... Bl . V = 0 for X E oV . l.n As in the proof of Theorem 3 , define sequences {u } and {v J so that n n $ s: v1 s: v2 s: . . . . s: u2 s: u1 s c$ on v, and un ,vr. E c 2 +a(v) for all ... n ? 1 . It may then be shown that lim u = u and lim v = v are both n n n....., solutions of ( 7 ) , thus proving the theorem ; note that in contrast to ... ... Theorem 3 we do not necessarily have u = v on V in this case . Asymptotic Behaviour of Solutions : In order to apply the two preceding existence theorems to the problem of determining the connection between the boundedness over all time of the solution of ( 5 ) and the existence of positive solutions of ( 7 ) , we require next two theorems concerning the asymptotic behaviour of solutions of linear systems . These are analogous to theorems proved by Friedman [14 , Ch . 6 ] , but with different boundary conditions . THEOREM 5 : Suppose that the coefficients a . . , b . , c in Lu are l.J l. uniformly continuous and bounded in D, the coefficients d0 and d1 in Blin u are uniformly continuous and bounded in S , and for some 1-11 > 0 , d0 ( x , t ) ? 1-lt for al l ( x ,t ) E S . Suppose further that u( x , t ) satisfies the differential equation ou Lu - dt = g( x.1 t ) for ( x , t ) E D where g is continuous on D, together with the boundary condition B 1 . u = h ( x , t ) for ( x , t ) E S ?n where h is continuous on S . If lim g( x , t ) = 0 , lim h (x ,t ) = 0 and t-- 42 lim c ( x ,t ) ? 0 uniformly on V, oV and V respective ly, then lim u ( x . t ) = 0 t-- t_, uniformly on V . ' R AX1 Proof: Consider the function ? x) = e? - e ( x1 being the first component of x) , where R is any positive number satisfying R ? 2 x1 for all x E V, and A is a posit ive constant to be determined later . Then cp( x ) satisfies : 2 AX1 L? = - a11 C x , t ) X e AX1 AR AX1 - b 1 ( x , t ) Ae + c ( x , t ) ( e - e ) . Independent ly of the value of R , we choose A sufficiently large so that AX1 AR AX1 Lcp(x) < - 2e + c ( x ,t ) ( e - e ) , for ( x ,t ) E D. Since lim c ( x ,t ) s o , it follows that for some a sufficiently large , Letting 9 XR AX1 AX1 c ( x ,t ) ( e - e ) < e for t > cr , and all X E V. AX1 = inf e xEV we then have : Lcp( x) < -e for t > a and all X E V . . ? ? ? ? ? ? . ? ( 8 ) Also , B1. cp( x) ?n > ? for all ( x ,t ) E D, for some pos itive ? , if R is sufficiently large . Choose R so that this is the case . Now let 9 0 = in? cp( x ) , e 1 = suE_ cp( x ) . Cons id er the function xEV xEV W ( x ,t ) = e ?e x) + e cp( x) + A cp e ( x ) e-s ( t-cr) for t > 0 ? cr 1-12 0 where e , s are positive constants and A = sup l uC x ,cr ) l . xEV By ( 8 ) : LW( x ,t ) < - e - ? -1-"2 eA -s < t -o) 9 e , 0 for all x E V . ow -Also , ot - - e;Acp( x) -s( t-o) e e > - sAe 1 -s< t -cr) for all x E V . I f we -- e e o t ake s = j-, then 1 Clearly : 0 ?A e -? ( t-cr) , and so : 0 L$ ( x , t ) - ov < -e for t > a and x E V ot . . . . . . ( 9 ) $ ( x , cr ) > A for x E V . . . . . . . . . ? ? ? ? . . . . ( 1 0 ) Also : e? A? -?( t-a) B1 . ljl ( x , t ) > -- + e + -- e m e e 0 > e for all X E oV , t > a . . . . . . . . . . . . . ( 1 1 ) By hypothesis , for any e > 0 , there exists a( e ) such that 4 3 j g ( x ,t ) j < e and j h ( x ,t ) j < e for t > a ; we may assume a( e ) ? O? By two applications of Theorem 1 , using ( 9 ) , ( 10 ) and ( 11 ) , we have u( x , t ) < ljl ( x , t ) and -u( x ,t ) < ljl( x ,t ) for ( x ,t ) E V and t ? a . j u( x , t ) j < ?( x , t ) for ( x ,t ) E V and t ? a. For ( x ,t ) E V and t ? a , we have : j u( x , t ) l ? A1e + A2e -? ( t-a) , A1 and A2 positive constants , A1 depending only on ? This completes the proof of Theorem 5 . For the next theorem , we need the following standard result ( see , for example , the book by Ladyzenskaj a and Ural ' ceva[2 2 , pp . 1 37 , 1 3 8) ) . LEMMA : Suppose that the operators L and B1 . are as defined in Ch . 3, ?n with the additional assumptions that : ( i ) a . . = a . . E Ca( V) , b . E Ca(V) , c E Ca(V) and c ( x ) ? 0 for all X E V; ?J J ? ? ( ii ) d0 and d1 are of class c 1+a( oV ) , and there exists ?1 > o such that .. do ( x) ? ?1 for all X E oV . Then for any g E Ca(V) , the boundary value problem Lv = g( x ) for x E V .. Bl . V = 0 for X E oV ?n has a unique s olution v E c2+a(V) . Thus, certainly, v and al l its first and second partial derivatives are bounded in V. THEORE M 6 : Suppose that the operators L and Blin satisfy the hypotheses of Theorem 5, and the operators L and B 1 . satisfy the hypotheses of ?n the Lemma. Suppose also that c( x ,t ) - c( x ) , g( x ,t ) - g ( x ) , a . . ( x ,t ) - a . . ( x) , b . ( x , t ) - b . ( x) , ?J ?J ? ? d0 ( x ,t ) - d 0 ( x) and d1 ( x ,t ) - d 1 ( x) as t - CD, uniformly in V; here g is continuous on D and g E Ca(Y) . If u( x , t ) is a so lution of the boundary value prob lem ou Lu - ot = g( X, t ) for (X, t ) E D B1 . u = 0 for ( x , t ) E S ?n } and v( x ) is the unique so lution of the boundary value prob lem Lv = g ( x) for x E V '"' ' Bl . V = 0 ?n for X E oV then u( x , t ) - v( x) as t - ?, uniformly in V. } Proof: Put w ( x , t ) = u( x ,t ) - v( x) , for ( x , t ) E D. Then : OW ou C L-L ) v - '"' Lw - -at = Lu - dt - LV = g( X, t ) - g( x) C L-L )v for ( x, t ) E '"' B . w = - B V = ( Bl . - B1 . ) v for ( x, t ) E l?n l in ?n ?n D . s . By virtue of the hypotheses of Theorem 6 and the boundedness of v and its first and second part ial derivatives on V ( see the Lemma ) , we may apply Theorem 5 and conc?ude that lim w( x , t ) = 0 , uniformly on V , t-- which proves Theorem 6 . We are now in a pos i? ion to make a first statement about the relationship between solut ions of the parabolic problem ( 5 ) and the elliptic problem ( 7 ) . We shall assume for the purposes of this discussion that the operators L , ? , B1 . and B1 . satisfy the ?n ?n hypotheses of Theorems 5 and 6 in addition to the conditions imposed when describing problems ( 5 ) and ( 7 ) . We assume also that f( x ,t ,u ) - f( x ,u) as t - ?, uniformly in x for x E V and in u on any bounded u-interval . Suppose that we have upper and lower solutions cp( x , t ) and 1jl ( x ,t ) for ( 5 ) , for all T > 0 , and upper and lower solutions qxx ) and * ( x) 44 for ( 7 ) , such that cp( x ,t ) - qx x) and 1jl ( x ,t ) - *( x) as t - ?, uniformly for x E V . Suppose also that the conditions for the monotone iteration theorems , Theorems 3 and 4, are s atisfied . It is clear from the . constructions used in these theorems ( applied for arbitrarily large T ) that we can , using induction , apply Theorem 6 to the function pairs u ( x , t ) , u ( x ) and v ( x ,t ) , v ( x ) , and deduce that for all positive n n n n integers n : u ( x , t ) - u ( x ) } n n as t - ?, uniformly for x E V. v ( x ,t ) - v ( x ) n n We now suppose further that u( x) is the only solution of ( 7 ) lying .\ between ?( x) _ and 4( x) . This would be the case , for example , if the lower solution ? ( x ) were positive and the hypotheses for result ( vi i i ) 4 5 of Keller and Cohen ( s ee Ch . 4 ) were satisfied (this means , in particular , that f ( x , u ) would be concave in u ) . S ince u( x) is the only solut ion of ( 7 ) between ? ( x ) and 4( x) , and the sequences {u ( x) } and {v ( x) } both converge uniformly to solut ions n n of ( 7 ) lying between ? (x ) and qx x) by Theorem 4 , it follows that u ( x ) - u(x ) and V ( x) - u( x ) as n - ., uniformly for X E V, and we n n know also that V ( x) ? u( x) ? u ( x) for all X E V and all positive n n integers n . Thus , given any e > 0 , there exists a positive integer N ( e ) , independent of X E V, such that l un( x) - u( x ) l < % and l vn ( x) - u( x ) I < ? whenever n ? N ( e ) . Further , there exists T( e , N ( e ) ) independent of X E V such that 1?1 ( x ,t ) - ? ( x ) I < ? and l vN( x ,t ) - vN ( x ) l < ? whenever t > T( e , N ( e ) ) . l?< x ,t ) u( x ) I < e and l vN( x ,t ) - u( x) I < e whenever t > -r ( e , N ( e ) ) . u( x ) - e < vN ( x , t ) ? u( x , t ) ? ?( x ,t ) < u( x ) + e whenever t > T( e , N ( e ) ) where u is the solution of ( 5 ) obtained in the proof of Theorem 3 ; note that by applying Theorem 3 for arbitrarily large T , we can show that u( x ,t ) exists for all t ? o . u( x , t ) - u ( x) as t - ?, uniformly for X E V. Thus , under the given conditions , the existence of exactly one solution to the steady-state problem ( 7 ) lying between ? ( x ) and ? x) imp lies that , for any init ial value u0 ( x) lying between ?( x , O ) and ?( x , O ) , the unique solution u( x , t ) of ( 5 ) ( for arbitrarily large T ) will tend to the steady-state solution u( x ) of ( 7 ) as t - ?, uniformly for X E V . Let us now consider the special case where the coefficients and the function f in the parabolic problem are independent of t , so that we are concerned with the problem ... ou ... Lu - ? + f( x ,u ) = 0 for ( x , t ) E D ... Blinu = 0 for ( x ,t ) E S u( x ,O ) = u0 ( x) for x E V ? ? ? ? ? ? ? ? ( 1 2 ) As a particular case of the preceding discussion , we obtain the following theorem . THEOREM 7 : Suppose that ( i ) the operators L and B1 . satisfy the hypotheses of Theo?m 6 in 1n addi tion to the conditions imposed when describing problem ( 7 ) ; ( ii ) the initial value u0 ( x) in ( 12 ) is non-negative for all x E V, and there exis ts a non-negative solution ? of ( 7 ) such that ?( x ) ? uo ( x) ? 0 for all X E V; ( ii i ) there is no so lution u of ( 7 ) , different from ?, such that 0 s u ( x) s ?( x ) for aLL x E V; ( iv ) f( x , O ) ? 0 for all x E V ; ( v ) L , Blin ' f , u0 and V satisfy the hypotheses of the monotone iteration theorems, Theorems J and 4 . Then there exists a unique solution u of ( 12 ) such that u( x , t ) - ?( x ) as t - "" ? uniformly for x E V. Proof: Bearing in mind our assumptions that ?( x ) ? u0 ( x ) ? 0 for all x E V and f ( x , O ) ? 0 for all x E V , it is obvious that ?( x ) is an upper solution and 0 a lower solution for both ( 7 ) and ( 12 ) . The theorem then follows from the preceding discussion . Note : A crucial hypothes is in Theorem 7 is the existence of a minimal non-negative solution ? of ( 7 ) ; as dis cussed in Ch . 4 , such a minimal non-negative solution does exist under a wide range of conditions . Let us now try to relate the preceding material to the thermal explosion problem with which we began . Cons ider the time-dependent problem "' ou ;\f( x ,u ) Lu - - + = 0 for ( x , t ) E D ot "' 46 B1 . u = 0 for ( x , t ) E s . ? ? ? . ? ? ? ( 1 3 ) 1n u( x , O ) = 0 for x E V and its related steady-state problem Lu + .. :\f( x ,u) = 0 for x E V } E ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ( 14 ) B1 ? u = 0 for x uV A A 1n where L , B1 . , f and V are assumed t o satisfy the hypotheses of 1n Theorem 7 . These problems differ from problems ( 12 ) and ( 7 ) only in the re-introduction of the parameter A and the fact that u0 = 0 , and represent a modest generalisation of the original heat-generation problem . If we adopt the orthodox approach and equate the existence of pos itive steady-state solutions with the absence of a thermal exp losion , then it is natural to define the critical parameter A* for the pair of 47 problems ( 1 3 ) , ( 14 ) to be the least upper bound of the set of positive values of ? for which positive solutions of ( 14 ) exist ( if the s et is not bounded above , we ? can take ? :': to be infinite ) ? This conforms with the notation used by Keller and Cohen[19] and is also the definition used by Boddington , Gray and Harvey[4] . As discus sed in Ch . 4 , if we assume that L and B1 . are of the rather special form described in that chapter , it ?n is known that if f is continuous , positive , and strictly increasing in u , for x E V and u ? 0 ( these being hypotheses H0 , H1 and H2 o f Keller and Cohen [ 19 ] ) , then pos itive steady-state solutions , if they exist at all , occur for all ? in( 0 ,?:':) and for no ? greater than ?:': . Thus , by the orthodox criterion , "explos ion" occurs if A > ?:': but not if 0 < A < Vi . Indeed , under the same conditions on f , it is known , as has been remarked , that a minimal positive solution of ( 14 ) exists if 0 < A < A:':. It then follows from Theorem 7 that if 0 < A < V: , ( 1 3 ) has a unique solution u( x , t ) which is bounded for t ? 0 and tends to the minimal positive solution of ( 14 ) as t - ?. What of the reverse implication? Suppose we know that ( 1 3 ) has a solution u( x , t ) which is bounded for t ? 0 ; does it tend to a solution of ( 14 ) as t - m? To deal with this question , we require two theorems analogous to Theorems 2 . 5 . 1 and 2 . 6 . 1 of Sattinger[32] , but with different boundary conditions . The proofs are included for the s ake of completeness ; they are analogous to those used by Sattinger . .. THEOREM 8 : Suppose that there exists a lower solution ? ( x ) for the e l liptic prob lem ( 7 ) , and a so lution u( x ,t ) of the particular parabolic prob lem ( 12 ) with u0 = ? . Suppose also that the coefficient d0 ( x) in Bl . is strictly positive for all X E oV , and the function f( x , u ) ?n satisfies a uniform Lipschitz condition in u on any finite u-interval, and has partial derivative f continuous for all x E V and all real u . Th ou u en et ? 0 for all ( x , t ) E D. u( x ,t+h ) - u( x , t ) Proof : For any ( x ,t ) E D, put wh( x ,t ) = h where h > 0 . Then if ( x ,t ) E D, we have : Now .. .. f( x ,u( x ,t+h ) ) - f( x ,u( x ,t ) ) owh -Lwh + h - Tt"" - 0 ? .. f( x ,a ) J b f( X ,b ) = f ( x ,u ) du u a J 1 .. = f (X, Tb+ ( 1-T )a ) (b-a )dT . u 0 Let b = u ( x ,t+h ) and a = u( x ,t ) , where ( x , t ) E D . Then : f( x ,u( x ,t+h ) ) - f( x ,u( x ,t ) ) = ? < t h ) ( ) h -:. x , , wh x, t where ?( x , t ,h ) satisfies : Also : 1 = J f ( x , 'TU( x , t+h ) + ( 1--r )u( x ,t ) ) dt . 0 u .. awh Lwh + swh - Tt = 0 for ( X , t ) E D . B linwh = ?Blinu( x ,t+h ) - Blinu( x ,t ) } = 0 for ( x , t ) E s . wh ( x , O ) = u( x ,h ) - u( x , O ) h u( x ,h ) - W ( x) = :::!:: 0 for x e h V , .... since ? is also a lower solution for ( 12 ) , and so u( x ,h ) ::2:: ${ x) for all x E V and h > 0 by Theorem 2 . Applying Theorem 2 again , we see that wh ( x , t ) ::2:: 0 48 for all ( x , t ) E D , and all h > 0 . It follows that ?u = lim wh ( x , t ) :::!:: 0 ut h-G+ for all ( x ,t ) E D. Note : An analogous theorem holds if we have an upper solution ? for ( 7 ) and a solution u of ( 12 ) with u0 = ?; in that case ** ? 0 for all C x ,t ) E D. THEOREM 9 : If we make the same assumptions as for raeorem 8, and suppose in addition that for some constant K , u( x , t ) s K for all ( x ,t ) E D, then lim u ( x ,t ) = u( x ) exists for all X e V , and u is equal a. e . to a t....., classica l so lution of the e l liptic prob lem ( 7 ) . Proof : We use the inner product notation to denote the usual L2 ( V ) inner product for real functions , i . e . ( f , g ) = J fg dx . Also , given V two functions f :V x [a ,b] - R and u :D - [a ,b] , we define the function f[u] :D - R by f[?] ( x , t ) = f( x , u( x ,t ) ) for all ( x ,t ) E D. Now consider the operator L1 = L - c ( x ) , understood to have as domain the set of all u E s 2 ,2 ( V ) satisfying B linu = 0 . Lions and Magenes [2 4 , Vol . ! , pp . 114- 12 1] describe the construction of the adj oint operator L? and also the adj oint domain consisting of all u E s2 , 2 ( V ) satisfying an appropriate adj oint boundary condition Cu = 0 . As shown by Lions and Magenes [24 , Vol . I , Ch . 2 , Theorem 2 . 1 , Corollary 2 . 1 and Remark 2 . 2 on pp . 119 , 120] , if u E domain 11 and v E domain Li , then ( L1u ,v ) = ( u , L{v) . Now let ? be in the domain of Li ? We know that u( x ,t ) is in the domain of L1 as a function o f x , for each t ? 0 . Write f1 ( x ,u ) = f( x ,u) + c( x) u . s ince .. ou f ( x , u) = 0 ( x , t ) E Then Lu - dt + for D we have ( L1u , ? ) + ( f 1 [u] , 1?) ( ? ,ut ) = 0 for all t > 0 . i . e . ( u ,L1? ) + ( f1 [u] , E; ) ( ? ,ut ) = 0 for all t > 0 . So for all T > 0 . Now let T - ?. Since u( x , t ) is non-decreasing in t by Theorem 8 , and bounded above by K for all ( x ,t ) E D by hypothes is , it follows that lim u( x , t ) = u( x) exists for all X E V , and hence that t-oa:> 1 J T lim T u( x , t ) dt = T-- 0 T u( x ) for all x E V. 49 So 1 T lim T J ( u , Li? ;) dt = T._, 0 lim ( f J u( x , t ) dt , Lis ) T-- 0 by interchanging the order of integrat ion , since L* is independent 1 of t Here the interchange of the order of integration follows by Fub ini ' s Theorem , and the final step follows from the Lebesgue dominated convergence theorem and the fact that $ ( x) ? u( x ,t ) ? K for all ( x , t ) E n. Similarly , using the fact that f1 ( x ,u) is continuous in u , we obtain Als o , . 