Dynamics of Nonlinear Parabolic Systems with Time Delays

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

198, 751]779 Ž1996.

0111

Dynamics of Nonlinear Parabolic Systems with Time Delays C. V. Pao Department of Mathematics, North Carolina State Uni¨ ersity, Raleigh, North Carolina 27695-8205 Submitted by William F. Ames Received May 1, 1995

The dynamics of a coupled system of semilinear parabolic equations with discrete time delays is investigated using the method of upper and lower solutions. It is shown that if the reaction function in the system possesses a mixed quasimonotone property and the corresponding elliptic system has a pair of coupled upper and lower solutions then there is a monotone iteration process which yields a pair of quasisolutions of the elliptic system and the sector between the quasisolutions is an attractor of the delayed parabolic system. Under some additional conditions this sector is a global attractor and the solution of the parabolic system converges to a true solution of the elliptic system. The same conclusions are obtained for a coupled system of parabolic-ordinary equations with time delays. Applications are given to three model problems arising from ecology and nuclear engineering. These model problems possess multiple steady-state solutions and sufficient conditions are given to ensure the stability and instability of these Q 1996 Academic Press, Inc. solutions.

1. INTRODUCTION Differential equations with time delays are traditionally formulated for spatially homogeneous systems, and various types of time-delayed equations in the framework of ordinary differential systems have been treated systematically Žcf. w3, 5, 6x.. In recent years considerable attention has been given to systems of semilinear parabolic differential equations with time delays. In this paper, we consider a coupled system of parabolic equations with discrete time delays which is given in the form

­ u ir­ t y L i u i s f i Ž x, u, u t . Bi u i s h i Ž x . u i Ž t , x . s hi Ž t , x .

Ž t ) 0, x g V . Ž t ) 0, x g ­ V . Ž i s 1, . . . , n . Ž 1.1. Ž yri F t F 0, x g V . , 751 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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C. V. PAO

where u ' Ž u1 , . . . , u n ., u t ' Ž u1Ž t y t 1 , x ., . . . , u nŽ t y tn , x .. for some positive constants t 1 , . . . , tn , and V is a bounded domain in R N with boundary ­ V. The operators L i , Bi are given by N

Li u i '

Ý

aŽjki. Ž x . ­ 2 u ir­ x j ­ x k q

j, ks1

N

Ý bjŽ i. Ž x . ­ u ir­ x j js1

Bi u i ' a i ­ u ir­n q bi Ž x . u i

Ž i s 1, . . . , n . ,

where ­r­n denotes the outward normal Žor conormal. derivative on ­ V. It is assumed that for each i s 1, . . . , n, L i is a uniformly elliptic operator in V ' V j ­ V, Bi is of either Dirichlet type Ž a i s 0, bi Ž x . ' 1. or Neumann]Robin type Ž a i s 1, bi Ž x . G 0., and it is allowed to be a different type for different i. The purpose of this paper is to study the dynamic property of the parabolic system Ž1.1. in relation to its corresponding elliptic system yL i u i s f i Ž x, u, u . Bi u i s h i Ž x .

Ž x g V. Ž x g ­V.

Ž i s 1, . . . , n . .

Ž 1.2.

In addition to the systems Ž1.1. and Ž1.2. we also investigate the dynamics of the coupled parabolic-ordinary system

­ u ir­ t y L i u i s f i Ž x, u, u t . ­ u ir­ t s f i Ž x, u, u t . Bi u i s h i Ž x .

Ž i s 1, . . . , nU . Ž i s nU q 1, . . . , n .

Ž t ) 0, x g V .

Ž i s 1, . . . , nU . Ž t ) 0, x g ­ V .

u i Ž t , x . s hi Ž t , x .

Ž 1.3.

Ž i s 1, . . . , n . Ž yg i F t F 0, x g V .

in relation to its corresponding steady-state problem yL i u i s f i Ž x, u, u . f i Ž x, u, u . s 0 Bi u i s h i Ž x .

Ž i s 1, . . . , nU . Ž i s nU q 1, . . . , n .

Ž x g V.

Ž 1.4.

Ž i s 1, . . . , nU . Ž x g ­ V . .

The systems Ž1.3. and Ž1.4. may be considered as special cases of Ž1.1. and Ž1.2., respectively, with L i s 0 and without the boundary condition for i s nU q 1, . . . , n. Parabolic systems in the form Ž1.1. have been treated by many investigators both in theory and in applications Žcf. w4, 8, 10, 12, 14, 16, 18]25x.. Most of the theoretical discussions in the earlier works consider the problem as an abstract functional differential equation using semi-group

NONLINEAR PARABOLIC SYSTEMS

753

theory Žcf. w10, 16, 18, 20, 22, 23x. while the various model problems in applications often involve mixed quasimonotone reaction functions Žcf. w16, 20, 21, 24, 25x.. Recently the method of upper and lower solutions and its associated monotone iterations have been used to investigate the existence and asymptotic behavior of a solution Žcf. w4, 8, 12, 14, 19, 21x.. The monotone method for the existence proof is constructive and can be used to compute numerical solutions of the corresponding discretized equations Žcf. w4, 8, 9, 13, 14x.. In this paper we use the method of upper and lower solutions and a similar idea as that in w15x to obtain local and global attractors for the parabolic systems Ž1.1. and Ž1.3., and to obtain conditions which ensure the convergence of the time-dependent solution to a steady-state solution. These results are independent of the time delays t 1 , . . . , tn , and are quite useful for the study of the persistence and stability of certain ecological and engineering problems. Some applications are given to a diffusion logistic model with time delays, a reactor model in nuclear engineering, and a Volterra]Lotka competition model with ncompeting species. In each of the three model problems, multiple steadystate solutions exist and sufficient conditions are given to ensure the stability and instability of these solutions. The plan of the paper is as follows: In Section 2 we imbed the system Ž1.1. into a system of 2 n equations and show that these two systems are equivalent. Section 3 is devoted to the monotone convergence of the time-dependent solution to the maximal or minimal steady-state solution of the extended problem. The main results are given in Section 4 where we show that the maximal and minimal solutions of the extended problem are quasisolutions of the elliptic system Ž1.2., and the sector between the quasisolutions is an attractor of the parabolic system Ž1.1.. Under some additional conditions this sector is a global attractor and the time dependent solution converges to a steady-state solution. Similar conclusions for the systems Ž1.3. and Ž1.4. are also given in this section. Applications of these results to three model problems are given in the final section.

2. AN EXTENDED PARABOLIC SYSTEM Let V be the closure of V, and for any finite T ) 0 and i s 1, . . . , n, we set D T s Ž 0, T = V ,

ST s Ž 0, T = ­ V ,

Q0Ž i. s w yt i , 0 . = V , Q0 s Q0Ž1. = ??? = Q0Ž n. ,

D T s w 0, T x = V ,

QTŽ i. s w yt i , T x = V , QT s QTŽ1. = ??? = QTŽ n. .

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C. V. PAO

Denote by C a Ž D T . the set of Holder continuous functions in D T with ¨ exponent a g Ž0, 1., and by C 1, 2 Ž D T . the set of functions which are once continuously differentiable in t g Ž0, T x and twice continuously differentiable in x g V. The above spaces for vector-valued functions Žwith n components. are denoted by C a Ž D T . and C 1, 2 Ž D T ., respectively. Similar notations are used for other function spaces and other domains. Throughout the paper we assume that for each i s 1, . . . , n, the coefficients of L i and the first partial derivatives of aŽjki. are in C a Ž V ., the boundary coefficient bi is in C 1q a Ž ­ V ., and ­ V is of class C 1q a . We also assume that h i and hi are Holder continuous on ­ V and Q0Ž i., respectively, and ¨ satisfy the compatibility condition at t s 0 when a i s 0. The function f i Ž x, u, v. is assumed Holder continuous in x and continuously differen¨ tiable in u and v for u, v in some bounded subset of R n. The above smoothness assumptions are used to ensure the existence of a classical solution to Ž1.1. by the method of upper and lower solutions when the vector function f Ž ?, u, v . ' Ž f 1 Ž ?, u, v . , . . . , f n Ž ?, u, v . . possesses a mixed quasimonotone property in some subset L of R n. Specifically, by writing u and u t in the split forms u ' Ž u i , wux a i , wux b i . ,

ut ' Ž wut x c i , wut x d i . ,

where a i , bi , c i , and d i are some nonnegative integers we have the following definition. DEFINITION 2.1. A vector function fŽ?, u, v. is said to be mixed quasimonotone in L if for each i s 1, . . . , n, there exist nonnegative integers a i , bi , c i , and d i with a i q bi s n y 1 and c i q d i s n such that for every u ' Ž u i , wux a i , wux b i . and v ' Žwvx c i , wvx d i . in L, f i Ž?, u, v. is monotone nondecreasing in wux a i and wvx c i and monotone nonincreasing in wux b i and wvx d i . The function fŽ?, u, v. is said to be quasimonotone nondecreasing in L if bi s d i s 0 for all i. To specify the set L in the above definition for the system Ž1.1. we rewrite the system in the form