1 J T T - ll.m T ( ? ,ut ) dt = lim ( ? ,} J ut ( x ,t ) dt ) T-- 0 T-- 0 = lim ( ? , u( x ,T ) ? u( x ,O ) ) T-- = ( ? , 0 ) = 0 s ince u( x ,T ) is bounded as T - ?, for all x E V . Here again we are using the Lebesgue dominated convergence theorem . So , tak ing limits as T - ?, we obtain finally : c G , Li? ) + C f1 [GJ , s ) = o . .. Now L1 is invertible , s ince for g E L2 ( V ) , the system L1w = g , Blinw = 0 .\ has a unique solution , by a result of Agmon , Douglis and Nirenberg( 1 ] . The same applies to Li , since the order of the adj oint boundary condition is 0 or 1 by Theorem 2 . 1 ( b ) on p . 11 5 of the book by Lions and Magenes [24] , and s o the uniqueness theorem of Agmon , Douglis and Nirenberg applies . Let G1 be the inverse of L1 . Then by a result of Riesz and Nagy[30 , p . 304] , Gi is the inverse of Li ? Put w = -G1f1 [u) . Then : ( G1f1 [G) ,Li? ) ( f1 (u] ,G!Li? ) ( f1 [G] , ? ) . Hence , from above , ( u ,Li? ) = - ( f1[G] , ? ) = (w ,Li? ) . Therefore ( u-w , Li? ) = 0 for all ? in the domain of Li ? But the invertibility of L! implies that the range of L? is all of L2 (V ) . Hence ( u-w ,n) = 0 so for all ? E L2 ( V) . Thus u = w a . e . , i . e . u = -G1f1 [G) a . e . So u i s a weak solution of the elliptic problem ( 7 ) . By Theorems 2 . 2 . 1 and 2 . 2 . 2 of Sattinger[ 3 2 ] , we conclude that G is equal a . e . to a class ical solution of ( 7 ) , as required . Applying Theorems 8 and 9 to problems ( 1 3 ) and ( 14 ) , we obtain the following theorem . THEOREM 10 : Suppose that ( i ) the coefficient do ( x) in B l . is s trictly positive for al l X E oV , A 1n and Af( x , O ) ? 0 for all x E V ; ( ii ) f satisfies a uniform Lipschitz condition in u on any finite u? interval, and has partial derivative f continuous for al l x E V and u all real u ; ( ii i ) there exists a so lution u( x ,t ) of ( 13 ) such that, for some constant K , u( x ,t ) ? K for al l ( x , t ) E D. Then there exists a non-negative solution of ( 14 ) which is equal a. e . to the limit as t - m of the solution of ( 13 ) . If in addition C iv) the coefficients c in 1 and a1 in B1in satisfy c( x ) ? o for al l X E V and d1 ( x) > 0 for al l X E oV ; ( v ) Af( x ,u ) > 0 for all x E V and u ? 0 then the so lution of ( 14 ) is strictly positive on V. Proof : S ince 0 is obvious ly a lower solution for both ( 1 3 ) and ( 14 ) , we can immediately apply Theorems 8 and 9 to deduce the exis tence of a 5 1 solution ?( x) o f ( 14 ) such that lim u( x ,t ) = ?( x ) a . e . Since u( x , t ) ? 0 t-- for all ( x , t ) E D by Theorem 2 , and ? i s continuous , it follows that IJ( x) ? 0 for all x E V, thus proving the first p?art . The second part follows from a form of the maximum principle . Clearly ? is not identically zero , by hypothesis ( v) . Suppose 1J( x0 ) = 0 for some x0 E V . I f x0 E V , we obtain a contradiction by using Theorem 6 of Ch . 2 of Protter and Weinberger( 2 8) ; if x0 E oV , we obtain a contradiction by us ing Theorem 8 of the same reference . So IJ( x ) > 0 for all x E V, completing the proof . We have now shown that , under quite wide conditions , the existence of a bounded s olution of ( 1 3 ) implies the existence of a pos itive s olution of ( 1 4 ) which is the limit a . e . of the solution of ( 1 3 ) as t - ?. Conversely , we showed earlier that , again under quite wide conditions , the existence of a (minimal ) pos itive solution of ( 14 ) imp lies the existence o f a bounded solution o f ( 1 3 ) which tends t o the (minimal ) positive solution of ( 14 ) as t - ?. Suppose we adopt the alternative approach to the study of thermal exp los ions , whereby one equates the boundedness over all pos itive time of the solution of the t ime-dependent problem with the absence of a thermal explosion . This would lead us to define the crit ical parameter A* for the pair of prob lems ( 1 3 ) , ( 14 ) to be the least upper bound of the set of positive values of A for which the solution of ( 1 3 ) is bounded . The results of this chapter establish fairly general conditions under which the two approaches to the problem and the two definitions of the critical parameter are equivalent . Certainly this is so for the cases discussed in Ch . 4 and also for most forms of the original heat ? generation problem as discussed b y Boddington , Gray and Harvey[4) . I t seems reasonab le t o suggest that the two approaches are i n fact equivalent under much wider conditions than those given in this chapter . For the remainder of this thesis , we shall treat the second approach to the thermal explosion problem as the fundamental one , and concentrate on describing the behaviour of the solution of the time? dependent problem under various assumptions . The information obtained will be compared with that obtained by studying the steady-state problem. 7 CONSTRUCTION OF UPPER AND LOWER SOLUTIONS FOR THE TIME-DEPENDENT PROBLEM 5 2 In this chapter we shall examine the behaviour o f the solution of the time-dependent problem under various assumptions as to the nature of the function f( x ,t ,u ) , in particular the rate of growth of f ( x ,t ,u ) as a function of u . We shall do this by constructing upper and lower solutions for various cases , and then applying a suitable comparison theorem . In all the theorems o f this chapter , the existence of a solution of the time-dependent problem will be taken as a hypothes is . However , in many cases we will construct both an upper and a lower solution , whereupon the existence of a solution will fol low from Theorem 3 if the conditions of that theorem are satisfied . In any event , it is quite suf?icient for our purposes in this chapter to show that if a solution exists , it must behave in such-and-such a way . We shall begin by working with a specific domain V ( described m below ) for which calculat ions are relatively simple , and later deal with the problem of e xtending our theory to general domains . We shall then discuss the results obtained in this chapter and the relationship between them and the results already known for the steady-state problem . The chapter will conclude with an examination of some theorems concerning the effect of reactant consumption . Notation : We write V m n 2m . = {x : r x . ? < 1 ) where the m . ( i = 1 , 2 , . . . n ) are i= 1 ? ? arbitrarily chosen positive integers . Lu and B1 . u will be as defined ?n in Ch . 3 . In the directional derivative ou h ' h . B an w ?c appears ?n linu ' we shall take the unit vector field n to be s uch that , for each 2m . -1 i = 1 , 2 , . . . n , n . ( x ) = a. ( x ) x . ? where a . is of class c 1+a( oV ) . ? ? ? ? m n 2m . Since the outward unit normal vector field v to oV = {x : m I: x . ? = 1 ) i= 1 ? is given by v. ( x) = ? 2m . -1 ? 2m . x . ? ? for all X E oV and i = 1 , 2 , . . . n m 5 3 the vector field n will be outwardly directed and nowhere tangential to oV provided that , for each x ? oV , m m n( x ) . v( x) = n 4m . -2 ? E 2m . a. . ( x ) x . i = 1 ? ? ? J n 2 4mi-2' E 4m . x . i= 1 ? ? is positive . We assume that this is the cas e . We then have that ou n 2mi- 1 ou On = :L a. ( x ) x . ? . i= 1 ? ? xi n 2m . ? We suppose further that the quantity E a . ( x) x . is bounded below i=1 ? ? and above on oVm by pos itive numbers 9 (a.1 , . ? . 0 and X e V , f( x ,t ,O ) ? 0 ; furthermore? f satisfies a m uniform Lipschitz condition in u on any finite u-interval . ( b ) u0 ( x) ? 0 for al l X ? Vm . ( c ) The coefficient d0( x , t ) of u in Blinu is strictly positive for all t > 0 and X E oV m . Then : ( i ) For any T > o , a lower solution for ( 15 ) is given by w( x ,t ) = 0 for all x E V , 0 s t s T . m ( ii ) For any T > o , if u( x ,t ) is a solution of ( 15 ) , u( x , t ) ? 0 for a l l X e V and 0 s t s T ? m Proof : ( I ) w( x , O ) = 0 s u0 ( x) for all X ? V m ( I I ) Bl . w = 0 for t > 0 and X E oV . ?n m Ow ( II I ) Lw - dt + Af( x , t ,w) = Af( x , t ,O ) ? 0 for t > 0 and x E Vm . This proves part ( i ) of the theorem . I f u ( x , t ) is a solution of ( 15 ) , it follows by Theoren 2 that u( x , t ) ? w( x ,t ) = 0 for all x E V and m 0 s t s T , thus proving part ( i i ) . THEOREM 1 2 : Suppose that ( a ) There exis t constants A . > 0 ( i = 1 , ? ? ? n ) , B ? ? 0 ( i = 1 , ? ? ? n ) and ? ? 5 4 C ? 0 such that, j b . ( x ,t ) I s B . ( i for c!l x E V and t > 0 , 0 < a . . ( x , t ) s A . ( i = 1 , . . . n ) , m ?? ? ? ? = 1 , . . . n ) and c( x , t ) ? -C . ( b ) There exists a constant M > 0 such that, for all t > O , x E V and m u ? 0 , f( x ,t , u ) ? M . Furthermore , f satisfies a uniform Lipschitz condition in u on any finite u-interval . ( c ) There exist constants D0 > 0 and 61 > 0 such that, for aU x E oVm and t > o , o < d0 ( x , t ) s D0 and 61 s d1( x ,t ) ; we require also that these constants be such that ( d ) uo ( x) ? 0 for all X E vm Then : ( i ) Fo? any T > 0 , a lower solution for ( 1 5 ) is given by r:. 2 w( x ,t ) = AK ( A - E x . ) ( 1 i = 1 ? e-t ) for all x E V , 0 s t s T m where A and K are constants chosen so as to satisfy . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 6 ) M 0 < K < ------------------------- . . . . . . . . . . . . . . ( 1 7 ) 2 n E A . + 2 i = 1 ? n E B . + ( C+ 1 ) A i= 1 ? ( ii ) If u( x ,t ) is a so lution of ( 15 ) , th?n for any T > 0 , u( x , t ) > 0 for 0 < t s T , and if lim u( x ,T ) = u( x ) exists , then for aU X E vm , T- 26 19 AM( V + -- - '?') Do u < x ) ? ----------------'::"":""--:-- > o . n n 26 19 2 E A . + 2 I B . + ( C+ 1 ) ( V + -0--) i=1 ? i = 1 ? 0 Proof: ( I ) w( x , O ) = 0 s uo ( x) for all X E vm ' by hypothesis ( d) . n 2m . -1 ::. l u'fl ( I I ) Bli.n;.) = d0 ( x , t ) ?...? + d 1 ( x , t ) E a . ( x) x . -0 -i= l l l xi 5 5 n 2 = d0 ( x ,t ) ).K( A - r x . ) ( 1 i = 1 ? n 2m . - 1 e-t ) + d1 ( x , t ) I: Cl ? . ( x ) x . ? {).K( -2x . ) ( 1-e-t ) } . 1 ? ? ? ?= n 2m . n 2 E x . ) . 1 ? ? = 2d1 ( x , t ) I: a. ( x)x . ? ) i = 1 ? ? ? ).K( 1 - e-t ) {D0 ( A- ? ) - 2 619 } < 0 for all t > .o by ( 16 ) ? ( I II ) Lw - %r + ).f( x , t ,w ) n n for t > 0 and X E oV m = L a . . ( x ,t ) ).K ( -2 ) ( 1-e-t ) + E b . ( x ,t ) ).K( -2x . ) ( 1-e-t ) i = 1 ?? i=1 ? ? . n 2 -t n 2 -t + c( x , t ) ).K ( A- I: x . ) ( 1-e ) - ).K( A- E x . ) e + ).f( x ,t ,w) i=1 ? i = 1 ? n n ? -2).K E A . - 2).K E B . - C).KA - ).KA + ).M i= 1 J. i=1 ? n n = ). {-K( 2 E A . + 2 E B . + CA + A) + M } i = 1 ? i=1 ? > 0 for all t > 0 by ( 1 7 ) . for t > 0 and x E V m Thus part ( i ) of the theorem is proved . By Theorem 2 , it follows that i f u( x , t ) is a solution of ( 15 ) , then u( x , t ) ? w( x ,t ) for x E V and m 0 ? t ? T . I t follows that u( x , t ) > 0 for x E V , 0 < t ? T . m n 2 Furthermore , w ( x ,t ) - ).K( A - E x . ) as t - ?, i=1 J. so if lim u( x , T ) T- = u( x ) e xi sts , then for all x E V , m n 2 u( x) ? ).K( A- I: x . ) ? ).K(A-?) . i = 1 J. S ince 2 619 A may be chosen arbitrarily close to ? + ---- and K may be chosen oo arbitrarily close to _____________ M _____________ , part ( ii ) of the n n 2 theorem follows at once . E A . + 2 E B . + ( C+1 )A i = 1 J. i = 1 J. Note : The condition that f should s atisfy a uniform Lipschitz condition in u on any finite u-interval may be removed if we alter hypothes is ( d ) to read "u ( x ) > 0 for all x E V . " The only change in the proof is 0 m that Theorem 1 rather than Theorem 2 is used in proving ( i i ) . THEOREM 1 3 : Hypotheses as for Theorem 12 except that &n hypothesis ( b ) , we suppose that there exis t constants M1 > 0 , M2 > 0 such that, for al l t > 0 , X E Vm and u ? 0 , f( x , t ,u ) ? M1u + M2 . We still assume that f satisfies a uniform Lipschitz condition in u on any finite u-interval . Then : ( i ) For any T > o , a Lower so lution for ( 15 ) is given by w ( x , t ) = t ( A n 2 :E X . ) for al l X e V ' 0 ? t ? T i= 1 ? m if A is a constant chosen so as to satisfy 2 619 'f < A < W + and if n n { 2 ( :E A . + :E B . ) i= 1 ? i=1 ? A > max M ( A _ 'f) 1 n n D o + CA ( ii ) If A > max 2 ( :E A . + :E B . ) i=1 ? i = 1 ? : 2 } . . . . . . . . . . . . . ( 1 8 ) and if u( x ,t ) i s a so lution of ( 1 5 ) , then u( x ,T ) - = as T - ? . uniformly for X e V . m Proof: ( I ) and ( I I ) are similar to the proof of Theorem 12 . ( I I I ) Lw - ? + Xf( x ,t ,w) n n = :E a . . ( x ,t ) ( -2t ) + :E b . ( x ,t ) ( -2tx . ) + c ( x ,t ) t (A i=1 ? ? i = 1 ? ? n 2 t X ? ) i=1 ? n n n 2 - ( A - :E x . ) + Af( x ,t ,w ) i = 1 ? 5 6 ? -2t :E A . - 2t :E B . - CtA - A + A[M1t ( A-'i') + M2) i = 1 ? i=1 ? for t > 0 and x E V m n n 2 { i?/i + i?1 BJ - CA] + AM2 - A . > 0 for all t > 0 by ( 18 ) . Thus part ( i ) of the theorem is proved . Now if X satisfies the condition 2 619 of part ( ii ) , then we can choose A sufficiently close to w + --D-- so that 0 ( 18 ) i s s atisfied and therefore part ( i ) will hold . It follows by Theorem 2 that if u( x , t ) is a solution of ( 1 5 ) , then u( x ,t ) ? w( x ,t ) for all X e V and 0 ? t ? T . Now w( x , t ) ? t ( A-'i') for all X E V ' so m m w( x ,t ) - ? as t - w, uniformly for x E V , which proves part ( i i ) . m 5 7 Note : As in the case of Theorem 12 , the requirement that f should s atisfy a uniform Lipschitz condition in u on any finite u-interval may be removed i f we alter hypothesis ( d ) to read "uo ( x) > 0 for all X E vm . " THEOREM 14 : Suppose that ( a ) As for Theorem 12 . ( b ) There exists a cons?t M > 0 such that? for all t > 0 , x E V and m u ? 0 , f( x , t ,u ) ? Mu . ( c ) There exis t constants D0 ? 0 and 61 > 0 suah that? for all x E avm and t > o , 0 ? d0 ( x ,t ) ? D0 and 61 ? d1 ( x , t ) ; if D0 > 0 , then we require also that 2619 '?' < * + Do ( d ) There exists a ccr.s?ant e > 0 suah that uo ( x ) > e for ?ll X E vm Then : ( i ) For any T > o , a strict lower so lution for ( 15 ) is given by t n 2 w ( X ' t ) = ?? ( A - . I: xi ) for al l X E V m ' 0 ? t s T ?= 1 if A is a constant ahcsen so as to satisfy 2 619 '?' < A < ? + Do C if D0 > 0 ; if D0 = 0 then A is chosen to satisfy A > '?') n n ( C+ 1 )A and if >.. > 2 ( I: A . + i = 1 ? I: B . } + i= 1 ? M (A - '?) . . . . ? . ? . . ? . ? ? ? . ? ( 19 ) ( ii ) If >.. > n 2 { I: A . i= 1 ? n + I: B . } + ( C+1 ) ( * + i= 1 ? ( if n0 > O ; in the case where n0 = 0 we require >.. > C?1 ) and if u( x , t ) is a solution of ( 15 ) , then u( x , T ) - m as T - m, uniformly for X E V . m Proof: ( I ) w ( x ,O ) = ?A A n 2 l: x . ) s e i= 1 ? for all x E V m for all x E V , by m hypothes is ( d ) ( I I ) is s imilar to the proof of Theorem 12 . = ( II I ) Lw - ? + Af( x , t ,w ) t n {-2ee t ) n -2x . ee ? t n . 2 I: a . . ( x ,t ) -A- + I: b . ( x ,t ) { ? } + c ( x ,t )A e ( A- I: x . ) i= 1 ?? i= 1 ? i= 1 ? t n ?A- L x: ) + Af( x ,t ,w) A i= 1 ? for t > 0 and x E V m e t n n = ? { -2 I: A . - 2 I: B . - ( C+1 )A + AM ( A-f) ) i= 1 ? i= 1 ? > 0 for all t > 0 by ( 19 ) . 5 8 Thus part ( i ) o f the theorem is proved . Now if A satisfies the condition of part ( ii ) , then we can choose A sufficiently close to 2 ?19 W + --D-- ( if D0 > 0 ) or sufficiently large ( if D0 = 0 ) so that ( 19 ) is 0 satisfied and therefore part ( i ) will hold . It follows by Theorem 1 that i f u( x ,t ) is a solution of ( 15 ) , then u( x ,t ) > w( x ,t ) for all t x E V and 0 s; t s: T . m ee -?kw w( x ,t ) ? ?A A-f) for all x E V , so m w( x ,t ) - = as t - =, uniformly for x E V , which proves part ( ii ) . m THEOREM 1 5 : Suppose that ( a ) As for Theorem 12 . ( b ) There exist constants M > 0 and a > 0 such that, for all t > O , 1+.!. x E Vm and u ? 0 , f( x ,t ,u) ? Mu a ( c ) As for Theorem 1 4 . ( d ) As for Theorem 1 4 . Then : ( i ) For any T such that o < T ( 15 ) is given by w( x , t ) ( t1: )ae ( A -= A (t 1:-t )a < t l': ' n 2 I: x . ) i= 1 ? if A is a constant chosen so as to satisfy 2619 f < A < ljl + Do a s trict lower solution for for aU X E vm ' O S: t :S: T ( if D0 > 0 ; if D0 = 0 then A is chosen to satisfy A > f) ?.\ 59 and if A > A 1/a n n _ ___;,_ ___ ....,.... 1 { ? + 2 I: A . + 2 I: B . + CA) . . . . . . . ( 2 0 ) 1+- i=1 ? i= 1 ? Me1/a.(A-'f) a ( ii ) If A > 2 619 1/a ( ? + --) D0 n ----_;._---- 1 {2 I: A . 6 . 1 ? Me1/a( ? + 2 19 - 'f) 1+-cx ?= Do n 2 619 + 2 I B . + C ( \1 + -D-) } i= 1 ? 0 ( if D0 > O ; in the aase where D0 = 0 we require A > ?/ ) and if u( x ,t ) Me a. is a solution of ( 15 ) , then there exists a finite number T > 0 suah that u( x , T ) - m as T - T- , uniformly for x E V m . In partiaular? given any positive e , this wil l be the aase for al l suffiaiently large A ; and given any positive A , this wi l l be the aase for all sufficiently large e . Proof: ( I ) is similar to the proof of Theorem 1 4 . ( I I ) i s similar to the proof of Theorem 12 . Ow ( I I I ) Lw - et + Af( x ,t ,w ) n - 2 ( t* ) ae n = L a . . ( x ,t ) + I: b . ( x ,t ) i= 1 ?? A( t*-t )a. i= 1 ? ( t* )a.e n 2 + c ( x ,t ) --'----'---- (A - I: x . ) A( t*-t )a i= 1 ? a.( t* )ae n 2 ____;;;...;....____;__..,.... ( A - I: X . ) + Af ( X ' t 'w ) A( t*-t )a+ 1 i= 1 ? {-2 ( t* ) a.e } n {- 2 ( t* ) a.e ] n ( t* )a.ec ? I A ? + I: B ? - ..;.