­ u ir­ t y L i u i s f i Ž x, u i , w u x a i , w u x b i , w u t x c i , w u t x d i . Bi u i s h i Ž x . u i Ž t , x . s hi Ž t , x .

in D T

on ST

in Q0Ž i. Ž i s 1, . . . , n . .

Ž 2.1.

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NONLINEAR PARABOLIC SYSTEMS

Then the mixed quasimonotone property of fŽ?, u, u t . leads to the following definition. DEFINITION 2.2. A pair of functions ˜ u ' Ž˜ u1, . . . , ˜ u n ., ˆ u ' Žu ˆ1 , . . . , uˆn . in C 1, 2 Ž D T . l C Ž D T . are called coupled upper and lower solutions of Ž2.1. if ˜ uGˆ u in D T , and if

­u ˜ir­ t y L i u˜i G f i Ž x, u˜i , w˜u x a i , wˆu x b i , ˜ut

ci ,

­u ˆir­ t y L i uˆi F f i Ž x, uˆi , wˆu x a i , w˜u x b i , ˆut

ci ,

Bi u ˜i G h i Ž x . G Bi uˆi , u ˜i Ž t , x . G hi Ž t , x . G uˆi Ž t , x .

ˆut

. ˜ut d . di

in D T

i

on ST in Q0Ž i. Ž i s 1, . . . , n . .

Ž 2.2.

For the parabolic-ordinary system Ž1.3. the definition of upper and lower solutions is the same as that in Definition 2.2 but with L i s 0 and without the boundary inequalities for i s nU q 1, . . . , n. In either case, inequalities between two vector-valued functions are always in the componentwise sense. It is clear from the above definition that upper and lower solutions are in general coupled, and a solution of Ž2.1. is not necessarily an upper solution nor a lower solution. However, if fŽ?, u, u t . is quasimonotone nondecreasing then upper and lower solutions are not coupled, and every solution of Ž2.1. is an upper solution as well as a lower solution. To distinguish their independence in this situation we refer to ˜ u, ˆ u as ordered upper and lower solutions. For a given pair of coupled Žor ordered. upper and lower solutions ˜ u, ˆ u we set ²ˆ u, ˜ u: '  u g C Ž D T . ; ˆ uFuF˜ u4

Ž 2.3.

and make the following basic hypothesis on f: ŽH. fŽ?, u, u t . is a C 1 function and possesses a mixed quasimonotone property in L ' ²ˆ u, ˜ u:. The above hypothesis implies that there exist constants K i G 0 such that f i Ž x, u, u t . y f i Ž x, v, vt . F K i Ž
Ž 2.4.

In the Lipschitz condition Ž2.4. it is understood that
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C. V. PAO

Ž1.3.. has a unique solution uU in ²ˆ u, ˜ u:. Moreo¨ er there exist sequences  uŽ m.4 ,  uŽ m.4 which con¨ erge monotonically from abo¨ e and below, respecti¨ ely, to uU as m ª `. To investigate the asymptotic behavior of the solution we consider an extended system of 2 n equations by introducing the function v ' M y u, where M ' Ž M1 , . . . , Mn . is a positive constant vector satisfying Mi G u ˜i in D T , i s 1, . . . , n. Since problem Ž1.3. may be considered as a special case of Ž2.1. with L i s 0 for i s nU q 1, . . . , n, we only need to consider the system Ž2.1.. Define Fi Ž x, u, v, u t , vt . ' f i Ž x, u i , w u x a i , w M y v x b i , w u t x c i , w M y vt x d i . Gi Ž x, u, v, u t , vt . ' yf i Ž x, Mi y ¨ i , w M y v x a i , w u x b i , w M y vt x c i , w u t x d i . hUi Ž x . ' Mi bi Ž x . y h i Ž x .

Ž i s 1, . . . , n . .

Ž 2.5.

Then an extended system of Ž2.1. is given by

­ u ir­ t y L i u i s Fi Ž x, u, v, u t , vt . ­ ¨ ir­ t y L i ¨ i s Gi Ž x, u, v, u t , vt . Bi u i s h i Ž x . , u i Ž t , x . s hi Ž t , x . ,

in D T

Bi ¨ i s hUi Ž x . on ST

¨ i Ž t , x . s hiU Ž t , x . in Q0Ž i. Ž i s 1, . . . , n . .

Ž 2.6.

We first show that the 2 n-vector function F Ž ?, w, wt . ' Ž F1 Ž w, wt . , . . . , Fn Ž w, wt . , G1 Ž w, wt . , . . . , Gn Ž w, wt . . Ž 2.7. is quasimonotone nondecreasing in ²w, ˆ w ˜ :, where w ' Žu, v. ' Ž u1 , . . . , u n , ¨ 1 , . . . , ¨ n . and ²w, ˆ w ˜: ' Ž u, v . g C Ž DT . = C Ž D T . ; ˆu F u F ˜u, M y ˜u F v F M y ˆu .

½

5

Ž 2.8. LEMMA 2.1. Let f i Ž x, u, u t . satisfy hypothesis ŽH., and let w ˜ ' Ž˜u, ˜v. and w ˆ ' Žˆu, ˆv., where ˜v s M y ˆu and ˆv s M y ˜u. Then F Ž?, w, wt . is quasimonotone nondecreasing for all w, wt g ²w, ˆw ˜:. Proof. In view of Ž2.8., u and M y v are in the sector ²ˆ u, ˜ u: whenever : Ž . w ' Žu, v. g ²w, w . Hence by 2.5 and the mixed quasimonotone property ˆ ˜ of fŽ?, u, u t ., the function Fi is monotone nondecreasing with respect to all the components of the 2 n vectors Žu, v. and Žu t , vt . except possibly the component u i . Similarly, Gi is monotone nondecreasing with respect to all

757

NONLINEAR PARABOLIC SYSTEMS

the components of Žu, v. and Žu t , vt . except possibly the component ¨ i . This shows that F Ž?, w, wt . is quasimonotone nondecreasing in ²w, ˆw ˜:. We next show an equivalence relation between the two pairs of upper and lower solutions Ž˜ u, ˆ u. and Žw, ˜w ˆ.. LEMMA 2.2. Let hiU s Mi y hi , i s 1, . . . , n. Then the pair w ˜ ' Ž˜u, M y ˆu., w ˆ ' Žˆu, M y ˜u. are ordered upper and lower solutions of Ž2.6. if and only if ˜ u and ˆ u are coupled upper and lower solutions of Ž2.1.. Proof. Suppose ˜ u ' Žu ˜1 , . . . , u˜n . and ˆu ' Ž uˆ1 , . . . , uˆn . are coupled upper and lower solutions of Ž2.1.. By Ž2.2., Ž2.5. and the mixed quasimonotone property of fŽ?, u, u t ., the function w ˜ ' Ž˜u, ˜v., where ˜v ' Ž ¨˜1 , . . . , ¨˜n . with ¨˜i s Mi y u ˆi , satisfies the relation

­u ˜ir­ t y L i u˜i G f i Ž x, u˜i , w˜u x a i , w M y ˜v x b i , ˜u t

ci ,

M y˜ vt

di

.

s Fi Ž x, ˜ u, ˜ v, ˜ ut , ˜ vt .

­ ¨˜ir­ t y L i ¨˜i G yf i Ž x, Mi y ¨˜i , w M y ˜ v x a i , w˜ ux b i , M y ˜ vt

ci ,

˜u t

di

.

s Gi Ž x, ˜ u, ˜ v, ˜ ut , ˜ vt . Bi u ˜i G h i Ž x . ,

Bi ¨˜i s Mi bi y Bu ˆi G hUi Ž x .

u ˜i Ž t , x . G hi Ž t , x . ,

¨˜i Ž t , x . s Mi y u ˆi Ž t , x . G hiU Ž t , x .