___;_...? a( t ;? ) a.e (t ;Lt )a+ 1 = A( t*-t )a. i= 1 ? A ( t*-t )a. i = 1 ? ( t*-t )a 1 ? 1? AM(t* )a.+1 e a. ( A-'f) a. + ???------?..;.__?- 1? A a. ( t*-t )a.+1 for x E V and 0 < t < t* m 1? ( t* )Ue n n AMt*a 1/a(A-'f) a . ( - ( 2 I A . + 2 I: B . + CA) (t*-t ) - aA + -??--:-:----) A(t*-t )a+1 i=1 ? i=1 ? A1/a 1? ( *)a n n ' Mt*e1/a(A-'f) a ? t e (- ( 2 I: A . + 2 I: B . + CA )t* - aA + .:.:;A;.;..;;..-=-._,...;:-;.;-_,;_-) A(t*-t )a+1 i= 1 ? i=1 ? A1/a for 0 < t < t* > 0 by ( 2 0 ) . Thus part ( i ) of the theorem is proved . Now if A satisfies the condition o f part ( ii ) , tten we can choose A sufficiently close to 2 619 W + ? ( if D0 > 0 ) or sufficiently large ( if D0 = - 0 ) so that A1/a n ---....,..----.,.-1 {2 E A . + . 1 l. 1+- 1.= Me1/a( A-'i') a n 2 E B . + CA} i= 1 l. and we may then choose t* large enough so that ( 20 ) is sat isfied and therefore part ( i ) will hold . It follows by Theorem 1 that if u( x ,t ) is a solution of ( 15 ) , then u( x , t ) > w( x , t ) for all X E vm and 60 0 :s: t :s: T . But w ( x , t ) ? ( t :': ) ae (A-'i') for all x E V , so w ( x ,t ) - CD A (t*-t )a m as t - t*- , uniformly for x E V . I t follows that there must exist m a T with 0 < T :s: t :': , st.:.c=: -:hat u( x ,T ) - CD as T -+ T- ; that this limit is uniform for x E V c?? be s een at once by redefining w ( x ,t ) with m t :': = 1'? Thus part ( ii ) is prove d . THEOREH 16 : Suppose that ( a ) As for Theorem 1 2 . ( b ) There exist cor;stcr:?s \ > 0 , M2 > 0 , a > 0 such that, for al l 1+! t > 0 , x E Vm and u ? 0 , =< x , t ,u) ? M1u a + M2 . Furthermore, f satisfies a uniform Lipschitz condition ?n u on any bounded u-interval . ( c ) As for Theorem 12 . ( d ) As for Theorem 12 . Then : ( i ) For any T such that o < T < t* , a lower solution for ( 15 ) is given by t n 2 w( x ,t ) = A - E x. ) for all x E V , 0 :s: t :s: T ( t*-t )a i= 1 1 m if A is a cons tant chosen so as to satisfy 2 619 'i' < A < w + --Do and if A is chosen sufficiently large, depending on the constants Ai , Bi ( i = 1 , . . . n ) , C , M1 , M2 , a , A , 'i' and t* . ( i i ) If A and ).. are chosen as in ( i ) , and if u( x ,t ) is a solution of ( 1 5 ) , then there exists a nwriber T with 0 < T < t :': such that u( x ,T ) - CD as T - 1'- , uniformly for x E V . m Proof : ( I ) 2nd (} I ) are simi lar to the proof of Theorem 12 . 6 1 ( II I ) Lw Ow \f( x ,t ,w) - ? + n -2t n -2x . t I: a . . ( x ,t ) + I: b . ( x , t ) ? = . 1 l.? ( t*-t )a . 1 l. ( t'Lt )a ?= ?= t ( A - n 2 + c ( x ,t ) I: x . ) ( t*-t )a i= 1 ? t* + (a-1 ) t n 2 --.....;....;;.'---':- (A - I: x . ) + U( x , t ,w ) (t*-t )a+1 i = 1 ? n n ? { -2t ) I: A . + { -2t } I: B . _ tCA ( t'':-t ) a i= 1 ? ( t1Lt )a i = 1 ? ( t'':-t )a A[t* + ( a- 1 ) t] ( t*-t )a+1 + ( t*-t )a+ 1 n n for x E V and 0 < t < t* m - ( 2 I: A . + 2 I: B. + CA ) t ( t*-t ) - A(t* + ( a-1 ) t] i=1 ? i=1 ? n n Now put K = 2 I: A . + 2 I: B. + CA i=1 ? i = 1 ? and g ( t ) = - Kt { tlLt ) - A(t'': + ( a.- 1 )t] = Kt2 + t {- Kt* A (a.-1 ) } - At* . Then the minimum value over all real t o f the quadratic g ( t ) is - 4KAt1': - {Kt* + A( a.-1 ) }2 4K {Kt* + A { a.-1 ) }2 At'': - .... - ---:-:-?----4K attained when have : Kt1: + A (a.-1 ) t = 2K Thus , for 0 < t < t* and x E V , we m aw Lw - at + \f( x , t ,w ) 2 1? 1? 1 {Kt* + A( a.-1 )} a. a a.+1 } ? (- At* - - 4K - + \{M1t (A-'i') + M2 (t 1(-t ) ) . ( t *-t )a.+1 1? 1? The expres sion M1t a. ( A-'i') a. + M2 ( t*-t ) a.+1 is obviously continuous and strictly positive for 0 ? t ? t* , so there exists 6 > 0 such that 1+1. 1+1. M1t a. ( A-'i') a. + M2 ( t*-t ) a.+1 ? 6 for 0 ? t ? t* . 2 1 {Kt* + A ( a.- 1 ) } J If we then choose \ ? ? [At* + - 4K , then Ow Lw - at + A.f( x ,t ,w ) ? 0 for 0 < t < t* and x E V . m Thus part ( i ) of the theorem is proved . It follows by Theorem 2 that if u ( x , t ) is a solution of ( 15 ) , then u( x ,t ) ? w( x ,t ) for all x E V m and 0 ? t ? T . The rest of the argument parallels the proof of Theorem 15 . Note : In the case of Theorem 16 where a = 1 , 6 is the minimum value on [0 , t1:] of the expression M1t 2 ( A-? ) 2 + M2 ( t*-t ) 2 = t2 [M1 ( A-? ) 2 + M2) - 2M2t*t + M2 ( t* ) 2 . This quadratic attains its overall minimum value at 2M t* 2 t = which is a value between 0 and t* Hence 6 is the overall minimum of the quadratic . Thus 6 = 4M2 ( t * ) 2 [M1 (A-Y) 2 + M2 ] - 4M? ( t * )2 4 [M1 ( A-?) 2 + M2) M M ( t ?': ) 2 { A-?) 2 1 2 = ??----?----2 M1 ( A-?) + M2 Using this we can prove the following : COROLLARY : Suppose that ( a ) As for Theorem 12. ( b ) There exist aonstants M1 > o , M2 > 0 suah that, for all t > o , 2 x E Vm and u ? o , f( x ,t ,u ) ? M1u + M2 . Furthermore, f satisfies a uniform Lipsahitz aondition in u on any bounded u-interval . ( c ) As for Theorem 1 2 . ( d ) As for Theorem 12 . 6 2 Then : ( i ) For any T suah that o < T < t* , a lower solution for ( 15 ) is given by t n 2 w( x ,t ) = t*-t(A - t x . ) for aU X e V , 0 ? t ? T i= 1 ? m if A is a aonstant ahosen so as to satisfy 2618 ? < A < t + --? Do A 1 ? n n 6 3 and if X ? M2) (t* + -a<2 I: A . + 2 I: B . + CA) ) i=1 ? i=1 ? ? ? ? ? ? ? ( 2 1 ) A > ( i i ) If n n (2 I: A . + 2 I: B . + C( ? + i= 1 ? i= 1 ? and if u( x , t ) is a solution of ( 1 5 ) , then there exists a finite number T > 0 such that u( x ,T ) - ? as T - T- , uniformly for x E V . m Proof: ( i ) follows from Theorem 16 with a = 1 , using the value of 0 obtained in the note above . If A satisfies the condition of part ( ii ) , then we can choose A sufficiently close to A > n (2 I: A . + i= 1 ? n 2 t B . + CA) (M1 ( A-f) 2 + M2) i= 1 ? and we may then choose t* large enough so that ( 21 ) is satisfieu and therefore part ( i ) will hold . The rest of the argument parallels the proof of Theorem 15 . Construction of Upper Solutions on V : m THEOREM 17 : Suppose that ( a ) There exist constants A . > O ( i = 1 , . . . n ) , B . ? O ( i = 1 , . . . n ) and ? 1 C such that, for all x E V and t > 0 , 0 < A . s: a . . ( x ,t ) ( i = 1 , ? ? ? n ) , m ? 1? jb . ( x ,t ) j s: B . ( i = 1 , . . . n ) and c ( x ,t ) s: c . 1 . 1 ( b ) For any bounded positive u-interval positive number M , depending only on I , x E V , t > 0 and u E I . I , there exists a corresponding such that f ( x ,t ,u ) ? M for al l m ( c ) There exist constants 00 > 0 and n 1 ? 0 such that, for all ( d ) We require in addition that, if C ? n 2D 8 2 I: B . + C ('i' + -1-) i=1 1 6o 0 , then n < 2 E A . ? i = 1 1 x e av m ( e ) There exists a constant ? > 0 such that uo ( x) < ? for aL L X e vm . Then : ( i ) For any T > 0 , a strict upper solution for ( 1 5 ) is _given by n ? . 2 -w( x ,t ) = n ( A - l: X . ) for aU X e V , 0 ? t ? T - i= 1 ? m . 6 4 (so that w ( x , t ) is actuaLLy independent of t ) if A is a constant chosen so as to satisfy n n 2 l: A . - 2 E B . 2 n e 'f + __ 1_ < A < 60 i= 1 ? i = 1 ? c if c > 0 A > max and if if c = 0 n n i= 1 ? i= 1 ? 2D1S 2 l: A . - 2 E B . } -6--, 'f + -----?c?----?o if c < 0 ? ? ? ? ( 2 2 ) e 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I ( 2 3 ) where m0 ( e ,A ) , apart from depending on e and A , depends on the coefficients in the operators L and B1 . , the quantities 'f and e? and ?n the nature of the function f , but not on T . ( ii ) If 0 < )., < m0 ( e ,A ) , and if u( x , t ) is a so lution of ( 1.? ) , then for aU T > 0 and X e V ' u( x ,T ) s ? m A-I Proof: ( I ) w( x , O ) hypothes is ( e ) . ( I I ) B1 . w ?n ? = p:::::y ( A n 2 E x . ) ? e > u0 ( x) i= 1 ? for all X e V ' by m n 2m . -1 = d0 ( x ,t )w + d1 ( x , t ) E a . ( x ) x . i= 1 ? ? ? e n 2 n 2mi- 1 -2exi = d0 ( x , t ) -:---m-A- ( A- 1: x . ) + d1 ( x ,t ) l: a . ( x ) x . ? ( A-'f ) I i =1 ? i= 1 ? ? e n 2 . n 2mi = p::::y {d0 ( x ,t ) ( A- 1: x . ) - 2 d1 ( x ,t ) l: a . ( x) x . } i=1 ? i= 1 ? ? > 0 for t > 0 and X E oV m 2D18 for all t > 0 , s ince A > 'f + --- by ( 22 ) . 60 Ow ( II I ) Lw - dt + ).,f( x ,t ,w ) n n = i?ta ii ( x ;t ) ( -2 ?1( A:'i') + 1 ? 1 b i ( x , t H -2xi ) ( A?'i') + c ( x , t ) w + ).,f ( x , t ,w) . eA Now for t > 0 and x E Vm , e ? w( x ,t ) ? A=Y' so by hypothesis ( b ) we have that for all t > 0 and x E V , f( x ,t ,w) ? M ( e ,A , f) . I f C ? 0 , m n n then by ( 2 2 ) and hypothesis ( d ) we have 2 E A . - 2 E B . - CA > O , i= 1 ? i= 1 ? n n while if C < 0 , then by ( 22 ) we have 2 E A . - 2 E B . - C ( A-f) > 0 . i= 1 ? i=1 ? Thus we define the positive number m0 ( e ,A) as follows : If C ? O , m0 ( e ,A ) = If c < o , m0 ( e ,A ) = n n e {2 E A . - 2 E B . - CA } i= 1 ? i= 1 ? (A-'f) M ( e ,A ,'f) n n e {2 E A . - 2 I B . - C ( A-?) } i= 1 ? i= 1 ? ( A-f) M( e ,A ,'f) Now if C ? O , then we have that for all t > 0 and x E V : m 0 0 0 0 . . ( 2 4 ) aw -2e n 2 e n CeA Lw - ? + Af( x ,t ,w) ? ( A_'?') E A . + (A=Y) E B . + p:::y + AM( e ,A , f) . i= 1 ? i= 1 ? If C < 0 , then we have that for all t > 0 and x E V : m 6 5 Ow In either case , it follows by ( 2 3 ) and ( 24 ) that Lw - ot + Af( x ,t ,w) < 0 for all t > 0 and x E V . m Thus part ( i ) of the theorem is proved . Now if A satisfies the condition of part ( ii) , then part ( i ) holds , and it follows by Theorem 1 that if u( x , t ) is a solution of ( 15 ) , then u( x , t ) < w ( x ,t ) for all x E V and 0 ? t ? T . It follows that for all m eA T > 0 and x E ?, u( x ,T ) < w( x , T ) ? A-'?' Thus part ( ii ) is proved . Note : That some such condition as hypothesis ( d ) , which places a definite upper bound on c( x ,t ) , is required in Theorem 1 7 , is shown by the example given in Ch . 2 . In that examp le , we have c ( x ,t ) = k for all x and t , and f( x ,t ,u ) = 1 for all x , t and u . It i s easily checked that hypotheses ( a ) , ( b ) , ( c ) and ( e ) of Theorem 17 are satisfied , with C = k . As 2 explained in Ch . 2 , if C = k ? n4 , the solution of the time-dependent problem is unbounded as t - oo, for any A > 0 . Thus Theorem 17 does 66 2 TT not hold in this case unless we require C < ?, i . e . unless we put an upper bound on c( x ,t ) as in hypothesis ( d ) of Theorem 17 . We may also observe in passing that , for the example o f Ch . 2 , we have n = 1 , B1 = 0 , A1 = 1 , ? = 1 and n1 = 0 , s o that hypothesis ( d ) of Theorem 17 reduces to C < 2 , which is than the weakest possible condition C not much stronger a condition TT2 < 4 F::S 2 . 47 . THEOREM 1 8 : Suppose that ( a ) As for Theorem 1 ?. ( b ) There exist constants M1 > 0 , M2 > 0 such that, for all t > 0 , x E Vm and u ? 0 , f( x ,t ,u) ? M1u + M2 ? ( c ) As for Theorem 1 ? . ( d ) As for Theorem 1 ? . ( e ) uo ( x) is bounded above for X E vm. Then : If u( x ,t ) is a so lution of ( 15 ) , and if A satisfies o < A < n 2 I: A . i = 1 ? n - 2 I: B . - C ( ? + i= 1 ? 2n e M1 ( ? + _ 1_) (>0 or o < A < .:? ( if c < o ) M1 2n e _ 1_) (>0 ( if c ? 0 ) ( 2 5 ) then there exists a constant K > 0 such that, for al l T > 0 and x E Vm ' u( x ,T ) ? K , where K depends on A. Proof: We may apply Theorem 1 7 in this case . In the notat ion of Theorem 17 , i f eA e < w < r::::'f' then eA f( x ,t ,w ) s M1 ( A_f) + M2 , so in the proof of Theorem 1 7 we may take Thus , if C ? 0 , then n n 2 I: A ? - 2 I: B ? - CA i= 1 ? i = 1 ? M1A + M ( A-f) 2 e n n 2 I: A . - 2 I: B . - C ( A-?) i=1 ? i= 1 ? while if C < O , then we have Now if C ? 0 and A satis fies ( 2 5 ) , then by choos ing A sufficiently 2D18 close to ? + --0-- and e sufficiently large ( as we may do , s ince e may 0 be chosen as large as we please ) we can ensure that o < X < m0 ( e ,A) < n 2 I: A . i = 1 ? n - 2 I: B . - C( 'f + i=1 ? 2D E) _1_) 60 67 and so part ( ii ) of Theorem 17 will hold , which proves Theorem 18 if we eA take K = A-'f ? If C < 0 and X satisfies ( 2 5 ) , then by choosing A sufficiently large and e equal to , say , A2 , we can ensure that m0 ( e ,A ) is as close -C as we please to ?' and so we can certainly ensure that So again with K = 1 part ( ii ) eA n ? of Theorem 17 will hold , and Theorem 18 is proved COROLLARY : Suppose we modify the hypotheses of Theorem 18 by requiring that f( x ,t ,u ) ? M2 for aZZ t > 0 , x E Vm and u ? 0 . Then if u( x ,t ) is a solution of ( 15 ) , we have that for any X > 0 there exists a constant K > 0 , depending on X, such that u( x ,T ) ? K for aZZ T > 0 and x E V . m Proof: For any X > O , we may choose M1 sufficiently small so that ( 2 5 ) holds , and the required result follows at once by Theorem 18 . Notes : ( i ) For the example discussed in Ch . 2 , with k = X , we have n = 1 , B 1 = O , A1 = 1 , 'f = 1 , n1 = 0 , C = 0 ; also f( x , t ,u) = u+1 for all x , t and u , so that M1 = M2 = 1 . Thus in this case Theorem 18 tells us that the solution to the time-dependent problem will be bounded above as t - m if 0 < X < 2 ; in fact , as explained Jn Ch . 2 , the solution n2 will be bounded above as t - m if 0 < X < 4 F::s 2 . 47 . ( ii ) Theorem 18 and its Corollary relate well t o results ( iv ) and ( v) of Keller and Cohen ( see pp . 15 , 16 ) . Keller and Cohen showed ( result .. ( v ) ) that if f( x ,u ) < F( x ) + p( x)u for x E V , u > 0 , then steady-state solutions exist if 0 < X < ? { p } ; Theorem 18 shows that if f( x , t ,u ) ? M1u + M2 for x E Vm ' t > 0 , u ? 0 then t ime-dependent solutions are bounded as t - m if 0 < X < ?( say ) where ? depends on M1 but not on M2 ? Keller and Cohen also show ( result ( iv ) ) that if f( x ,u ) < F( x) for x E V, u > 0 , then steady-state solutions exist for ?all X > 0 ; the Corollary to Theorem 18 shows that if f( x ,t ,u ) ? M2 for all X E Vm ' t > 0 , U ? 0 then time-dependent solutions are bounded as t - ? for all ? > 0 . THEOREM 19 : Suppos e that ( a ) As for> TheoY'em 1 7 . ( b ) TheY'e exists a constant M > 0 such that, for> aLl x E V , t > 0 and m u ? 0 , f( x , t , u ) ? Mu . ( c ) , ( d ) , ( e ) as for' TheoY'em 1 7. ( f ) For> al l t > 0 and x E V , f( x ,t , O ) ? 0 ; furthermoY'e, f satisfies a m unifor'm Lipschitz condition in u on any finite u-interval . ( g ) u0 ( x ) ? 0 for' a l l x E vm . Then : ( i ) For' any T > 0 , a str'ict upper' solution for' ( 1 5 ) is given by ' n 2 w( x ,t ) = e -??.t ( A - I: x . ) for' all x E V , 0 ? t ? T A-'f e i= 1 l. m if A is a constant chosen so as to satisfy 2n e 'f + 1 < A < ? A > 'f n n 2 I: A . - 2 :E B . i= 1 l. i= 1 l. c 2 n e 1 + --{)0 n n if c if c > 0 = 0 { 2n e A > max 'f + ---1-60 2 I: A . - 2 I: B . } i= 1 l. i= 1 l. c if c < 0 n n - CA and if 0 < ? < 2 I: A . - 2 I: B . i= 1 l. i= 1 l. M(A + 1) if c ? 0 n n 0 < ? < 2 I: A . - 2 I: B . - C (A-'f) i= 1 l. i=1 l. M(A + 1) if c < 0 ( ii ) If u( x ,t ) is a so lution of ( 15 ) , and if ? satisfies 0 < ? < n 2 I: A . - i=1 l. n 2 I: B . i= 1 l. 2n e - C ( 'f + _1_) {)0 or' o < ? < ?c < if c < o ) then u ( x ,T ) - 0 as T - ""? uniformly for' x E V m ( if c ? 0 ) Proof : ( I ) and ( I I ) are s i?i lar to the proof of Theorem 17 . ? ? ? ? ( 26 ) ( 2 7 ) 6 8 69 ( II I) Lw - ? + A.f( x ,t ,w) n -2ee-A.t = ? r a ? . ( x , t ) '? n :r: b . ( x ,t ) i=1 ? -2x . ee ? -A.t -At n 2 + c ( x , t ) .:::.e..;;..e -......- ( A- E x . ) i= 1 u A- + A-'f .' -'f ' "' . 1 ... ?= + -' t A.ee 1\ A-'f n 2 ( A- :E x . ) i= 1 ? + A.f( x , t ,w ) -At n :s: C2ee ) E A . A-'? . ? ?= 1 + -At n ( 2ee ) E B . A-'? ? ? ?= 1 + -At n 2 c ( x ,t ) ee ( A- E x . ) A-'? i= 1 ? -A.t n + + ee 2 A.M -;::::r ( A- :r: X. ) i=1 ? -At n n n 2 ee { - 2 E A . + E B . U? } :s: A-'f 2 + c ( x ,t ) ( A- E x . ) + AA + i=1 ? i= 1 ? i= 1 ? for all t > 0 ' Now if c : 0 and x E V : Til Ow -A.t n n Lw - ?t + A.f( x ,t ,w) :s: ?A { - 2 E A . + 2 E B . + CA + A.M( A+ 1) } Ol: ? - 't i = 1 ? i = 1 ? < 0 for all t > 0 by ( 2 6 ) . If C < O , then we have that for all t > 0 and x E V : m X E ? - -At n n L vw , ( ) ?e f ? 't"' ( \!/ ) ( ) } w - ::.t + 1\f X , t , w s ?A - 2 L. A . + 2 L. B . + c A- ? + A. M A+ 1 V A - I i = 1 ? i = 1 ? < 0 for all t > 0 by ( 26 ) . Thus part ( i ) of the theorem is proved . V . m Now if A. satisfies ( 2 7 ) , then by choosing A sufficiently close to 20 e 'f + ---1- ( if C : 0 and x E V : m m ee-A.T n 2 ? Aee-A.T u( x ,T ) < w( x ,T ) = ( A I: x ) "" A-'f - i I A-'f i = 1 Also , b y hypotheses ( f ) and { g ) , it follows from Theorem 1 1 that u( x , T ) :<1:: 0 for all T > 0 and x E V . Since A. > 0 , part ( ii ) of the m theorem follows at once . Note : Theorem 19 relates well to result ( vi ) of Keller and Cohen ( see p . 16 ) . Keller and Cohen 3howed that if there exists a posit ive p ( x ) ,\ 70 such that f( x ,u ) < p( x )u for x E V , u > 0 , then no positive steady-state solutions exist if 0 < A < ? { p } . Theorem 19 shows that , if f( x , t , u) ? Mu for all x E V , t > 0 , u ? 0 , and if 0 < A < ?( say ) where m ? depends on M , then all solutions of the time-dependent problem ( provided the initial function is bounded ) tend to zero as t - =, so that no positive steady-state solutions will exist if 0 < A < ?? THEOREM 20 : Hypotheses as for Theorem 1 ? except that, in hypothesis ( b ) , we suppose that there exis t constants M1 ? o , M2 > o and a such that 0 < a ? 