Ž i s 1, . . . , n . . Ž 2.9. In view of Ž2.7. and the quasimonotone nondecreasing property of F Ž?, w, wt ., the above inequalities imply that w ˜ is an upper solution of Ž2.6.. Ž A similar argument shows that w ˆ ' ˆu, M y ˜u. is a lower solution. Since ˜u G ˆu implies w ˜ G w, ˆ the pair w, ˜w ˆ are ordered upper and lower solutions. Conversely if w ˜ and w ˆ are ordered upper and lower solutions of Ž2.6. then, by definition, the pair ˜ u ' Žu ˜1 , . . . , u˜n . and ˜v ' Ž ¨˜1 , . . . , ¨˜n ., where ¨˜i s Mi y u ˆi , satisfy all the inequalities in Ž2.9.. Similarly the pair ˆu ' Žu . , . . . , u ˆ1 ˆn and ˆv ' Ž ¨ˆ1 , . . . , ¨ˆn . with ¨ˆi s Mi y u˜i satisfy all the equalities in Ž2.9. in reversed order. This implies that ˜ u and ˆ u satisfy the inequalities in Ž2.2., and therefore they are coupled upper and lower solutions of Ž2.1.. This proves the lemma. As a consequence of Lemmas 2.1 and 2.2 and the existence-uniqueness result in Theorem 2.1 we have the following equivalence relation between the two systems Ž2.1. and Ž2.6.. THEOREM 2.2. Let the conditions in Theorem 2.1 be satisfied, and let hiU s Mi y hi , i s 1, . . . , n. Then uU is the unique solution of Ž2.1. in ²ˆ u, ˜ u:

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if and only if Žu, v. ' ŽuU , M y uU . is the unique solution of Ž2.6. in ²w, ˆ w ˜:, where ²w, ˆw ˜: is gi¨ en by Ž2.8.. Proof. It is easy to see that if uU is a solution of Ž2.1. then Žu, v. ' Žu , M y uU . is a solution of Ž2.6. when hiU s Mi y hi . The uniqueness property of Ž2.6. ensures that ŽuU , M y uU . is the unique solution in ²w, ˆw ˜:. Conversely, if Žu, v. is a solution of Ž2.6. in ²w, ˆw ˜: then, since ŽuU , M y uU . is also a solution, the uniqueness property of Ž2.6. implies that Žu, v. s ŽuU , M y uU .. This shows that u s uU , v s M y uU , and uU is the unique solution of Ž2.1.. U

3. MONOTONE CONVERGENCE OF TIME-DEPENDENT SOLUTIONS In view of Theorem 2.2 the dynamic property of the system Ž2.1. can be determined by the behavior of the solution of the extended problem Ž2.6.. This leads to the consideration of a coupled system of N parabolic equations:

­ wir­ t y L i wi s Fi Ž x, w, wt .

Ž t ) 0, x g V . Ž t ) 0, x g ­ V .

Bi wi s h i Ž x .

Ž i s 1, . . . , N . Ž 3.1. Ž yt i F t F 0, x g V . ,

wi Ž t , x . s hi Ž t , x .

and its corresponding elliptic system yL i wi s Fi Ž x, w, w . Bi wi s h i Ž x .

Ž x g V. Ž x g ­V.

Ž i s 1, . . . , N . ,

Ž 3.2.

where w ' Ž w 1 , . . . , wN .. It is obvious that the extended system Ž2.6. may be considered as a special case of Ž3.1. with N s 2 n, w s Žu, v. and L nqi s L i ,

Bnqi s Bi ,

h nqi s hUi ,

Fnq i Ž x, w, wt . s Gi Ž x, u, v, u t , vt .

hnqi s hiU

Ž i s 1, . . . , n . . Ž 3.3.

Suppose a pair of ordered upper and lower solutions w ˜s ' Ž w ˜1 , . . . , w ˜N ., w ˆs ' Ž w ˆ1 , . . . , w ˆN . to the elliptic system Ž3.2. exist and the N-vector function F Ž x, w, wt . ' Ž F1 Ž x, w, wt . , . . . , FN Ž x, w, wt . .

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NONLINEAR PARABOLIC SYSTEMS

is quasimonotone nondecreasing in ²w ˆs , w ˜s :, where ²w ˆs , w ˜s : '  w g C Ž V . ; w ˆs F w F w ˜s 4 . Recall that w ˜s ' Ž w ˜1, . . . , w ˜N . is an upper solution of Ž3.2. if it satisfies yL i w ˜i G Fi Ž x, w ˜s , w ˜s .

in V

Ž i s 1, . . . , N .

on ­ V

Bi w ˜i G h i Ž x .

Ž 3.4.

and w ˆs ' Ž w ˆ1, . . . , w ˆN . is a lower solution if it satisfies Ž3.4. in reversed order. It is known that given any pair of ordered upper and lower solutions w ˜s , w ˆs , the elliptic system Ž3.2. has a maximal solution ws and a minimal solution ws in ²w ˆs , w ˜s : Žcf. w7, 11x.. It is easy to see from Ž3.4. and Definition 2.2 Žwith bi s d i s 0 for all i . that for any h ' Žh1 , . . . , hN . with w ˆi F hi F w˜i in Q0Ž i. the pair w ˜s , w ˆs are upper and lower solutions of Ž3.1.. By Theorem 2.1 and the arbitrariness of T, problem Ž3.1. has a unique solution wŽ t, x . such that w ˆs Ž x . F w Ž t , x . F w ˜s Ž x .

Ž t ) 0, x g V . .

Ž 3.5.

Denote the solution by wŽ t, x . when h Ž t, x . s w ˜s Ž x . in Q0 , and by wŽ t, x . Ž . Ž t, x . and wŽ t, x . converge when h Ž t, x . s w x . Our aim is to show that w ˆs monotonically to the maximal and minimal solutions ws Ž x . and ws Ž x ., respectively, as t ª `. To achieve this goal we first prepare the following positivity lemma for any function z g C 1, 2 Ž D T . l C Ž D T . satisfying the relation N

­ z ir­ t y L i z i G

N

Ý bi j z j Ž t , x . q Ý c i j z j Ž t y t j , x . js1

in D T

js1

Bi z i G 0 zi Ž t , x . G 0

on ST

in Q0Ž i. Ž i s 1, . . . , N . ,

Ž 3.6.

where bi j ' bi j Ž t, x . and c i j ' c i j Ž t, x . are given functions in C Ž D T .. LEMMA 3.1. Let bi j , c i j g C Ž D T . such that bi j G 0 for j / i and c i j G 0 for all i, j s 1, . . . , N, and let z ' Ž z1 , . . . , z N . satisfy the inequalities in Ž3.6.. Then z G 0 in D T . Proof. Since by hypothesis t U s mint 1 , . . . , tn4 ) 0, the last inequality in Ž3.6. implies that z j Ž t y t U , x . s z j Ž t y t j , x . G 0 for t g w0, t U x, x g V, and j s 1, . . . , N. Using this property and the hypothesis c i j G 0 in Ž3.6.

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C. V. PAO

we obtain the relation N

­ z ir­ t y L i z i G

Ý bi j z j Ž t , x .

in Ž 0, t U = V

js1

on Ž 0, t U = ­ V

Bz i G 0 z i Ž 0, x . G 0

Ž 3.7.

in V .

It follows from the hypothesis bi j G 0 for j / i that z i Ž t, x . G 0 in w0, t U x = V for i s 1, . . . , N Že.g., see w11, p. 564x.. This shows that z G 0 in w0, t U x = V, and therefore z i Ž t y t U , x . G 0 in w0, 2t U x = V for all i. Again by Ž3.6. and c i j G 0, the inequalities in Ž3.7. hold when the interval Ž0, t U x is replaced by Ž0, 2t U x. This leads to z i Ž t, x . G 0 in w0, 2t U x = V for i s 1, . . . , N. A continuation of the same process yields z i Ž t, x . G 0 on w0, mt U x = V for every m s 1, 2, . . . , and i s 1, . . . , N. This proves zŽ t, x . G 0 in D T . As a consequence of the above lemma we have the following monotone property of the solutions w and w. LEMMA 3.2. For any constant d ) 0 the solutions wŽ t, x . and wŽ t, x . of Ž3.1. possess the monotone property w ˆs Ž x . F w Ž t , x . F w Ž t q d , x . F w Ž t q d , x . F w Ž t , x . F w ˜s Ž x . in Rq= V .

Ž 3.8.