1 , such that, fer all x E V , t > 0 and u ? 0 , m 1-CI f( x ,t ,u) ? M1 + M2u . Then : ( i ) For any T > o , a strict upper solution for ( 15 ) is given by w( x ,t ) = KA11?;/a( A - i ? 1 xf ) for all X E vm , 0 ? t ? T (so that w( x ,t ) is actually independent of t ) if A > O , A is a constant chosen so as to satisfy n n 2n e 2 E A . - 2 E B . i= 1 ? i=1 ? 1 'f + -- < A < {>0 c if c > 0 2n e A > 'f + 1 ? if c = 0 n n 2 E A . . 1 ? ? = - 2 E B . } i= 1 ? c if c < 0 cmd K is a constant chosen so as to satisfy 1 { [ A 1-?; 1/? 1 + A 1 -CI ?1/CI K > max 1 , n n ' 2 I: A . - 2 E B . - CA i=1 ? i= 1? l. 1 K > max {1 , [ n >. 1-a;.; 1/? 1 + A1-a ]1/a , 2 I: A . - 2 E B . - C ( A-'f) i=1 l. i= 1 l. ( ii ) For any A > o , if u( x , t ) is a so lution of ( 15 ) , and K and A are as above , then u( x ,T ) ? KA11??/aA for al l X E vm and T > 0 . ( 2 8 ) ( 2 8 ) , and hypothesis ( e ) . . ( I I ) is similar to the proof of Theorem 11 : . . Ow ( II I ) Lw ? :... at + )..f( x ,t ,w) n . . 1 la..1 /a n 1la..11a = E a . . ( x ,t ) ( -2K).. -M2 ) ? + I: b . ( x ,t ) ( -2 x . K).. --M2 ) ' i = 1 1.1. i=1 l. l. + c ( x ,t )w + )..f( x , t ,w ) . n n n ? -2K)..1 10M11a I: A . + 2K)..11?1Ia I: B . + c ( x ,t )K)..11?121 a(A- I: x? ) . 2 i?1 l. 2 i = 1 l. i=1 l. ? !_ 1 .!_ _ 1 n + ).. {M1 + M2K 1-?a M? ( A- I: x? ) 1-a} . i =1 l. Thus , for all t > 0 and x E V : m. Ow Lw - at + H ( X 't 'w ) 1 I I n rt n 2 ? K -?1 ?1 a{-2KCI I: A . + 2KCI I: B . + c ( x ,t ) KCI( A- L x . ) 2 i = l l. i = 1 l. i=1 l. 1 I I n n n 2 ? K -?1 ?1 CI{Ka[-2 I: A . + 2 I: B . + c ( x , t ) ( A- I: x . ) ] 2 i=1 l. i = 1 l. i=l l. 1 1 -?.-11a.. 1-a} + A -M2 --M1 + A a- 1 since K > 1 ( by ( 2 8 ) ) and a- 1 ? O , so K ? 1 . Now if C ? 0 , then we have that for all t > 0 and x E V : m Ow Lw - et + )..f( x ,t ,w ) < 0 for all t > 0 by ( 2 8 ) ? If C < 0 , then we have that for all t > 0 and x E ? V : m Ow Lw - ot + )..f( x ,t ,w) < 0 for all t > 0 by ( 2 8 ) . Thus part ( i ) o f the ?heorem is proved . .\ 7 1 72 Now if X > O , and u( x ,t ) is a solution of ( 15 ) , and K and A are as above , then it follows by Theorem 1 that u( x, t) < w( x , t) for all x E V m - 1/?.1/a and 0 ? t ? T . Hence , for all T > 0 and x E Vm ' u ( x ,T ) < w( x ,T ) s KX -M2 A . This proves part ( ii ) o f the theorem . Extens ion to Other Domains : n 2m . Theorems 12 to 20 apply only to the domain V = {x : I x . 1 < 1 } m i= 1 1 ( the m . be ing arbitrary positive integers ) ; Theorem 11 , though stated 1 for the domain V , will obviously hold for any domain V . I f one wishes m to extend Theorems 12 to 20 in a constructive way to some other specific domain V* , this may be pos s ible if one can explicitly construct a diffeomorphism from V* to V , i . e . !'!1 if one can find open sets 0 1 ? V* , o2 ? vm and a homeomorphisr. of v* onto vm which can be extended to a differentiable function g : 01 - o2 with differentiable inverse . It i s necessary ?also that the second partial derivat ives of g should exist on 01 . The construction o? such a diffeomorphism is only possible in certain s imple cases . Then if u( x ,t ) satis:'ies ( 1 5 ) on the region { ( x , t ) : X E V?': ' t ? we can define v( x , t ) = u( g- .:. ( x ) , t ) on the region { ( x , t ) : X E v m ' t ? and use standard calculus te c!1niques to transform ( 1 5 ) into the o } , o } , corresponding initial-boundary value problem satisfied by v . It may then be poss ible to apply Tr.eorems 12 to 20 to this problem . The next chapter includes one s imple example of this technique . However , if one is pre?ared to abandon to some extent the explicitly construct ive approach used in Theorems 12 to 20 , one may prove a collection of theorems s imilar to Theorems 12 to 20 but applying to an arbitrary domain V . The method of proof that will be used here requires that we restrict ourse lves to a t ime-independent differential operator . ,. We shall denote by L1u the expres sion where ... ,. E ca( V) for i , j a . . = a . . 1 ] ] 1 i = 1 , 2 , ? . ? n . The differential elliptic . We shall be concerned value problem = n ,. o + I b . ( x ) u . 1 1 'dx':"" 1= 1 1 , 2 ' . . . n , and b . E Ca(V) for 1 ... operator 11 is assumed to be uniformly with the parabolic init ial-boundary A OU L1u + c ( x , t ) u - dt + ?f( x ,t ,u ) = 0 for ( x ,t ) E DT Blinu = 0 for ( x , t ) E ST u( x , O ) = u0 ( x) for x E V ? ? ( 2 9 ) where f is continuous for x E V, 0 ? t ? T and all u , u0 E c2+a(V) and the parameter ? is assumed to be positive . Now consider the eigenvalue problem .. L1cp + ? = 0 for x E V } . . . . . . . . . . . . . ( 30 ) acp + b ? = 0 for X E oV where a and b are positive constants . I f A denotes the inverse of the .. operator L1 with boundary condition as in ( 30 ) , then A is a compact ( i . e . completely cont inuous ) operator on the Banach space C (V) of continuous functions defined on V, with the supremum norm ( see Browder[7] ) . Further : E v .. + b ? = Ah = g for x ? L1g = h for X E V and ag 0 for x E oV . Suppose now that Ah = g for X E V, where h ( x) :<: 0 for all X E V. If g attains its maximum M in V , then it follows by Theorem 5 of Ch . 2 of Protter and Weinberger[28] that g ( x ) = M for all x E V. But then 73 ? = 0 for all X E oV , and so from ag + b ? = 0 for all X E oV , it follows that g ( x ) = 0 for all X E oV . Thus M = o , and g is identi cally zero on V . So if g is not identically zero on V ( equivalently , i f h is not identically zero on V) , then the maximum of g on V is attained at a point P on oV . Then , by Theorem 7 of Ch . 2 of Protter and Weinberger[2 8] , ? > 0 at P . Since ag + b ? = 0 at P , it follows that g ( x) < 0 at P , so g ( x ) < 0 for all x E V. Thus we have shown that if h ( x ) :.: 0 for all x E V , and h is not identically zero on V, then -Ah > 0 for all x E V. Now the set of non-negative functions on V is a cone in the Banach space C (V) , with interior the set of strictly positive functions on V. It follows from the above discussion that the operator -A is strongly positive with n = 1 , with respect to this cone ( us ing the terminology of Krein and Rutman[20 , p . 2 6 6] ) . It follows by Theorem 6 . 3 of Krein and Rutman[20 , p . 2 6 7] that -A has a unique normalised e igenfunction which is strictly positive on V, and the corresponding e igenvalue is real , posit ive and s imple . Since ( 30 ) mayJ be written 74 ( -A )? = ! ?, it follows that ( 30 ) has a unique normalised e igenfunction I.L ?1( x ; a ,b ) which is strictly positive on V, and the corresponding e igenvalue IJ1 ( a ,b ) is real , positive and simple . Obviously , there exist positive constants a1 ( a ,b ) , ?( a ,b ) such that a1 ( a ,b ) ? ?1( x ; a ,b ) ? ?( a ,b ) for all X E V . S ince acp1 + b Oq>1 On < 0 on oV . Oq>1 an- = 0 on oV and ?1 > 0 on oV , it follows also that We are now in a pos ition to state and prove a theorem analogous to our earlier Theorem 12 . A THEOREM 12 : Suppose that ( a ) There exis ts a constant C ? 0 such that c ( x ,t ) ? -C for all x E V and t > 0 . ( b ) There exists a constant M > 0 such that, for all t > 0 , x E V and u ? o , f( x ,t ,u ) ? M . Furthe?ore, f satisfies a uniform Lipschitz condition in u on any finite u-interval . ( c ) There exist constants D0 > 0 and 61 > 0 such that, for all x E oV and t > o , o < d0 ( x , t ) s D0 and 61 s d1 ( x ,t ) . ( d ) u0 ( x) ? 0 for all x E V. Then : ( i ) For any T > o , a lower solution for ( 2 9) is given by w ( x , t ) where K is a constant chosen so as to satisfy M 0 < K < ? ( D0 ,?61 ){C + ?( D0 ,S61 ) + 1] . ? ? . . . ? . ? ? ( 3 1 ) ( ii ) If u( x ,t ) i s any solution of ( 29 ) , then for any T > O , u( x ,t ) > 0 for 0 < t ? T , and if lim u( x ,T ) = u( x ) exists, then for all T-c? X E V, XMa1 C D0 ,?61 ) u (x ) ? { ( !;; ) J > o . a2CD0 ,?61 ) c + ? n0 , 261 + 1 Proof: ( I ) w( x ,O ) = 0 ? uo ( x) for all X e V, by hypothes is ( d ) . Ow ( I I ) Blinw = d0 ( x ,t )w + d1 ( x , t ) On Ocp1 = XK( 1-e-t ) {d0 ( x ,t )cp1 C x ; D0 ,?61 ) + d1 ( x ,t ) On } Ocp1 s XK( 1-e -t ) { D0cp1 ( x ; D0 ,?o1 ) + d1( x ,t ) On } for t > O , x e -ov . Oq>1 -t { ) = ?K( 1-e ) - ?61 + d1 ( x ,t ) On 75 < 0 for all _ t > 0 since d1 ( x , t ) ? 61 > ?61 for t > 0 , Oq>1 X E av , and also On < 0 for X E oV . .. Ow ( I I I ) L1w + c ( x ,t )w - ? + ?f( x ,t ,w) = ?K( 1-e-t ) {- ?( D0 ,?61 ) + c( x , t ) }?1- ?K?1e -t + ?f( x ,t ,w ) ? ?K{- IJ1 ( D0 ,?61 ) - C )? ( D0 ,?61 ) - ?K?( D0 ,126 1 ) + )..M > 0 for t > 0 and x E V , by ( 31 ) . Thus part ( i ) of the theorem is proved . B y Theorem 2 , i t follows that i f u( x , t ) is a solution of ( 29 ) , then u( x , t ) ? w( x ,t ) for all ( x , t ) E DT . It follows that u( x , t ) > 0 for x E V, 0 < t ? T . Furthermore , w( x ,t ) - )..K?1 as t - ? , so if lim u( x ,T ) = ?( x ) exists , T-oa:> then for all x E V, arbitrarily close to u( x ) ? ?K?1 ? )..Ka1 ( D0 ,?61 ) . Since K may be chosen M l } , part ( ii ) of the ? < ?o '?61 ) {c + u1 C Do ,?61 ) + theorem follows at once . Note : As for Theorem 1 2 , the condition that f should s a? isfy a uniform Lipschitz condition in u on any finite u-interval may be removed if we assume that u0 ( x ) > 0 for all x E V. In a s imilar fashion , one can state and prove Theorems 1 3A to 16A analogous to Theorems 1 3 to 16 . The remaining "lower solut ion" theorem , Theorem 1 1 , holds for arbitrary domains V in any case , as already remarked . Next we state and prove a theorem analogous to our earlier Theorem 17 . THEOREM 17A : Suppose that ( a ) There exist constants 60 > 0 and D1 > 0 such that, for al l x E oV and t > o , d0 ( x ,t ) ? 60 and o ? d1 ( x ,t ) ? n1 . _ ( b ) There exists a constant C < IJ1(?60 , n1 ) such that c ( x ,t ) ? C for al l x E V and t > 0 ? ( c ) For any bounded positive u-interval I , there exists a corresponding positive number M , depending only on I , such that f ( x ,t ,u) ? M for a l l x E V , t > 0 and u E I . ( d ) There exists a cons tant ? > 0 such that uo ( x ) < ? for all X E V. Then : ( i ) For any T > o , a strict upper solution for ( 2 9 ) is given by w( x , t ) = (?6 e D ) cp1( x ;?60 , D1 ) for aU ?x E _DT (11 0 ' 1 (so that w ( x ,t ) is actually independent of t) if e(?(?60 , D1 ) - c] 0 < A < M[e ,a1(?60 ,D1) ,a2(?60 , D1)J ? ? ? ? ? ? ? ? ? ? ? ? ( 32 ) 76 where M( e ,a1 ,? ) is defined in the proof be low, and is independent of T . e[?1 (?60 ,n1 ) - c] ( ii ) If 0 < A < and if u( x ,t ) is M[e ,a1(?60 ,n1) ,a2(?60 ,D1)] ' ea2 (?60 , n1 ) a solution of ( 2 9 ) , then for aU T > 0 and x E V, u( x ,T ) < a1 (?60 , D1 ) Proof: ( I ) ( I I ) w ( x , O ) ? e > u0 ( x) for all X E V, by hypothes is B 1 . w = d0 ( x , t ) w + d1 ( x ,t ) Ow ln On = = (1 e ) {dO ( x ,t ) - ?60 } q>1 ( x ;?60 , D1 ) Cl1 '2{)0 ,D1 ( d ) . > 0 for all t > 0 and X E oV s ince do ( x ,t ) ? 60 > ?60 for all t > 0 and X E oV . .. Ow ( I I I ) L1w + c ( x , t )w - dt + Af( x ,t ,w ) = a1 (?;0 , D1 ) {- ?1(?60 , D1 ) + c( x ,t ) } cp1 ( x ;?60 , D1 ) + Af( x ,t ,w ) ? e {- ?(?60 ,D1 ) + C} + Af( x ,t ,w ) for all t > 0 and x E V , using the fact that C - ?1 (?60 ,D1 ) < 0 , and so c( x ,t ) - ?1 (?60 , D1 ) < 0 for all t > 0 and x E V . Now for t > 0 and x E V , e ? w ( x ,t ) ? ea2 (?60 , D1 ) ) ' so by a1(?60 , D1 hypothesis ( c ) we have that for all t > 0 and x E V , f ( x ,t ,w) ? M[e ,a1 (?60 , D1 ) ,? (?60 , D1 ) ] . Hence , for all t > 0 and x E V : A Ow L1w + c ( x ,t )w - dt + Af( x ,t ,w) ? e {- ?(?60 ,o1 ) + c ) + AM(e ,a1 (?60 ,o1 ) ,? (?60 , n1 ) ] ? < 0 for all t > 0 and x E V by ( 32 ) . Thus part ( i ) of the theorem is proved . 7 7 Now if A satisfies the condition of part ( ii ) , then part ( i ) holds , and it follows by Theorem 1 that i f u ( x , t ) is a solution of ( 29 ) , then u( x , t ) < w ( x , t ) for all ( x , t ) E DT . It follows that for all T > 0 and X E V, ea2 (?60 , o1 ) u( x ,T ) < w( x , T ) ? a1 (?60 , D1) Thus part ( ii ) is proved . In a s imilar fashion , one can state and prove Theorems 1 8A t o 2 0A analogous to Theorems 18 to 20 . Thus the picture built up in Theorems 12 to 20 of the behaviour of the s olution of the time -dependent problem for different classes of functions f holds not only for the special domain V but for any domain V . m We shall be discussing that picture shortly . However , it should be pointed out that both sets of theorems , A A Theorems 1 2 to 20 and Theorems 12 to 20 , are of interest . Theorems 12A to 2 0A are indeed more general as regards the domain V , but the constructive nature of Theorems 12 to 20 allows us to use these theorems to obtain , for example , e as ily calculated bounds for the critical parameter A* , as well as other quantitative informat ion should we need it . This aspect of the matter will be examined in the next chapter . Discussion : We shall now summarise and e xamine one of the most important aspects o f Theorems 1 1 to 20 and 12A to 2 0A from our point of view , namely the information they give concerning the relationship between the nature of the function f and the behaviour of the solut ion u of the t ime-dependent problem as t - m, To avoid undue complication , we shall here assume that we are dealing with the t ime-dependent problem A ou A Lu - d-t + Af( x ,u ) = 0 for ( x , t ) E D A B1 . u = 0 for ( x , t ) E S 1n u( x , O ) = u0 ( x) for x E V and its related steady-state problem Lu + Af( x ,u ) = 0 for x E ... Bl . u = 0 for X E oV 1n . . ? . ? ? . . . ( 3 3 ) ? ? ? ? ? ? ? ? . . ( 34 ) 78 We shall s uppose that ( 33 ) and ( 34 ) are such that the discussion at the end of Ch . 6 applies , so that the existence of a positive sol?tion of ( 34 ) is equivalent to the boundedness over all positive time of the solution of ( 33 ) in the case u0 = 0 . This means , in particular , that f( x ,u ) > 0 for all x E V and u ? 0 , and u0 ( x) ? 0 for all x E V. First let us review what is known about ( 34 ) from the results of Keller and Cohen , and Amann ( see Ch . 4 ) . We suppose here that ( 34 ) is such that the theory discussed in Ch . 4 applies ; this requires in particular that ( 34 ) be self-adj oint . It is convenient to consider three categories of functions f . ( 1 ) f monotone increasing and concave in u (but not asymptotically linear ) : In this case , Keller and Cohen show ( result (viii ) ) that either there e xists A* > 0 such that a positive solution of ( 34 ) exists for 0 < A < A* but not for A ? A* , or else a positive solution of ( 34 ) exists for all A > 0 . ( 2 ) ? - monotone increasing and asymptotically linear in u : I n this case , Amann has shown that there exists a finite A* > 0 such that positive solutions of ( 34 ) exist for 0 < A < A* but not for A > A* ; a posit ive solution of ( 34 ) may or may not exist for A = A* ? ( 3 ) f monotone increasing and convex in u (but not asymptotically linear ) : In this case , Keller and Cohen show ( result (vii ) ) that there exists a finite A* ? 0 such that positive solutions of ( 34 ) exist for 0 < A < A* but not for A > A* ; a positive solution of ( 34 ) may or may not exist for A = A* ? Note that , as far as is known from the results of Keller and Cohen , it is poss ible that A* = 0 , i . e . that for certain f there are no positive solutions of ( 34 ) . Now suppose that ( 33 ) is such that the theorems of the present chapter apply . In considering the relation between the nature of the function f and the behaviour of the solution of ( 33 ) as t - =, we again find three categories of functions f appearing , which while not identical to the above , clearly correspond closely to them. ( 1 )A f( x ,u ) ? M1 + M2u 1-a for some M1 > 0 , M2 > 0 , 0 < a ? 1 : For any A > 0 , the solution of ( 3 3 ) is in this case bounded above 79 A as t - ? (Theorems 20 , 20 ) . In the light of the discuss ion at the end of Ch . 6 , this means that positive solutions of ( 34 ) exist for all ? > 0 . Thus the possibility of a bounded spectrum , which was left open by Keller and Cohen , can be ruled out in this case . A partial result in this direction was in fact obtained by Keller and Cohen(19 , Theorem 4 . 2] , who showed that if lim fu( x ,u) = 0 then u- positive solutions of ( 34 ) exist for all ? > 0 . An earlier paper by Hudj aev(16] , which has not yet been mentioned , deals more fully with this point . Hudj aev ' s paper has much in common with the paper of Keller and Cohen , but he requires that the coe fficient a0 ( x) of u be zero ( us ing the notation of Ch . 4 ) , and also that f( x ,u) can be written in the form a(x ) F ( u ) . With these restrictions , he proves(16 , Theorem 2 ] that a necessary condition for positive solutions o f ( 34 ) to exist for all A > 0 is that lim inf F( u ) = 0 , while a sufficient condition is u u - Cl) lim F ( u ) = 0 . These conditions are obviously closely related to our u u-- ( 2 ) A M1u + M2 ? f( x ,u) ? w':u "1 + M ?'? 2 for some positive M1 ' M2 ' W': 1 ' M?': . 2 ' ... and f satisfies a uniform LiEschitz condition in u on anz bounded u-interval : ( i) I f A is sufficiently small , then the solution of ( 3 3 ) is bounded above as t - m ( Theorems 18 , 1 8A ) , and so ( 34 ) will have a pos itive solution . ( ii ) If A is sufficiently large , then the solution of ( 3 3 ) tends to m either as t - m or as t tends to some finite value ( Theorems 1 3 , 1 3A ) , and so ( 34 ) will have no positive solution . Thes e facts are in general accord with the results proved by Amann for asymptotically linear f , but deal with a much wider class of functions . ( 3 )A f( x ,u) ? Mu1+a for some M > 0 , a > 0 ; and f satisfies a uniform Lipschitz condition in u on anz bounded u-interval : ( i ) I f ? is sufficiently small ( depending on the initial function u0 ) , then the solution of ( 33 ) is b ounded above as t - ? ( Theorems 17 , 17A ) , and so ( 34 ) will have a positive solution . This suggests that in the ,. case of convex f discussed by Keller and Cohen , positive solutions of ( 34 ) will exist for all sufficiently small A , so that the spectrum is always non-empty . 80 ( i i ) J?'o:-:- auy i'., > 0 ? if t.10 is sufficiently large ( depending c,l ' ) ? ??he?d the solution of ( 33 ) tends to a? as t tends to some finite val11e ( T?h?c.rems A 15 , 1 5 ) . ( ii i ) For any positive u0 , if X is sufficiently large ( depending 011 u0 ) p then the solution of ( 33) tends to ? as t tends to some finite value ( Theorems 15 , 1 5A ) . .. 1+a ( iv) I f we impose the stronger condition that f( x ,u) ? M1u + M2 for some M1 > O , M2 > 0 , a > 0 ( as well as satisfying the Lipschitz condition ) , then for any non-negative u0 , if X is sufficien i.. J_y large ? the solution of ( 33) tends to m as t tends to some finite value ( Theorems A 16 , 16 ) ? Thus , for sufficiently large }. ? ( 3lf ) has no positlve solution , Thus the information obtained by studying the time-dependen?c proble-u parallels in many respects that obtained by studying the steady? ?si:< :;. e problem ; however , the study of the time-dependent problem doeE, ::;eeiJI t0 have some advantages . ( a ) The conditions which need to be imposed on the function f c-.Pc much less stringent ; there are no requirements involving differen d a1J 1 . 1 .:, .y , munotonici ty , concavity or the like , only fairly crude inequ2..L:i.. '"'(. ' ( b ) There is no requirement that the problem be self-adj oint _ ( c ) Taking the theorems of this chapter in their full generali'cy _ o, , _; can deal with t ime-dependent different ial and boundary operato;_'s nf? ?ell as a time-dependent f; indeed , one can handle oscillat ing systEms wh.:re no related steady-state problem exists . ( d ) As already mentioned , it is possib le to extract interesting quantitative data fairly easily from Theorems 12 to 20 of this chapter ; Ch . 8 deals with this point . The Effect of Reactant Consumption : In discuss ing the heat -generation problem in Ch . 1 , we assumed that there was no consumption of reactant . S ince this hardly seems a realistic assumpt ion , it is t ime we considered the e ffect of reactant consumption . One way of doing this is to suppose that , for any fixed x and u , the heat generation function f( x , t ,u) decays to zero in a suitably well-behaved manner as t - m, We obtain the following theorem : THEOREM 2 1 : Suppose that ( a ) As for Theorem 1 ?. ( b ) For al l x E V , t ? 0 and u ? o , f( x ,t ,u) ? M( u ) F( t ) , where : m ( 1 ) M( u ) is boun\ded and positive on any fini te positive u-interval ; ( 2 ) F ( t ) - 0 as t - m, F( t ) > 0 and bounded above on {t : t ? 0 } ; ( 3 ) F is differentiable for aU t ? 0 , and there exist positive constants r1 , r2 , y such that lr;(\t)) I :s: r1 for aU t ? 6 , and F( t ) ? F(Xt) :s: r2 for all t ? 0 tf 0 < ? < y . Note that these are not severe restrictions, since they are satisfied by, for example F ( t ) -kt = Ae (A > 0 , Ae 0 k > 0 ) :s: t :s: and by F( t ) { -k (t- 1 ) ( - At-k ( t > 1 ) ( c ) , ( d ) and ( e ) as for Theorem 1 7 . ( f ) and ( g ) as for Theorem 1 9 . 1 ) ( A > O , k > 0 ) . Then : ( i ) For any T > o , a strict upper solution for ( 1 5 ) is given by 8 1 e n 2 w ( x , t ) = F(OHA-'f) (A - E x . ) F ( At ) for az:l x E V , 0 :s: t :s: T i=1 ? m if A is a constant chosen so as to satisfy and if n 2n e 2 E A . i = 1 ? 'f + 1 < A < 60 2D 8 1 A > 'f + ? { 2D18 A > max 'f + ? , 'f + n 2 E A . i = 1 ? O < X < n0 ( e ,A ) n - 2 E B . c i = 1 ? if c > 0 if c = 0 n - 2 E B . } i = 1 ? c if c < 0 ? ? ? ? ? . ? ? ? ? ? ? ? . ? . ? ? . ? ? ( 35 ) where n0 ( e , A ) , apart from depending on e and A , depends on the coefficients in the operators L and B1 . , the quantities 'f and 8, and ?n the nature of the functions M and F , but not on T . ( ii ) If o < ), < n0 ( e ,A ) , and if u( x , t ) is a solution of ( 15 ) , then u( x , T ) .... 0 as T - eo, uniformly for x E V . m Proof: ( I ) and ( I I ) are similar to the proof o f Theorem 17 . ( I I I ) By hypothes is (b ) ( 2 ) , F ( ?t ) is bounded above on {t : t ? 0 ) , so w( x , t ) is bounded above on { ( x ,t ) : x E Vm , t ? 0 ) . By hypothesis ( b ) ( 1 ) , there exists a positive number N ( e ,A ) such that M (w ) :s: N( e ,A ) for all x E V , t ? 0 . m = Ow H( x ,t ,w ) Lw - - + at n -2e E a . . ( x ,t ) F O.t ) i= 1 u. F(O)(A-'i') eF( Xt ) n 1 + c( x ,t ) F(O)(A-'i') ( A- E x . ) i=1 ? + Xf( x , t ,w) n -2x . e ? + E b . ( x ,t ) F(O)(A-'i') F( Xt ) i=1 ? Xe n 2 F(o)(A-'i') (A- E x . ) F' ( Xt ) i=1 ? -2eF( Xt ) n ? ( F( O ) ( A-P) ) i?lAi + 2eF( Xt ) n eF( Xt ) n 2 ( F(O)(A-'f) ) . E Bi + c ( x ,t ) F(O)(A-f) (A- E x . ) ?= 1 i = 1 ? Xer1r< Xt )A + F(OHA-'f) + ).N ( e ,A ) F( t ) for t > 0 and x E V , m using hypothesis ( b ) n n n 2 ? F( Xt ) (F(o)(A-'i'){- 2 E A . + 2 E B . + c ( x ,t ) ( A - E x . ) } i= 1 ? i = 1 ? i=1 ? if 0 < X < y , by hypothesis ( b ) ( 3 ) . We now define the positive number n0 ( e ,A) as follows : n n e [2 E A1 - 2 E B . - CA] 82 er1? ? ! r( o ) (?:il?< e ,Alr2 } ( 36 ) n n { e [2 E A . - 2 E B . - C (A-Y) ] i= 1 ? . 1 ? } I f C < 0 , nO ( e , A ) = min y , -e....,r=- 1 -A_+_F-:-( -0 ?.,....) ?.,...A-_...,.,'f,..,..)--N-:-( e-, A,....,)""r=-2- Now if C ? 0 , then we have that for all t > 0 and x E V , if 0 < X < y : m Ow Lw - at + H ( X , t , w ) If C < 0 , then we have that for all t > 0 and x E V , if 0 < X < y : m aw Lw - dt + H ( x , t , w ) n n er1A ? F( Xt ) [F(O) 0 and x E V . Thus part ( i ) o f the theorem is proved . m Now if X satis fies ,1the condition of part ( i i ) , then part ( i ) holds , 8 3 and it follows by Theorem -? 1 that if u( x , t ) is a solution of ( 1 5 ) , then u( x ,t ) < w( x ,t ) for all x E V and 0 ? t ? T . It follows that for all m T > 0 and x E V , m eAF O ,T ) u( x ,T ) < w( x ,T ) ? F(o)(A-f) Also , by hypotheses ( f ) and ( g ) , it follows from Theorem 1 1 that u( x ,T ) ? 0 for all T > 0 and x E V . Since , by hypothesis ( b ) ( 2 ) , F( ?T) - 0 as T - ?, part ( ii ) m is proved . Note : As in the case o f Theorems 12 to 20 , one can state and prove Theorem 2 1A , analogous to Theorem 2 1 but applying to an arbitrary domain V . Thus , i f we allow for reactant consumption , then part ( ii ) of the theorem shows that , in terms of the criterion for thermal explosion that we are working with at present , thermal explosion cannot occur . Since thermal explosions undoubtedly do occur , there appears to be something wrong with our criterion . However , a recent paper by Sattinger[ 3 3] sheds some light on this matter . Sattinger discusses a model of combustion with reactant consumption which involves two simultaneous partial differential equations , and so is rather more complex than the one we have used . However , he reaches the same conclusion - that the solution u( x ,t ) t ends to zero as t - ?, regardless of the initial condition , at least for the particular system he is dealing with . He then points out that whether or not thermal explosion takes place depends , not on the final state reached by the system , but on the manner in which that state is attained . I f the system is initially in what he calls a "subcrit ical" state , the combustion proceeds very slowly , with the temperature reaching an almost steady value , which it holds for a considerable time before ultimately falling to the ambient value . Alternatively , the react ion may aft er a certain time begin to proceed very quickly , with a rapid rise to a very high temperature , and this is what in practice constitutes an explosion . The fact that , in this case also , the temperature will in theory ultimately fall to the ambient value , is irrelevant , s ince the explosion will already have taken place at some finite time . It therefore appears that , if reactant consumption is to be taken into account , new criteria for thermal explosion must be used , relating to the e arly behaviour of a reacting system rather than to its final state . However , as Sattinger shows for the particular problem discussed in his paper , the early behaviour of the system when reactant consumption 84 is taken into account is related to the existence of steady-state s olutions for the equation in the form we have treated it , ignoring reactant consumption . Thus it may well be that , in many cases at least , . there are two equivalent ways of deciding whether or not explosion takes p lace - one may ignore reactant consumption and adopt one or the other of the ( equivalent ) criteria used in this thesis , or one may take reactant consumption into account and adopt Sattinger ' s criterion . The two approaches may give the same conclusion even though the latter is based on a model much closer to the actual s ituation than is the former . It must be said at once that there is a further complication . The problem dealt with by Sattinger corresponds to the problem without c reactant consumption in which f ( x ,u ) = c1exp ( - u+? ) , the Arrhenius a formula ( see Ch . 1 ) . Our Theorems 2 0 , 2 0A apply to this function , and show that the solution u( x ,t ) of problem ( 15 ) is in this case always bounded as t - ?, so that , in terms of our criteria , explosion will never take place . But Sattinger shows that it can take place in terms of his criterion , if A is sufficiently large , and it is apparent from ,. his discussion that , at least for this part icular f , neither of the criteria adopted in this thesis is appropriate ( this possibility was also suggested in a private communication by Dr . G . C . Wake ) . c2 For the case where f ( x ,u) = c1exp ( - u+T ) , there are positive a steady-state solut ions for all A > 0 , as is shown by our Theorems 2 0 , 20A and 10 . I f one analyses the s ituation more deeply , as has been done , for example , by Parter[2 7] , one finds that there exist two finite values A1 , A2 > 0 such that the steady-state problem has one positive solution for 0 < A < A1 , two for A = A1 , three for X1 < X < A2 , two for A = A2 and one for A > A2 , as illustrated : l luA I Ic (V) = suE_ j u( X ;x ) I where u( A ;x ) is a xEV steady-state solution 8 5 A s ? passes through the value ?2 , the number o f positive steady-state solutions changes from three to one , and (more important , perhaps ) the s ize of the minimal positive solution increases by an abrupt j ump . Sattinger shows that it is the value ?2 which is critical in the sense that for ? > ?2 , thermal explos ion takes place in terms of his criterion , taking reactant consumption into account . Thus the orthodox criteria for thermal explosion , which we use in this thesis and which are used by many other authors , seem not to apply to the Arrhenius funct ion , at any rat e , though they appear to work well enough if one uses instead the Frank-Kamenetskii approximation for which f ( x , u ) = eu , as is done , for example , by Boddington , Gray and Harvey[4] ( the use of the Frank-Kamenetskii approximation is commented on in Ch . 1 of this thesis ) . There is evidently plenty of scope for more detailed investigat ion of the steady-state problem for different functions f , looking not merely at whether or not posit ive solutions exist but at the number and s ize of such solutions for different values of A . The problems involved seem likely to be difficult - so far only some simple special cases have been stud?ed . In this connection , the theorems we have proved for the t ime -dependent problem , giving constructive bounds for the solution , may prove helpful in determining the size of possible steady-state solutions , though it is equally likely that these bounds will turn out to be too crude to be useful for this purpose . 8 BOUNDS FOR THE CRITICAL PARAMETER 86 In this chapter , we shall apply the results obtained in the previous chapter to the problem of finding bounds for the critical parameter ?* ( as defined at the end of Ch . 6 ) . The steady-state theory reviewed in Ch . 4 gives us upper and lower bounds for ?* in certain cases , these bounds generally involving the principal eigenvalue of some related linear problem. Also , Wake and Rayner(36] have recently developed a variational method for estimating ?* , again working with the steady-state problem. The theorems we have proved in Ch . 7 provide another means of obtaining rigorous bounds for ?* . As compared to the steady-state methods , our method has the advantage that it gives bounds for ?* which are easily computable by elementary methods . We shall shortly illustrate this by calculating these bounds in the cases of two important functions f . However , these bounds have no pretensions to being highly accurate estimates ; some idea of their closeness to the exact value of ?* will be obtained later in this chapter by comparing them with the results obtained by Boddington , Gray and Harvey(4] using an empirical formula for ?* . Preliminaries : Consider first the original heat-generation problem described in Ch . l . Using the notation defined in that chapter , this problem is of the form for ( x ,t ) E D oT K av + Hg2 ( T ) = 0 for ( x ,t ) E S T ( x ,O ) = T for X e ? V. a We shall assume that the boundary condition is linear , corresponding to heat loss following Newton ' s law of cooling , so that g2 ( T ) = T -Ta . If we divide the differential equation by K and change to a new t ime scale , which we may do without loss of generality , the problem reduces to 3 o2T oT 1 I: -2 - a:t + K gl ( T ) = 0 for ( x ,t ) e o i=l ox . ? K oT + H ( T-Ta) = 0 for ( x ,t ) e s dv T( x , O ) = T for x E V. a tl 87 We shall examine two important possibilities for the function g1 ? 1 ) The Arrhenius formula g1 (T ) = qpA exp( - ?) , where q is t?e exothermicity per unit mass of the reactant , p is again the density of the reactant , A is a proportionality constant , E is the activation energy of the reaction and R is the universal gas constant . We follow here the study by Boddington , Gray and Harvey(?] ; they use the common procedure of replacing the Arrhenius function by the Frank-Kamenetskii approximation , and so we shall do the same . The difficulties associated with the use of the Arrhenius function itself were discussed at the end of the previous chapter . E (T-T ) a Accordingly we make the change of variable u = ----?- RT2 exp ( - .f..) = RT E exp ( - it? T a E RT u 1':$ exp ( - ?1 - _a_} ) RT E a E u = exp ( - RT )e ? a a RT u ? f a ? 11 ? --E- ?s sma so that This is the Frank-Kamenetskii approximation , and using this we obtain finally the initial-boundary value problem 3 o2u ou E E u E - - ? + -- qpA exp ( - -)e = 0 for ( x ,t ) E D i=1 ox? ? KRT2 RTa ? a Hu + K ou = 0 for ( x ,t ) E S 0\1 u( x , O ) = 0 for x E V . We then write A = qp? exp( - Ri ) , this being the same as the parameter KRT a a y in the notation of Boddington , Gray and Harvey(?] . 2 ) The modified Arrhen?us formula g1 ( T ) = qpAT exp ( - ;T r ' the constant A being not necessarily the same as in the previous case . As mentioned in Ch . 1 , recent theory suggests that reaction rates may well be governed by formulae of this kind rather than by the original Arrhenius formula . Making the same change of variable as before gives : T exp ( - ;T ) = ( T a RT?u + -E-) exp (----..;;;.E--=---::--) R {T + RT2E-1u } a a 2 RTa E E 2 1 = T< u + RT )exp ( - ( RT ) ( E ) ) a a u + ? a Ta ?2 = -f 0 i=1 ? ox? ? m ? 3 2m . -1 ( 37 ) Hu + K I a. ( x) x . ? ou - 0 for X E oV ' t > 0 i=1 ? ? < u + ?)exp(u-; ?) = 0 ( u > -?)} , the modified Arrhenius ( u :c -?) form . Note that this function is non-decreasing and asymptotically linear in u, and also satisfies a uniform Lipschitz condition for all u , since its derivative is bounded . Since the values o f u that we work with are always positive , we may define f2 ( u) as we please for u ? -? , 8 9 so we define i t in such a way as to make f2 suitably well-behaved . Bounds for the Critical Value ?* : Let u( x ,t ) be the s olution of ( 37 ) . For our present purposes , we shall define the critical value ?* to be the value of ? below which u( x ,t ) is bounded as t - ?, and above which u( x ,t ) is unbounded as t - ? or as t tends to some finite value . As shown in Ch . 6 , this i s the same as the steady-state critical value above which no positive steady-state solutions exist . ( a ) f(u) ? f1 ( u ) : For the special problem ( 3 7 ) , Theorem 17 t ells us that the solution u( x ,t ) is bounded as t - ? if 0 < ? < 3 3 2? I: A . i = l ? (A-'f)M(e ,A ,'f) Thus 2e I: A . i = l ? ( A-'f)M( e , A , '?) is a lower bound for :\?': , for any choice of ? > 0 . Now M ( e , A , '?' ) may be taken as the least upper bound of f ( u ) on the interval eA Thus , for f ( u) - f1 ( u) , we have M ( e ,A , '?' ) = e A7f Thus we have that for any choice of ? > 0 , the following is a lower bound for A.* : say . From the graph , it is clear that the best lower bound for X* , namely cl 1 ?? will be obtained by choosing ? = c-? which we may do . We obtain 2 2 cl 2 3 therefore as a lower bound for A.* the value Ae t A . . c2e i = l ? Now ( i ) of Theorem 17 tells us that A may be arbitrarily ?hosen 90 greater than 'f + 2?9 = 'f + ?(3 where h = ? ? Since A may be chosen as close to this value as we please , it follows that - a lower boUnd for X* is given by : 3 X* = 2 I: A . i= 1 ? e ( 'f + 28) h ? ? ? . ? ? ? ? ? . ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ( 38 ) 2 Further , s ince f1 ( u ) = e u ? 1 + ? for all u ? 0 , we can apply the Corollary of Theorem 16 to ( 37 ) in the case f(u) = f1 ( u) . It follows from this Corollary that , provided X is sufficiently large , u( x ,t ) is unbounded as t tends to some finite value . This allows us to obtain an upper bound for X* in the case f( u) = f1 ( u) . In fact , the Corollary 29 tells us that , as long as $ + 11 > 'f , an upper bound for X* is given by 3 E A . [? , + ? - 'f) 2 + 1 ] i=1 ? 2 h ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ( 39 ) ( b ) f ( u ) - f2 ( u) : W? have that f2 ( 0 ) = Ta . As u - ?, f2 ( u) is T ef! T e f! a a asymptotic to (-?-) ( u+s ) , and f2 ( u) < (-?-) ( u+s) for all u ? o . 2 2 T e? f 2 - L - ?..} Further , for u ? o , f; ( u) = (?)l? e u+s + e u+? . T ef! a - -- as f! U ... m, Also , for u ? O , T a ? - lf2 T e ? 4- ? = (-a-) e; e u+? > o . ? ( u+? ) 3 - - - - - - - - - - - - - -- - - - - - - - - - - - T a --+-------?c?l ________________________________________________ ? u (\ Thus , for all u ? 0 , Ta( 1+? ) Tae ? Ta + ? u ? f2 ( u) ? (-?-) ( u+? ) . It follows from Theorem 18 that a lower bound for ?* is given in this case by v: = 3 2? E A . i=1 l. , 2El ? T e ":> ( 'i' + -) a h 29 It also follows from Theorem 1 3 that , as long as * + 1l > Y, an upper bound for ?* is g iven in this case by * + -w-} , T ? 'i' ) a It will be seen from these illustrat ions that the calculations involved in determining ?* and f* are quite simple . One could readily perform s imilar calculations for other forms of the function f( u ) if desired . Comparison with KnowT. Values : 9 1 While we do not know in general how close the bounds ?* and f* are to the true value of )J: , it is poss ible to get some feeling for this by comparing these bounds with the known value of A* in certain special cases . Us ing the Frank-Kamenetskii approximation in the original heat ? generat ion problem , Boddington , Gray and Harvey[4) have obtained an empirical formula for )J: which appears to agree well with all known information ; we shall denote the value obtained using the ir formula by \ ... 11. " est ' Values of A* t for various special regions V are given in Table 1 es on p . 9 2 , us ing the notation of ?oddington , Gray and Harvey . We shall first compare , for each of the special regions in Table 1 , our lower bound A* with A* ? ? the size of the upper bound f* relat ive to A* - est ' est will be invest igated at the end of the chapter . Apart from giving some feel for the s ize of A* and f* , our calculations will also serve to illustrate the technique ment ioned in Ch . 7 ( p . 72 ) of transforming from a region V* to the region V for which Theorems 12 to 20 hold , and for m which formulas ( 3 8 ) and ( 39 ) for ?* and A* were calculated . ( eontinu?d on p . 93 ) REGION j F ( j ) Sphere 2 1 . 1 1 1 ( radius = a ) Infinite cylinder _ 1 1 . 000 "' ( radius = a ) Infinite slab 0 0 . 857 ( thickness = 2a ) Equicylinder (heig.ht = 2a) 2 . 72 8 1 . 178 rad1us = a Thin circular disc (thickness = 2? 0 . 4 37 0 . 9243 radius = 10a Long circular cylinder (heig?t = 1 0a) 1 . 41 8 1 . 0 50 rad1us = a Cube 3 . 2 80 1 . 222 ( s ide = 2a ) Infinite square rod 1 . 44 3 1 . 0 5 1 ( s ide = 2a ) ?EMENO\ RADIUS Rs a 3a T 3a a Sa 2 15a 1 1 a 3a 2 RECIPROCAL SQUARE MEAN RADIUS -2 Ro 1 2 a 2 3a2 1 3a2 1 + fl - 2 . 4142 3a2 3a2 ? 1 + 0 . 02 ] = 1 . 0020 pa2 l'i(rr 3a2 ? .l.. + 2JH.) = 2 . 0012 ?a2 2 5 26 3a2 1 + 2 ? 2 . 1027 n = 3a2 3a2 2 1 + - 1 . 6 366 n - 3a2 3a2 1 R2 [ 1 ? 1 ) ) h . . A* = O 3F( j ) + j + 1 Bi w ere B1 = hRS 1s the est B . mb 1ot nu er 2 [ 1 ? .l..) ] = 2 [0 . 9061 + ( 0 . 3000 ) BiJ a 3 . 333 + 3 Bi a B1 3a\.!. ? .l..) ] = 2 [ 2 . 0387 + ( 0 . 500 0 ) B i] 2 3 + 2 Bi a B1 3 2 [ 1 ? .l..) ] = 2 [ 8 . 1 548 + ( 1 . 1 669 )BiJ a 2 . 571 + 1 Bi a Bi ' 3a2 [ 1 + e ( .l_) ) = a2 [0 . 906 1 + ( 0 . 35 16 )Bi] 2 . 4142 3 . 5 34 3 . 72 8 Bi Bi 3a2 [ 1 + e c.l..) ] = a2 [ 5 . 66 36 + ( 1 . 0797 )Bi] 1 . 0020 2 . 773 1 . 437 Bi Bi 3a2 [ 1 + e ( .l_) ) = a2 ( 1 . 68 5 3 + ( 0 . 4759 ) B i) 2 . 00 1 2 3 . 1 5 0 2 . 4 1 8 Bi Bi 3a2 [ 1 + e ( _.;._) ) = a2 [0 . 9061 + <.o . 3892 )Bi] 2 . 1 027 - 3 . 666 4 . 2 80 B1 B1 3a2 [ 1 + e (__!_) ) = a2[2 . 0 396 + ( 0 . 5 8 14 )Bi) 1 . 66 36 3 . 1 5 3 2 . 443 Bi Bi TABLE 1 : Values of Al': for various regions . est ' lO N ( eontinu?d 6?om p . 9 1 1 We begin by working with the heat-generation problem 3 o2u r 2 i=1 oy . ? ou + ot )..e u = 0 for y E V* , t > 0 9 3 Hu ou + K dv = 0 for y E ov?': , t > 0 ? ? ? ? ( 40 ) u( y , O ) = 0 for y E v?': y1 2m1 y2 2m2 y3 2m3 where Vi: = {y : (-) + (-) + (-) < 1 } ; this is the t ime-a1 a2 a3 dependent version of the problem considered by Boddington , Gray and Harvey , apart from our choice of the region V* . Following the remarks y . on p . 72 , we make the change of coordinates x . = -2:.. ( i = 1 , 2 , 3 ) . This ? a . transforms V* into ? the region V with dimension n = 3 . m 2 2 We then have : ou - 1 ou cy. - a:- ox:-' o u = .1._ o u ( i 2 2 2 = 1 , 2 , 3 ) . Also , v(y ) = = = for the region ? ? ? 3 oy . a . ox . ? ? ? ou ou v?: r: vi ( y ) where : ' 0\) = dY-' i= 1 ? outward uni-: normal to V'"''? 1 ( 2m1 y1 2m1- 1 --,?---,?---.... ? a-) , 4m . y . 4m . - 2 1 1 --i:-< -2:..) ? 2 a . a . ? ? ou Thus , changing coordinates , dv transforms into (\ 3 r i=1 2m . -1 2m . x . ? ? ? a . ? 4m . -2 l. 1 ou a:- ? ? ? = 3 2m . - 1 :::. ? uU r a . ( x ) x . ? . 1 1 1 ox . 1= 1 where a . ( x ) = 1 2 a . 1 4m . -2 1 Results concerning as for the problem f( u ) :: eu . A* for the problem ( 40 ) will therefore be the same ( 37 ) with Ai = ? ( i = 1 , 2 , 3 ) , ai( x) as above and a . ? 94 The easiest cases to deal with are those for which 3 2 m . = 1 ( i = 1 , 2 , 3 ) . ? In that case V is the spherical region { x : E x . < 1 ) , m i=1 ? and so , in the notat ion of Ch . 7 , * = Y = 1 . Further , 9 and 8 are the extreme values 3 2 flxi of 1: a. ( x ) x . = 1: 2 i= 1 ? ? i= 1 a . on oV m It is eas ily shown , using Lagrange ? multiplier techniques , that 9 = rnaxla . } ' e = ? formula ( 3 8 ) , we obtain : e ( 1 3 1 2 E 2 i= 1 a . ? 2 + m in {a . }h ) ? 1 Thus , us ing rnin {a . } ? We now apply this to cases where V* is one of the first three regions listed in Table 1 . 1 ) . Sphere , radius a : Here a1 = a2 = a3 = a , and so number for the sphere . i . e . From Table 1 : Thus x?': est/ V: A:': = 6 2 a 2 e ( 1 + ah ) 6 Bi = where Bi = ah is the B iot a2e ( 2 + Bi ) A:': = ' ?'? 11." est A?'? e st = ? = 1 B ' ?0 . 90 6 1 + C0 . 45 30)Bi] . a 1 B . ?0 . 9 0 6 1 + (o . 3ooo ) Bi] . a 0 . 90 6 1 + ( 0 . 45 30 ) Bi 0 . 9061 + (0 . 3000)Bi ' 1 . 5 1 - - - - - - - - - - - - - - - - - - - - - - 1 Sphere --?--------?----?--?------?--------?--------?-----------+Bi 1 2 3 4 5 2 ) . Infinite cylinder , radius a : Here we take a1 = a2 = a and let a3 ? ?, and so 4 2 a e ( 1 + a?) 4 B i = Where Bi = 3ah ? the B iot -2- l.S number for the From Table 1 : ' ?'? 11." est/).?': 1 . 36 1 infinite cylinder . i . e . Thus ).)': .. )._?'? est ' ?'? 11." est = = ? = 1 B ' -1i2 . 0 387 (o . 6 796)B) ? + a 1 B ' -1i2 . 0 387 ?0 . 5000 ) Bi) . + a 2 . 0 387 + ( 0 . 6 796 ) Bi 2 . 0 387 + (o.5ooo)Bi ? Infinite cylinder 1 2 3 4 5 3 ) . Infinite slab , thickness 2a : Here we take a1 = a and let a2 - ?, a3 - ?, and so ).?'; = 2 2 a 2 e ( 1 + ah ) = 2 B i where Bi = 3ah is the Biot number for the infinite slab . i . e . From Table 1 : Thus x??: = )._i; = est )._)': est --p: = __!_r Bi ) ' ?8 . 1548 + (1 . 3591)Bi . a 1 B ' ?8 . 1548 + (1 . 1669)Bi) . a 8 . 1548 + ( 1 . 35 9 1 )Bi 8 . 1 548 + (1 . 1669)Bi 0 3 5 1 . 16 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 1 Infinite s lab --+--------+--------+--------+--------+-------?--------? B i 1 2 3 4 5 9 6 The next cases we wish t o consider are those where V* i s one of the three finite cylindrical regions . To obtain these we take a2 = a1 , m1 = m2 = 1 and let m3 - =. In the limit , as m3 - =, Vm becomes the right circular cylindrical region fx : 2 2 < 1 , l x3 l < 1 } . x1 + x2 ? = 1 , 'f = 2 . Further , on av : m 2 2 2m3 x1 x2 m3 x3 - + - + 3 2m . 2 2 2 I: a . ( x ) x . ? a1 a1 a3 = i=1 ? ? 2 2 4m -3 x2 m3 x3 - + 2 2 a1 a 3 So in the limit , as m3 - =, we have : 3 2mi 1 2 I: a . ( x ) x . = where x? + x2 = 1 , l x3 1 < 1 i = 1 ? ? a1 = where I x3 1 = 1 . Thus Thus 9 = min {.1..., a1 1 and s imilarly El = min{a1 ,a3} So , using formula ( 38 ) , we obtain : 2 (2. + .1...) 2 2 a1 a3 We now apply this to the three finite cylindrical regions listed in Table 1 . 4 ) . Equicylinder , height 2 a , radius a : Here a3 = a1 = a , and so A* = = 6 2 a 2 e ( 2 + ah ) 3 Bi where Bi = ah is the Biot number for the equicylinder . i . e . From Table 1 : Thus 2 . 5 8 1 1 B . A* = ?0 . 90 6 1 + Co . 906 1)Bi] . v? est ' ?'? 11." est a 1 B . = ?0 . 9061 + ?0 . 3516 )Bi) . a A?': = 0 . 9061 + ( 0 . 90 6 1 )Bi 0 . 9061 + (0 . 3 5 1 6 ) B i . Equicylinder --+-------?------??------?-------+--------?--------? B i 1 2 3 4 5 5 ) . Thin circular disc , thickness 2a , radius 10a : Here a1 = 10a , a3 = a , and so = 2 ( 2 1ooi e ( 2 + 1 + -) 2 a 2.) ah ( 2 . 04 ) B i where Bi = 5?h is the Biot number for the thin circular disc . i . e . Ai: = From Table 1 : A ... est = A1? Thus est )J? = 1 ' -?6 . 66 2 5 a 1 -?:5 . 66 36 a 6 . 662 5 + 5 . 66 36 + (\ B . (2 . 665 0 ) BiJ . + B . C1 . 079 7)Bi] . + ( 2 . 6 650 )Bi ( 1 . 0 79 7 ) Bi 97 2 . 47 1 . 18 x?t: . e s t / )J: 1 2 Thin circular disc Bi 3 4 5 6 ) . Lon? circular czlinder , height lOa , radius a : Here a1 = a , a3 = Sa , and so = 2 (2. + _1_) a2 2 Sa2 2 e ( 2 + ah ) ( 2 2 . 44 ) Bi where Bi a2e ( 1 5 + 11 Bi ) 1 5ah . the = -u 15 Biot number for the long circular cylinder . i . e . From Table 1 : Thus "-:': est/ Al': 2 . 80 1 . 0 8 x?': = "-?'? = est \ ... /\est "-:': = 1 B ' ? 1 . 8 170 + t1 . 3 32 5 ) Bi] . a 1 B . ? 1 . 6 8 5 3 + C0 . 475 9)Bi] . a 1 . 8 170 + ( 1 . 3 32 5 ) B i 1 . 6 8 5 3 + ( 0 . 47 5 9 ) Bi Long circular cylinder --?-------+--------+-------?--------?--------?--------? B i 5 1 2 3 4 9 2 9 9 Finally , we wish to consider the cases where V* is one of the last two regions listed in Table 1 , namely the cube and the infinite square rod . To obtain these we take m1 = m2 = m3 = m and let m - =. The region V* then tends to the rectangular prism with dimensions 2a1 X 2a2 X 2a3 , and Vm tends to the cubical region {x : l xi l < 1 , i = 1 , 2 , 3 ) . Thus ? = 1 , l = 3 . Further , on oV : m 3 2m . ? L a . ( x) x . = . 1 ? ? ?= 3 1: i = 1 2m x . ? 2 a . ? So in the limit , as m - =, we have : 3 2m . ? l: a.. ( x ) x . = i = 1 ? ? I: ( .2...) i such that a? l x . l = 1 ? ? Thus 9 = 1 and e max{a . ) =t:i ? So , us ing formula ( 3 8 ) , we obtain : ? A-;': = e( 3 3 2 E 1 2 i = 1 a . ? We now apply this to the last two regions listed in Table 1 . 7 ) . Cube , s ide 2a : Here a = a 1 2 = a3 = a , and so 6 >..?': = = number for the cube . From Table 1 : i . e . Thus l\ )._1: = est >.,?'? est ---rr = 2 a e ( 3 + ij!) 6 Bi where Bi = ah is the B iot a2e ( 2;-:J + 3 B i ) 1 B . ?1 . 5 6 94 + (1 . 35 9 1 ) B i] . a 1 B " ?0 . 90 6 1 + (o . 3B92)Bi] . a 1 . 5 6 94 + ( 1 . 3 59 1 ) B i 0 . 9 061 + ( 0 . 3 892 ) B i .. . 100 A?'; est/ X* 3 . 49 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 1 . 73 Cube ??-------+--------+--------+--------+-------?---------4 B i 1 2 3 4 5 8 ) . Infinite square rod, s ide 2a : Here we take a1 = a2 = a and let a3 - ?, and so 4 = 2 a 2/T. e ( 3 + ah ) 4 Bi where Bi the Biot number for the infinite square rod . 1 Bi i . e . ).,?'? = -?2 . 8 831 + ( 2 . 0 3 8 7 ) Bi) . .. a 1 B ' ).,:'{ = ?2 . 0 396 + (o . 58 14)Bi) . est From Table 1 : a X-'? Thus est 2 . 8 8 3 1 + ( 2 . 0 38 7 ) Bi --... - = 2 . 0 396 + (0 . 5 814)Bi A" 3 . 5 1 1 . 41 Infin it e square rod 1 2 3 4 5 3ah . = -- lS 2 Bi 1 'J 1 From . the foregoing calculatio-ns , it can be seen that in the cases V considered , the ratio ??t lies between 1 and about } in all cases , and is much less than this in some cases . It is clear that in general ?* cannot be regarded as in any sense an approximation to A* t ; nevertheless es it may be of interest as being a rigorous lower bound for A* which is not too remote from the true value , particularly for small values of the Biot number and certain types of region . The upper bound given by ( 39 ) , though of theoretical interest s ince it shows that A* is finite , is not as useful for estimat ion purposes as the lower bound ?* already discussed . We can illustrate this by cons idering the case where V* is a sphere of radius a ( see p . 9 4 ) . In 1 1 this case * = ? = 1 , 8 = a and A1 = A2 = A 3 = :2' and so we obtain : r}? = = ?-2- + 2 2h2 a a 4 a2h2 1] ?2 + ( B i ) 2] 4a a where B i = ah is the B iot number for the sphere . Table 1 , we obtain : Thus , using the appropriate value of A* t from es A?': v- = est 3[2 + ( Bi ) 2](0 . 90 6 1 + ( 0 . 3 ) B i] 4 B i = J11 ???2 2 + 0 . 6 + ( 0 . 9 06 1 ) Bi + ( 0 . 3 ) ( Bi ) 2 ] = ? g( Bi ) , say . Evidently g ( B i ) - ? as Bi - 0+ or as B i - m, Further : g ' ( B i ) = - 1 ? 81 2 2 + 0 . 906 1 + ( 0 . 6 )Bi ( B i ) 2 = 0 when Bi = 1 . 0799 ( to four places ) , the solut ion being obtained using the Newton-Raphson method . Thus g ( B i ) attains its minimum for Bi > 0 when Bi = 1 . 0 799 , whence g ( B i ) ? 3 . 6 for all Bi > 0 . r}?, Hence ? ? 2 . 7 for all Bi > 0 . Thus the upper bound r* is typically est very much larger than A* , which is why it is of theoret ical rather than practical interest . (\ 9 NON-LINEAR BOUNDARY CONDITIONS 102 In this final chapter , we shall examine to what extent the methods of Chs . 7 and 8 can be adapted to non-linear boundary conditions . We shall begin by working with the initial-boundary value problem Lu - ou Af( x , t ,u) 0 for E V 0 < t :S: T rt + = X m ' B u = 0 for x e ov , 0 < t :S: T ( 41 ) gen m u( x , O ) = uo ( x) for x E V m where , as in the problem ( 1 5 ) discussed in Ch . 7 , f is continuous for X E vm ' 0 :S: t :S: T and all u , uo E c 2+a(Vm ) and A is taken to be positive . ou Recall that Bgenu = d0 ( x ,t ) g( u) + d 1( x ,t ) an' where g( u ) is strict ly increas ing for all u ( for those functions g which occur in physical problems , we are only concerned with u ? 0 , and we may extend the definition of g (u ) to negat ive u so as to sat isfy this condition without loss of applicability ) . As in the case of problem ( 1 5 ) in Ch . 7 , we shall suppose that the derivative ? appearing in B u is of the form cm gen ou - an- - n 2m . - 1 ou L a. ( x ) x . l ox . , and shall likewise follow in other respects the i = 1 l l l notation used in studying problem ( 1 5 ) ( see p . 5 3 ) . I f we assume further that g( O ) :S: 0 (which is certainly true in physical applications ) then Theorem 1 1 extends at once to problem ( 41 ) . Theorems 17 to 2 0 , on the construction of upper solut ions , and also Theorem 2 1 on reactant consumpt ion , also extend at once to problem ( 41 ) if we assume that g ( u) ? u for all u ? 0 ; however , this is certainly not true of the funct ion g( u) = u5 14 which occurs when cooling at the boundary i s by natural convect ion . However , ?a s lightly different condition on g takes account of the case g ( u ) = u5 14 and still allows us to extend the important Theorems 17 , 18 and 2 0 , where the upper solut ion constructe d is independent of t . Theorems 19 and 2 1 have so far proved impossible to extend using this condition on g , because the fact that the upper solution tends to zero as t ? ? creates technical difficulties of an apparently insuperable nature in the construction of the proof . However , extending Theorems 17 , 1 8 and 2 0 leads to the following theorems . . B THEOREM 17 : Suppose that 1 0 3 ( a ) As for Theorem 1 ? except that we suppose c ( x ,t ) s 0 for all x E Vm ' t > o . ( b ) , ( c ) As for Theorem 1 ? . ( d ) n n E B . < E A . ? i=1 l. i=1 l. ( e ) As for Theorem 1 ? . ( f ) There exist constants N > 0 and p > 1 such that g (u ) ? NuP for a l l u ? o . Then : ( i ) For any T > 0 , a s trict upper so lution for ( 41 ) is given by e n 2 w( x , t ) = y;::r ( A - E x . ) for aU x E V , 0 s t s T ? i= 1 l. m if A is a constant chosen so as to satisfy 2D18 A > 'f + ----'7" No e P-1 0 ? . ? ? . . . . ? ? ? . . . ? . . ? . ( 42 ) and if o < A < m0 ( e ,A ) where m0 ( e ,A ) does not depend on T . ( ii ) If 0 < A < m0 ( e ,A) , and if u( x ,t ) is a solution of ( 4 1 ) , eA then for al l T > 0 and X E vm , u( x ,T ) s A=Y . Proof: ( I ) As for Theorem 1 7 . ( I I ) B w gen n 2m. - 1 = d0 ( x ,t ) g ( w ) + d1 ( x ,t ) E a . ( x ) x . i= 1 l. l. l. n n 2m . - 1 -2ex . e p 2 p 1. 1. ? d0 ( x , t ) N ( A_f) ( A- L x . ) + d1 ( x , t ) E a . ( x ) x . ( A-'f ) i=1 l. i = 1 l. l. for t > 0 and X E oV m > 0 for all t > 0 by ( 42 ) . ( I I I ) As for the case of Theorem 17 where C = 0 . THEOREM 18B : This is to Theorem 1 tB as Theore? 18 is to Theorem 1 ? . The conclusion of the theorem is : If u ( x , t ) is a so lution of ( 41 ) , and if A satisfies o < A < n n 2 E A . - 2 E B . i = 1 l. i= 1 l. M1'f (\ 104 then there exists a aanstant K > 0 such that, for aU T > 0 and x E ?, u( x , t ) :!i: K , where K depends on A. COROLLARY : Goes through exactly as it does for Theorem 18 . THEOREM 20B : This is to Theorem 1 ?B as Theorem 20 is to Theorem 1 ??. The conclusion of the theorem is analogous to that of Theorem 20 in the case C = 0 , the only change being that the condition that A must satisfy becomes A > y + r. 2D? !2.:? 1 1 /p . . . . . . . . . . . . . ( 4 3 ) l6oNKp- 1 ).. a M2 a j Proof: Similar to the case C = 0 of Theorem 2 0 . ( I I ) is slightly modified : B w gen ox . ? ? 6 NKp)..p/?1p/a( A-'i')p - 2D K).. 11?11Cl.e for t > 0 and X E oV 0 2 1 2 m > 0 by ( 43 ) . Extending the "lower solution" theorems , Theorems 12 to 16 , to problem ( 4 1 ) poses far more problems . Certainly the extension is immediate if we assume g ( u ) :!i: u for all u ? 0 , but unfortunately this condition is not satisfied by the non-linear funct ions g which arise in applications . If we require g to sat isfy some more realistic condition , then we immediately run into technical difficulties unless W = 'i' , i . e . unless the region V is spherical . So let us take V to be the sphere m m {x : n 2 r x . < 1 } , so that m . = 1 for i = 1 , 2 , . . . n , and w = ? = 1 . i=1 ? ? In that case we obtain the following extension of Theorem 12 . THEOREM 1 2B : Hypotheses ( a ) , ( b ) , ( c ) and ( d ) are as for Theorem 12, 2 6 9 except that the condition 'i' < w + + is omitted in ( c ) . In addition, 0 we suppose that n 2 ( e ) V {s the sphere {x : E x . < 1 } . m i= 1 ? ( f) There exists a constant Q > 0 such that g ( u ) :!i: Queu for al l u ? 0 (this condition is satisfied by the functions g which arise in app l.ications ) . Then : ( i ) For any T > 0 , a lower solution for ( 4 1 ) ?s given by tl n 2 -t w ( x ,t ) = )J<(A - I: x. ) ( 1 - e ) foro aU x E V , 0 :S: t :S: T i=1 1 m where A and K arae constants chosen so as to satisfy A > 1 2 6 9 ' ( A- 1 ) eAK( A-1 ) < QD 1 0 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ( 44) 0 < K < M n n 2 I: A . t 2 I: B . + ( C+ 1 ) A i = 1 1 . i=1 1 105 (note that by choosing A sufficiently close to 1 , ( 44) can always be satisfied; the choice of A wi ll depend upon the value of A? but this is of no consequence foro this theoroem) . ( ii ) If u ( x , t ) is a solution of ( 41 ) , then foro any T > 0 , u( x ,t ) > 0 foro 0 < t :S: T , and if lim u ( x , T ) = u( x) exists? then foro all X E V , T-= m u( x ) : 0 by ( 44 ) . for t > 0 and X E oV m ( I I I ) As for Theorem 1 2 . This proves part ( i ) . Part ( ii ) follows as in the proof of Theorem 1 2 , if we observe that A-1 may be chosen arbitrarily close to a . Unfortunately , theorems parallel to Theorems 1 3 to 16 cannot be obtained by this method , because in each case a condition on A is obtained which involves the value of A , which in turn depends on A , and so a vicious circle results . 1 0 6 While the preceding theorems are rather incomplete , they do at least show that certain qualitative aspects of the behaviour of solutions of ( 15 ) still hold for ( 41 ) . It may well be that in other respects there are qualitative changes in behaviour when we change from a linear to a non-linear boundary condition . The extension to more general domains V which was carried out , in the case of the linear boundary condition , in Theorems 12A to 20A , has not proved possible so far in the case of the non-linear boundary condition . The difficulty is the unavailability of a theory for non? linear operators comparable to the theory for linear operators which was used in proving Theorems 12A to 20A . In Ch . 8 we used Theorem 1 7 to obtain a lower bound X* for the critical value X* in the heat -generat ion problem , which turned out to be quite close to X* in certain cases where a good approximation to the value of X* was known . It is o f interest to see whether we can s imilarly use Theorem 17B to obtain a lower bound for X* in the case of certain non-linear boundary conditions , and if so , what information we can deduce about the size of V= in the case of the non-linear boundary condit ions , compared to its size in the case of the linear boundary condition . We shall consider a modification of problem ( 37 ) on p . 8 8 , the modifications consisting of the introduct ion of a non-linear boundary condit ion and the assumption that f( u) = eu , giving the following problem : 3 2 r A o u -i :.. 2 i= l vx . l. ou ' eu = 0 ot + 1\. 3 2m . -1 g ( U) + K L a. ( X ) X. l. :..OU = 0 i= 1 1 1 vxi for x E V , t > 0 m for X E oV , t > 0 _ m u( x , O ) = 0 for x E V m We shall consider particularly two possibilities for g (u ) : ( a ) The natural convect ion boundary condition : ( 4 5 ) In this case g (u) = Hu514 , so in the notation of Theorem 17B we 5 have N = H , p = 4' 60 = 1 . ( b ) The thermal radiat ion boundary condition : 4 4 4 In this case g (u ) = ae[ ( u+T ) - T J ? creu for all u ? 0 , so we a a may takr, N = cre , p = 4 , 60 = 1 (where here and here only , ? denotes the 1 0 7 emissivity of the surface ) . Reasoning exactly as for the linear boundary condition on? p . 89 , we obtain from Theorem 1 7B that a lower bound for X* (more precisely , a number such that for all X smaller than this , all solutions of ( 45 ) are 3 2e 1: A . eA i= 1 ? - A=Y bounded as T - ?) is given by A-f e 2K8 Now A may be arbitrarily chosen greater than 'f + --':"" so if we p-1 ' Ne put n = A - ( 'f + 2KE> ) , then both e and n are arbitrary positive NeP- 1 numbers . The expression for the lower bound on X* now becomes : 3 2K8 2 e r A . exp f e < n + l + -1 )) K1e exp fe ( 1 + i= 1 ? NeP l 1 2K8 2K8 = 1-p 1-p n + p-1 n + p- 1 n + K2e n + K2e Ne Ne 3 2K8 where K1 = 2 E A . , K2 = -r . 1 ? ?= = t3( e ,T\) , say . We define ?( e , n) for e = 0 or ? = 0 so that the funct ion ? is continuous on the set { C e ,n) : e , ? ? 0 } ; it is eas ily verified that this is possible . We wish to find the best possible lower bound for A* that Theorem 17B will give us , so the next step is to find , if possible , a point ( e ,n) with e , n ? o which maximises ?( e ,n) . Now + exp Le e 1 + L n + = = 0 if and only if e {-K1? 2-K1'fn} + e 2-P {-2nK1K2 -pK1K2 'f} + e 3-2P { -K1K; } 1 -p { } 2 -2p { 2 } 2 = + e PnK1K2 +nK1K2 + e pK1K2 + K1n 0 n + . ? ? ? ? ( 46 ) 1 0 8 Also , 1311( e ,11) = = = 0 if and only if 1-p 11 = e'?' - K2e . . . . . . . . . . . ( 4 7 ) I f ( 4 6 ) and ( 47 ) hold simultaneously , then substitution from ( 47 ) into ( 46 ) gives the equation K1e 3'?'2 = 0 , which yields e = 0 . Now 13( e ,11) ? 0 as e ? 0 , regardless of the value of n , so e = 0 clearly does not give the des ired maximum for l3( e ,11) . For e ,n > 0 , we now know that the graph of 13( e ,n) has no points where the tangent plane i s horizontal . Further , it is easy to see that as vfe2 + n2' - ? in the first quadrant , 13( ? , 11) - 0 . Hence 13( e ,n) must attain its maximum value in the first quadrant on the n = 0 axis . So our problem reduces to finding a positive value of e which maximises the value of 13( e , O ) . Now , putting 11 = 0 in ( 46 ) , we obtain : h' ( e ) 13? ( ? , 0 ) = 0 if and only if ?2 -p ( -pK1K2'?') + ? 3-2p ( -K1K? ) + ?2-2p (pK1K? ) = 0 . . . . . . . . . . . . . ( 4 8 ) I f we write h ( ? ) = - p'?'eP - K2e + pK2 , then h( O ) = pK2 > 0 and p 2'?'ep-l - K < 0 for all ? > 0 , so h i s strictly decreasing 2 for e > 0 . Since h ( e ) < 0 for e sufficiently large , it follows that ( 48 ) has exactly one positive solution . We denote the un ique positive solution of ( 48 ) by e* . p Since 13( e ,O ) - . o as e ? 0 and as e - ?, it follows that the maximum value of 13C e , O ) for e > 0 is attained when e = e* ; so we have finally p that the desired first quadrant maximum value o f 13C e ,n) is 13( e??: o ) p ' ? ? . ? ? ? ? ? ? ( 49 ) (\ 109 Since ? may be chosen arbitrarily close to 0 , it follows that ?( e* , O ) is a lower bound for ?* which is the largest that can be obtained p B from the data of Theorem 17 . 1t is of interes t to compare ?( e* ,O ) with the lower bound ?* that p - was obtained in the case where the boundary condition was linear , and also with the empirical estimate ?* t of Boddington , Gray and Harvey(4] , es to see whether we can establish a difference between the value of ?* for the linear boundary condition and the value of \* for the non-linear boundary condition . Before doing this , we discuss a special case which we shall use for illustrative purposes . Example : Sphere of radius 1 : In order to give some feel for the numbers involved in the problem under discussion , we shall illustrate the discussion at each stage by taking the special case where V is a three-dimensional sphere of radius m 1 ; we shall consider the two values of p that are of specia_ practical 5 significance , namely p = 4 and p = 4 . As discussed on p . 94 , for a spherical region of radius 1 ' we have 3 2K I: A . = 3 , so K1 = 6 . Also y = e = 1 , and so K2 = . I f we write N i=1 h = p = p = ? N analogy with linear then 2 ( 4 8 ) reduces K by the cas e , K2 = - and t o : h 5 - phep - 2e + 2p = o . 5 For the special values p = 4 and p = 4 , this becomes : 4 5he 5 /4 Be + 10 = 0 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ( 5 0 ) 4 2h? 4 4 0 ? ? ? ? ? ? ? ? . ? ? ? ? ? ? ? ? ? ( 5 1 ) - e + = Also , in this case : -e?'c { 1 + ? * )p-1 } ?( e1: 0 ) = 3h'( e 1: )P e P 2 e p p ' p From p . 94 we have finally that in this special case : ' * h h = 0 . 9061 + (0 . 45 30)h ' h ??st = 0 . 9061 + (0 . 3000)h ? Relationship between ?( e* ,O ) and ?* : We have from ( 38 ) on p . 90 that ?:_?': for the linear boundary condition ( if we take H = N so as to relate it to the non-linear boundary condition ) is given by "-* We now consider the ratio Since e* satisfies ( 48 ) , we have : p 1 10 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ( 52 ) r (K2 , p ) Now it is easily = 'i' + K2 ( e,': )P K2 p e -e ,'( ( 1 p 1 - -) p 'i' + K2 e,': -e,': ( 1 1 - -) - .:.E..) e p p = '? ( 1 p seen from ( 4 8 ) that as K - 0 2 ( i . e . as Bi - a:> 1 where Bi is the B iot number , s ince Bi is proportional to ?) , e* - O , 2 p while as K2 - a:> ( i . e . as Bi - 0 ) , e? - p . We illustrate this in the case of the spherical region of radius 1 by giving the values of eg/4 and E:'i': for various values of h , where in this h Bi 2 The case = - K2 . 4 equations involved were solved us ing the Newton-Raphson method . h( = B i ) eg14 ( equation ( 5 0 ) ) e, ?: 4 ( equation ( 5 1 ) ) 0 . 00001 not calculated 3 . 994906 0 . 0001 1 . 249917 3 . 9 5 12 5 1 0 . 00 1 1 . 249 175 3 . 646415 0 . 0 1 1 . 241807 2 . 789 326 0 . 1 1 . 1 73651 1 . 8175 2 3 1 0 . 7 86836 1 . 0975 7 2 10 0 . 2 33816 0 . 640207 100 0 . 042 5 39 7 0 . 367118 1 , 000 0 . 00690082 0 . 208661 10 , 000 0 . 00109 779 0 . 1180 34 100 , 000 0 . 000 174091 0 . 06659 39 It follows in general that , as K2 - 0 , r ( K2 , p ) - r( O+ ,p ) = 1 , while as K2 - m, r (K2 ,p ) - r( m,p) = pP e 1-P . The value of r( ?,p ) increases very rapidly with increasing p , as the following table illustrates : p r( oo,p ) = pp e 1-p 5 /4 1 . 029 2 1 . 472 4 12 . 745 8 1 . 5 30 X 10 4 16 5 . 643 X 10 12 1 1 1 We can thus assert that for large values of K2 , i . e . for small Biot number , the value of 13( e:?': , 0 ) is greater then the value of ).,?': , and p - that the ratio between them for small Biot number increases very rapidly as p increases , i . e . as the non-linearity becomes more pronounced . On pp . 94-100 , we discussed in detail the relation between )..* and the empirical )..* for est various regions , and determined in part icular ' ... the limit of the rat io /\est --v- as Bi - 0 , as shown in the next table . A-;': Region lim est ).,?': Bi-oQ - ( 1 ) Sphere , radius a 1 ( 2 ) Infinite cylinder , radius a 1 ( 3 ) Infinite slab , thickness 2a 1 ( 4 ) Equicylinder , height 2a , radius a 1 ( 5 ) Thin c ircular disc , thickness 2 a , radius lOa 1 . 1 8 ( 6 ) Long circular cylinder , height lOa , radius a 1 . 0 8 ( 7 ) Cube , s ide 2a 1 . 7 3 ' ( 8 ) Infinite square rod , s ide 2a 1 . 41 )._?'-est . I f we compare the value of lim "*"" w?th that of the quantity Bi-o I\ r( a> ,p ) = lim Bi-.Q 13( e:??: 0 ) p , 'J. 11. " we see that for regions 1 , 2 , 3 and 4 , the value ll 5 of S( &* , O ) exceeds ?* ? when the Biot number is small , both for p = 4 p est 1 1 2 and for p = 4 , and when p = 4 the excess i s quite large . This .shows that , for small Biot number , the true value of A* in the case of a non? linear boundary condition exceeds the approximate estimate ?* t obtained e s for a linear boundary condition . The excess is certainly large when 5 p = 4 , but may be rather small when p - 4 For regions 5 , 6 , 7 and 8 , the above remarks apply only to the case p = 4 . When p 5 = S( e?? 0 ) does not exceed A'': t and so no firm 4' p ' e s conclusion can be drawn . We refer several times above to the condition that the Biot number be "small" . To give an idea of the s ize of Biot number for which the above remarks apply , we again illustrate by considering the case of the spherical region o f radius 1 , where Bi = h in our notation . In Tables 2 and 3 on p . 1 1 3 , we tabulate , for both p = ? and p = 4 , values of S < e :'; , o ) aC e * O ) A* A* P p ' ' - ' est ' A:'; these tab les we have : and S( e ?? o ) p ' ??-?-- for various values of h . A'}'? est From ( a ) p In this case , the value of the Biot number h at which SC e :': o ) starts to exceed A?: t is slightly less than o . 1 . p ' es ( b ) p = 4 : In this case , the value of the Biot number h at which S( e* 0 ) starts to exceed A* t is slightly less than 1 . p ' es It appears from Tables 2 and 3 that as h decreases ( i . e . as K2 increases ) the rat io r(K2 ,p ) = S( e? , O ) --??-- decreases at first to a value A'}t; rather less than 1 , and then increases to its limiting value r( ?, p ) = pp e l-p as h - 0 ( i . e . K2 - ?) . We shall show that this is the case in general . We have that e? is defined implicit ly as a function of K2 by equation ( 48 ) ; if we differentiate ( 48 ) with respect to K2 , we obtain ( Qo?nued on p . 1 1 4 ! {\ h 1 0 0 , 000 10 , 0 0 0 1 , 0 0 0 10 0 10 1 0 . 1 0 . 0 1 0 . 00 1 0 . 0001 TABLE 2 : h 1 00 , 000 10 , 0 0 0 1 , 0 0 0 1 00 10 1 . 0 . 1 0 . 01 0 . 0 0 1 0 . 00 01 0 . 00001 TABLE 3 : 1 1 3 13(e't; 0 1 13( * 0 ) "-* "-:st 13( e* O ) p , - e l2 ' - p ' ).? ).* est 2 . 207 5 3 . 3 332 2 . 2074 1 . 0 0 0 0 . 662 2 . 207 1 3 . 3 32 3 2 . 2048 0 . 999 0 . 66 2 2 . 20 3 1 3 . 3233 2 . 1922 0 . 99 5 0 . 660 2 . 1642 3 . 2 35 6 2 . 1 14 3 0 . 977 0 . 65 3 1 . 8396 2 . 5 6 0 1 1 . 712 6 0 . 9 3 1 0 . 669 0 . 73 5 8 0 . 829 1 0 . 69 8 7 0 . 95 0 0 . 84 3 0 . 105 1 0 . 1068 0 . 1066 1 . 0 14 0 . 99 8 0 . 0 1098 0 . 01100 0 . 0 112 8 1 . 02 7 1 . 02 6 0 . 001103 0 . 001103 0 . 0 01 1 35 1 . 0 29 1 . 0 29 0 . 0001 104 0 . 0 0 0 1 104 0 . 0 001 1 36 1 . 02 9 1 . 029 5 Values of parar.eters for different value s of h , with p =4 . ?( e :': O ) 13( e:': O ) "'?'? A.J. 13( e:': O ) p ' 12 ' .. est )!? ?.-?? - p ' .. .. - est 2 . 2 0 7 5 3 . 3332 2 . 0 648 0 . 9 35 0 . 620 2 . 2071 3 . 3323 1 . 9606 0 . 88 8 0 . 5 88 2 . 2 0 31 3 . 32 33 1 . 7 891 0 . 812 0 . 5 3 8 2 . 1642 3 . 2 356 1 . 5 220 0 . 70 3 0 . 470 1 . 8396 2 . 560 1 1 . 14 70 0 . 624 0 . 448 0 . 73 5 8 0 . 829 1 0 . 7031 0 . 9 5 6 0 . 848 0 . 1 0 5 1 0 . 1068 0 . 3081 2 . 9 32 2 . 88 5 0 . 01098 0 . 0 1 100 0 . 0 8245 7 . 5 09 7 . 49 6 0 . 001 103 0 . 001103 0 . 01268 11 . 496 11 . 49 6 0 . 0 001104 0 . 00 0 1104 0 . 0 0 1389 12 . 5 82 12 . 5 82 0 . 0 0001 104 0 . 00001104 0 . 0 00 1406 12 . 726 12 . 72 6 Values o f parameters for different values o f h , with p=4. ( continULd 6?om p . 1 1 2 1 de* Since , by ( 52 ) , e * cannot exceed p , it follows that ? > 0 for all p 2 K2 > O , so e? increases steadily from 0 to p as K2 goes from 0 to m. Also , r ( K2 , p ) = and so : -e* ( 1 - ?) e* -e*( 1 ? e P p + j(1 - f> e p ?'?( 1 1 ) -E:" - - ( p - e* ) e P P = P2 -1 { ( 'f + K2 ) ( e* - p ) ( 1 ( K + p '?'( e:?': ) P ) p '?' p ?) ( '?' + K2 ) 2 p + ( K2 + p2 '?'( e? ) p- 1 ) } = p ( K2 )A ( K2 ) where p (K2 ) is pos itive for all K2 > 0 , and : = '?'( e;1: p = '?'( e;1: p Now as K - 0 e* - 0 so A( O+ ) 2 ' p ' by ( 5 2 ) . = - 'fp < 0 . Also , as K2 - CD' e1: - p , so p > 0 for all K2 > 0 since p > 1 and , for all K2 > 0 , Thus , as K2 goes from 0 to m, A ( K2 ) increases steadily from - 'fp to 'f(pP- 1 ) , so rK ( K2 ,p ) increases steadily from a negative value to a 2 pos itive value . Hence , as required , we have shown that as K2 goes from (\ 1 1 5 ?( e* , O ) 0 to eo, r (K2 ,p ) = f* initially decrease s and then increases . Since the B iot number is proportional to i , it follows that as the Biot number 2 increases from 0 to ?, r(K2 ,p ) initially decreases and then increases . We have now shown that i f the B iot number is small , the non-linear boundary condition gives a value of X* which is higher than the estimate X* t o f Boddington , Gray and Harvey[4) for the linear boundary condition , es at least when p = 4 . When p = ?, this conclusion has been proved only for some of the regions considered by Boddington , Gray and Harvey . For larger Biot number , no firm conclusion can be drawn , s ince we have no means of knowing how close our lower bound ?( e* 0 ) is to the true value p ' of x.??? . However , one may speculate that the behaviour of X* for the non- linear boundary condition may perhaps correspond in a qualitative sense to the behaviour of P( e:'= O ) i . e . if we write x??? for the crit ical value .. p ' , p of A. in the of the non-linear boundary condition ( p > 1 ) and x??? case .. 1 for the critical value in the case of the l inear boundary condition , ).,1: then the ratio rt may decrease initially as the B iot number increases 1 from near zero , then increase to a finite l imiting value as the B iot number tends to infinity . One may also ask whether , for all p > 1 , ? > 1 for sufficiently small Biot number . From the evidence given in 1 this chapter , this seems quite likely , but there is evidently plenty of scope for further research on the case of a non-linear boundary condition . 1 1 6 APPENDIX We give here the details of the example discussed in Ch . 2 . First we need to calculate several Fourier series and invers e Laplace trans forms . LEMMA 1 : If f ( x ) = -f( 2-x) for 1 < x ? 2 , then J 2 nnx {2 t f ( x ) cos n? dx f ( x ) cos --2- dx = 0 0 0 (n odd ) (n even) . Proof : 2 J f ( x ) cos nTTX dx = 0 2 1 2 J =< x ) cos ? dx + J {-f ( 2 -x ) }cos ? dx 0 2 1 2 1 = f=< x ) cos 0 nnx dx 2 0 I f( u ) cos nn( 2 -u) ( -1 ) du 1 2 1 = f=c x ) cos 0 nnx d --2- X 1 J f (u ) cos (nrr 0 nnu -2-)du . The lemma now follows s ince cos ( nn - nnu ) = cos nnu if n is even , and 2 2 Cos ( nn - ?) nTTU . ? . dd 2 = - cos --2- l= n ?s o . LEMMA 2 : Suppose that k '1 ( 2r:+1 ) 2 ti for a:ny integer n . 4 defined by cosli 0 ) OX2 - dt Cons ider the equation where we assume k > 0 , A > 0 ; further , u( x , t ) sat isfies the initial? boundary conditions u( x , O ) = 0 for - 1 s x s 1 u( - 1 ,t ) = 0 for t ? 0 u( 1 ,t ) = 0 for t ? 0 . Let y ( s ,x ) be the Laplace transform with respect to t of u . Taking Laplace transforms , the problem becomes : ? - sy + ky + ? = 0 , with y ( s , - 1 ) = y ( s , 1 ) = 0 . dx2 s The equation is n2y - ( s -k )y = - A s s-k > 0 . A particular integral is 1 ?) - A = s - s(s -k) 2 D - ( s -k ) t\ We may assume s > k , so that A A k(s-k) - ks Thus the general solution is y ( s ,x ) . y ( s , -1 ) 0 = = y( s , 1 ) = 0 = ( 1 ) - ( 2 ) : ;s:l( X = Ae + - /S-'1( X ). Be + k(s-k) - ./s-k' Js-k ). Ae + Be + k(s-k ) js::1( - js::1( ). Ae + Be + k(s-k) ( A ) ( - JS-K - B e - ers::K) ). - ks ). - ks = 0 . ). - ks . . . . . . . . . . . . . ( 1 ) . . . . . . . . . . . . ( 2 ) Hence A = B , and substituting back into ( 1 ) gives at once : ). ). A = B = ks k(s-k) Js-k - Js-k e + e Thus the required solution is y( s , x ) = { ). ). } cosh x.rs=K + ks - k(s-k) cosh /s-k' ). k(s-k) - ks ? 12 2 Now kt = e and Using these and Lemmas 6 and 7 , we may take inverse Laplace transforms to obtain a formal solution u( x , t ) . Case 1 : u( x , t ) ( 2n+1 ) 2n2 k -1 4 for any n = 0 , 1 , 2 , . . . . : CD ( - l )n ( 2 1 ) ( 2n+1 )nx ).n ? n+ cos 2 = k E 2 2 n=O k _ ( 2n+1 ) n 4 e 2 2 (k _ ( 2n+1 ) n )t 4 - 1 ) 4). CD ( - 1 )n+1 - - E ...;......,. ..;._- cos nk n= O 2n+1 ( 2n+1 )nx 2 2 2 (k _ ( 2n+1 ) n )t 4 e 2 2 CD ( k - ..:..< ;:..;2n;.;..+??;;..:;)___:.;,n_) t {:-('-2_E_+?2_'1!_2-::-:::-= ? E ( - 1 )ncos ( 2n+1 )nx e kn n=O 2 ( 2n+1 ) 2n2 4 + -- 4 } 2n+1 CD 4). = n E n=O CD + ).n E k n=O ( -1 )n+1 ( 2n+1 ) ( 2n+1 ) 2n2 4 k - {k - 2 2 cos ( 2n+?) n ) ( 2n+1 ) CD + ? E n=O ( -1 )n+1 ( 2n+1 ) 2 2 ( 2n+1 ) n k - 4 cos ( 2n+1 )nx 2 ( 2n+1 )nx 2 2 ( k _ ( 2n+1 ) n ) t 4 2 e cos ( 2n+1 )nx 2 ). - k Now from Lemma 2 we can see that the second of these sums is discontinuous at x = ?1 . We must therefore redefine it at x = ?1 to make it continuous ; again using Lemma 2 and remembering that in this problem we are only concerned with the interval - 1 ? x ? 1 , we can see that the second sum must be replaced by ? cos? k cos . This gives the formal solution : u( x , t ) CD = 4? n E n=O ( -1 )n -------- ??2?2?----- cos ( 2n+?) n } ( 2n+1 ) {k - ( 2n+1 )TTX 2 e 2 2 (k _ ( 2n+1 ) n )t 4 ? cos fix ? + k Jt< - -k ? ? ? ? ? ? ? ? ? ? ( S 1 ) cos It is necessary to verify that this is indeed a solution of the problem . Clearly u( -1 ,t ) = u( 1 ,t ) = 0 for t ? 0 . 4? CD ( -1 )n Further , u ( x , O ) = - E --------..;__?-.,..------ cos n n=O ( ) 2 2 (k - 2n+t n } C 2n+1 ) + = ? cos JKx 1 ) A cos fix A - k cos 11< - + k cos Jk - k = o . ( 2n+1 )nx 2 A cos Ji 0 and n sufficiently absolute value than large , the general term of each series is smaller in 2 2 ( k _ ( 2n+1 ) n )t e 4 0 2 2 CD ( k _ ( 2n+1 ) n ) t ou at + ku + ? = 4? n ( 2n+1 )TTX 4 n n ? 0 ( -1 ) cos 2 e X 2 2 - 2n+1 + l - ( 2n+1 )n2 1 4 (k - ( 2n+t) n } .< 2n+1 ) {k 2 2 k I _ ( 2n+?) n } = 0 as required . Thus ( 51 ) is an actual solution of the problem . I t i s clear f?om the 2 1 2 4 form of ( 51 ) that if 0 < k < : then u( x , t ) is bounded as t - m, and in fact u ( x ,t ) - ?cos ./Kx - 1 ) k cos Jk' as t - CD, TT2 I f k > 4 then u( x , t ) is unbounded as t - CD, Case 2 : u ( x ,t ) ( 2N+1 ) 2TT2 k = 4 for some N = 0 , 1 , 2 , . . ? ? : ( -1 )n ( 2n+ 1 ) cos ( 2n+?)TTX 2 2 2 2 ( k _ ( 2n+?) TT ) t e - 1 } k _ ( 2n+1 ) TT 4 + ?- 1 )N ( 2N+1 )t cos 4A CD ( -1 ) n+1 - - I: ....:.......,,......:......,...-- cos TTk n= O 2n+1 ( 2N+1 )TTX 2 ( 2n+1 )TTX 2 e 2 2 ( k _ ( 2n+1 ) TT )t 4 A - k Calculat ing as in the previous case , this gives : u( x , t ) = 4A I: TT n#N + ( - 1 )n {k - 2 2 ( 2n+t) TT } C 2n+ 1 ) 4A( -1 )N ( 2N+1 )TTX TTk(2N+1) cos 2 cos ( 2n+1 )TTX 2 e 2 2 (k _ ( 2n+1 ) TT ) t 4 + ATT I: k nr!N ( - 1 )n+ 1 ( 2n+1 ) 2 2 ( 2n+1 ) TT k - cos ( 2n+ 1 )TTx 2 A - j( ? 4 From Lemma 4 we see that , in order to make the second sum in this expres sion continuous at x = ?1 , we must replace it by leading u( x , t ) = N ? ( 1 )N . ( 2N+1 )TTX ( -1 ) ( 2N+1 )TTX) ](1- - X Sl.n 2 - ( 2N+1 )TT cos 2 to the formal 4A I: TT n#,N {k - solution ( - 1 )n 2 2 ( 2n+?) TT ) ( 2n+1 ) cos ( 2n+1 )TTx 2 e ( k - 2 2 ( 2n+1 ) TT ) t 4 + 3A( - 1 ) N ( 2N+1 )TTX N A ( - 1 ) X sin ( 2N+1 )TTX TTk ( 2N+1) cos 2 + k 2 A - k . . . . . . . . . . . . . . . . . . ( 52 ) As be fore , it is necessary to verify that this is indeed a solution of th? problem. Clearly u( - 1 ,t ) = u( 1 ,t ) = 0 for t ? 0 . Further , u( x ,O ) 41 ( -1 ) n = - E -----=-_..::.::--::-- -- cos TT n"'N ( 2 1 ) 2 2 r {k - n+4 TT } ( 2n+1 ) + 3l ( - 1 ) N ( 2N+1 )TTX TTk ( 2N+1 ) cos 2 + ( 2n+1 )TTX 2 1 2 5 + 3H - 1 ) N TTk(2N+1) cos ( 2N+1 )TTX 2 + N )..{ -1 ) X k s in ( 2N+1 )TTX 2 1 - k = 0 for -1 ? x ? 1 , by Lemma 5 . Thus the boundary conditions are satisfied . Checking the differential equat ion , we have : + + ( - 1 )n+ 1 ( 2n+1 )TT2 2 2 {k _ ( 2n+1 ) TT ) 4 4 31( -1 )N+l ( 2N+1 )TT 4k cos cos ( 2n+1 )TTX 2 ( 2N+1 )TTX 2 + e 2 2 ( k _ ( 2n+?) TT )t 1 ( - 1 )N ( 2N+1 )TT ( 2N+1 )TTX k cos 2 ( 2N+1 )TTx 2 + ? ( -1 ) N+ l ( 2N+1 ) 3TT 3 t 4k cos ( 2N+1 )TTX 2 ou = 41 r ( -l ) n ot TT n;tN 2n+1 cos ( 2n+1 )TTX 2 e 2 2 ( k _ ( 2n+?) rr )t ?TT N ( 2N+1 )rrx + -r -1 ) ( 2N+1 ) cos 2 ou ot + ku + 1 = o , as is readily checked . actual solution in this case ; evidently , we h ave in this case that u( x , t ) is always unbounded as t - ?. The Corresponding Steady-state Problem : d2u This is the problem -2- + ku + 1 = 0 ( - 1 < x < 1 ) , where k > 0 , dx 1 > 0 , and the boundary conditions are u( - 1 ) = u( 1 ) = 0 . The general solution of the equat ion is u( x ) = A cos Jkx + B sin Jkx - ? ? u( -1 ) u( 1 ) = 0 = A cos Jk - B s in jk - 1 k = 0 = A cos Jk + B sin Ji< - ? . 1 2 6 Adding : A cos Jk = ? ? Subtract ing : B sin ./K = 0 . Now if .!k # nn, i . e . k # n2n2 for n = 1 , 2 , 3 , ? . . . , we must have B = 0 . 2 2 If k = n n for some n = 1 , 2 , 3 , . . . . , then B is arbitrary . If ? ..? ( 2n+ 1 )n . k ..? .; ?<.. r 2 , 1 . e . r ( 2n+ 1 ) 2n2 f 4 or ? n = 0 , 1 , 2 , . ? ? ? , then A = :-k-..;.;.._711 __ ? cos f/ 1\. ( 2n+1 ) 2n2 f I f k = 4 or some n = 0 , 1 ,2 , . . . ? , no solution is possible . we have the following cases : 2 2 m n I f k # -4-- for any m = 1 , 2 , 3 , . . . . , the solution is u ( x ) = ?cos /Kx _ 1] . )(1. cos j1( If k = n2n2 for some n = 1 , 2 , 3 , . . . . , the solution is u ( x ) = ffk? cos ? - 1] + B s in /kx cos where B is arbitrary . ( 2n+1 ) 2n2 If k = 4 for some n = 0 , 1 ,2 , ? . . . , no solution exists . So TT2 In particular , if 0 < k < ? then the steady-state problem has the positive solution u ( x) = ?cos J!x - 1] which is also the limit of k'- cos fi( ' the solution u( x ,t ) of the t ime-dependent problem as t - ?. For larger values of k, posit ive solutions of the steady-state problem do not exist . LIST OF REFERE NCES 1 . S . Agmon , A . Douglis and L . Nirenberg , Estimates near the boundary 1 2 7 foro solutions of e lliptic partial differential equations satisfying general boundary aonditions, I , Commun ications on P ure and Applied Mathematics 12 ( 195 9 ) , 62 3-727 . 2 . H . Amann , Multiple positive fixed points of asymptotical ly linear maps , Journal of Funct ional Analys is 17 ( 1974) , 174-21 3 . 3 . H . Amann , Supersolutions, monotone iterations and stabi lity , Journal of Different i al Equations , to appear . 4 . T . Boddington , P . Gray and D . I . Harvey , Thermal theory of spontaneous ignition : criticality in bodies of arbitrary shape , Philosoph i cal Trans actions of the Royal Society of London , Serie s A( Mathemat ical and Physical Sciences ) , 270 ( 197 1 ) , 467-506 . 5 . M . Boudart , Kinetics of chemical processes , Prent ice -Hall , Englewood Cliffs , New Jersey , 196 8 . 6 . J . N . Bradley , Flame and combustion phenomena , Methuen , London , 1 9 69 . 7 . F . E . Browder , Linea? parabolic differential equations of arbitrary order; general boundary-value prob lems for e lliptic equations , Proceedings of the Nat ional Academy of Sciences of the U . S . A . 39 ( 19 5 3 ) , 1 85-190 . Also : Errata t o the above , same j o urnal , s ame vo lume , p . 12 9 8 . 8 . N . Chafee and E . F . Infante , Bifurcation and stabi lity for a nonlinear parabolic partial differential equation , Bullet in of the Ameri can Mathemat i cal Society 80 ( 1974 ) , 49-52 . 9 . C . Y . Chan , Positive solutions for nonlinear parabolic second initial boundary value problems , Quarterly of App lied Mathemat ics 31 ( 1974) , 443-454 . 1 0 . D . S . Cohen , Multiple stable solutions of non- linear boundary value prob lems arising in chemical reactor theory , SIAM Journal of Applied Mathemat ics 2 0 ( 19 71 ) , 1-1 3 . 1 1 . R . Courant and D .Hilbert , Methods of mathematical physics , Vol . I I , Inters cience , New York , 1962 . 1 2 8 12 . L . Fe j er , Uber die Eindeutigkeit der Losung der linearen partiel len Differentialgleichung zweiter Ordnung , Mathemat is che Zeitschrift 1 ( 19 1 8 ) , 7 0 - 7 9 . 13 . D . A . Frank-Kamenetskii , Diffusion and heat exchange in chemica l kinetics , Princeton Univers ity Press , Princeton , 19 5 5 . 1 4 . A . Friedman , Partial differential equations of parabolic type , Prent ice-Hall , Englewood Cliffs , New Jersey , 1964 . 15 . S . Glasstone and D . Lewis , Elements of physical chemistry , 2nd edit ion , MacMillan , London , 1964 . 16 . S . I . Hudj aev , Boundary prob lems for certain quasilinear e l liptic equations , Soviet Mathemat ics : Doklady 5 ( 19 6 4 ) , 1 8 8 - 192 . 1 7 . E . D . Kaufman , Advanced concepts in physical chemistry , McGraw-Hil l , New York , 1 9 6 6 . 1 8 . J . P . Keener and H . B . Keller , Positive solutions of convex non- linear eigenvalue problems , Journal of Di ffe rent ial Equat ions 16 ( 1 9 7 4 ) , 1 0 3 - 12 5 . 19 . H . B . Keller an d D . S . Cohen , Some positone prob lems suggested by non? linear heat generation , Journal of Mathemat ics and Mechan ics 1 6 ( 196 7 ) , 1 3 6 1 - 1 376 . 20 . M . G . Kre in and M . A . Rutman , Linear operators leaving invariant a cone in a Banach space , American Mathemat ical Society Translation s , Series 1 , 10 ( 1 9 6 2 ) , 1 9 9 - 3 2 5 . 21 . O . A . Ladyzenskaj a , V . A . S o lonnikov and N . N . Ural ' ceva , Linear and quasi linear equations of parabolic type , American Mathemat ical Society Trans lat ions of Mathemat i cal Monographs 2 3 , American Mathematical Society , Providence , Rhode Is land , 1 96 8 . 22 . O . A . Ladyzenskaj a and N . N . Ural ' ceva , Linear and quasi linear e l liptic equations , Academic Press , New York , 19 6 8 . 2 3 . T . W . Laetsch , The number of solutions of a non- linear two point boundary value prob lem , Indiana Univers ity Mathemat ics Journal 2 0 ( 1970 ) , 1 - 1 3 . 24 . J . L . Lions and E . Magene s , Non-homogeneous boundary value prob lems and app lications , Vols . I and I I , Springer-Verlag , Berlin , 1 9 72 . 1 2 9 2 5 . A . McNabb , Comparison and existence theorems for mul tiaomponent diffusion systems , Journal of Mathemat ical Analysis and Applicat ions 3 ( 1 9 6 1 ) , 1 33-144 . 2 6 . M . N . Ozisik , Boundary value problems of heat conduction , Internat ional Textbook Company , S cranton , Pennsylvan ia , 1968 . 2 7 . S . V . Parter , Solutions of a differential equation arising in chemical reactor processes , S IAM Journal of Applied Mathemati cs 2 6 ( 1 9 7 4 ) , 6 87-71 6 . 2 8 . M . H . Protter and H . F . We inberger , Maximum principles in differential equations , Prent i ce -Hall , Englewood Cliffs , New Jersey , 1 9 6 7 . 2 9 . A . Reynolds , Asymptotic behaviour of solutions of nonlinear parabolic equations , Journal o f Di fferent ial E quat ions 1 2 ( 1 9 7 2 ) , 2 5 6 - 2 6 1 . 30 . F . Riesz and B . S z . Nagy , Functional analysis , Frederick Ungar , New York , 1 9 5 5 . 31 . D . H . Satt inger , Monotone methods in non- linear elliptic and paraboli c boundary value prob lems , Indiana Uni vers ity Mathematics Journal 2 1 ( 1972 ) , 9 7 9 - 1 0 00 . 32 . D . H . Satt inge r , Topics ?n stabi lity and bifurcation theory , Lecture Notes in Mathemat ics 309 , Springer-Verlag , Berlin , 19 7 3 . 3 3 . D . H . Satt inger , A non-linear parabolic system in the theory of combustion , Quarterly of App lied Mathemat ics 3 3 ( 1 9 75 ) , 47-6 1 . 34 . M . R . Sp iege l , Theory and prob lems of Laplace transforms , S chaum , New York , 1 9 6 5 . 3 5 . G . C . Wake , Non- linear heat generation with reactant consumption , Quarterly Journal of Mathematics , Oxford Second Series , 2 2 ( 1 9 71 ) , 5 8 3 -59 5 . 36 . G . C . Wake and M . E . Rayner , Variational methods for nonlinear eigenvalue prob lems associated with thermal ignition , Journal o f Di fferent ial Equat ions 1 3 ( 19 7 3 ) , 2 4 7 -2 5 6 .