Proof. Let zŽ t, x . s wŽ t q d , x . y wŽ t, x .. By Ž3.1. and the mean-value theorem, z ' Ž z1 , . . . , z N . satisfies the relation

­ z ir­ t y L i z i s Fi Ž w Ž t q d , x . , wt Ž t q d , x . . y Fi Ž w Ž t , x . , wt Ž t , x . . N

s

­ Fi

js1

ž

q

Ý

Ý

­ wj

N js1

ž

/

Ž j . zj Ž t, x .

­ Fi ­ wXj

Ž jX . z j Ž t y t j , x .

/

Bi z i s h i Ž x . y h i Ž x . s 0

in D T on ST

z i Ž t , x . s wi Ž t q d , x . y hi Ž t , x . s wi Ž t q d , x . y w ˆi Ž x . in Q0Ž i. ,

Ž 3.9.

where ­ Fir­ wXj , j s 1, . . . , N, denote the partial derivatives of Fi with respect to the components of wt , and j and jX are some intermediate

NONLINEAR PARABOLIC SYSTEMS

761

values in ²w ˆs , w ˜s :. Since by Ž3.5., wi Ž t q d , x . G w ˆi Ž x ., and by the quasimonotone nondecreasing property of F Ž?, w, wt .,

­ Fi ­ wj

Ž j . G 0 for j / i

and

­ Fi ­ wjX

Ž jX . G 0 for all i , j

we conclude from Lemma 3.1 that zŽ t, x . G 0 in D T . The arbitrariness of T ensures that zŽ t, x . G 0 for all t ) 0, x g V. This proves the relation wŽ t q d , x . G wŽ t, x . in Ž3.8.. A similar argument gives the relation wŽ t, x . F wŽ t q d , x . in Rq= V. Finally by the mean-value theorem, the function zŽ t, x . ' wŽ t, x . y wŽ t, x . satisfies the equations in Ž3.9. with z i Ž t, x . s w ˜i Ž x . y w ˆi Ž x . in Q0Ž i. and with possibly some different intermediate values X j and j in ²w ˆs , w ˜s :. It follows again from w ˜i Ž x . G w ˆi Ž x . and Lemma 3.1 that zŽ t, x . G 0. This gives wŽ t, x . F wŽ t, x . in Rq= V which completes the proof of the lemma. Based on the property Ž3.8. we have the following monotone convergence theorem. THEOREM 3.1. Let w ˜s Ž x ., w ˆs Ž x . be a pair of ordered upper and lower solutions of Ž3.2., and let wŽ t, x ., wŽ t, x . be the solutions of Ž3.1. corresponding to h Ž t, x . s w ˜Ž x . and h Ž t, x . s w ˆŽ x . in Q0 , respecti¨ ely. Assume that F Ž?, w, wt . is quasimonotone nondecreasing in ²w ˆs , w ˜s :. Then as t ª `, wŽ t, x . con¨ erges monotonically from abo¨ e to the maximal solution ws Ž x . of Ž3.2., and wŽ t, x . con¨ erges monotonically from below to the minimal solution ws Ž x .. Moreo¨ er, for any initial function h Ž t, x . with w ˆs F h F w ˜s in Q0 the corresponding solution wŽ t, x . of Ž3.1. satisfies the relation wŽ t , x . F wŽ t , x . F wŽ t , x .

Ž t ) 0, x g V . .

Ž 3.10.

Proof. By the monotone property Ž3.8. the limits lim w Ž t , x . s w U Ž x . ,

lim w Ž t , x . s w U Ž x . as t ª `

Ž 3.11.

exist and satisfy the relation wŽ t, x . F w U Ž x . F w U Ž x . F wŽ t, x . in Rq= V. The same argument as that in w11x shows that w U and w U are solutions of Ž3.2.. To show that w U and w U are the respective maximal and minimal solutions in ²w ˆs , w ˜s : we consider the function zŽ t, x . ' wŽ t, x . y ws Ž x ., where ws Ž x . is the maximal solution of Ž3.2.. It is easy to see from Ž3.1. and Ž3.2. that z ' Ž z1 , . . . , z N . satisfies the differential and boundary equations in Ž3.9. for some intermediate values j, jX in ²w ˆs , w ˜s :. Since zŽ t, x . s w ˜s Ž x . y ws Ž x . G 0 in Q0 , Lemma 3.1 implies that zŽ t, x . G 0 in D T . The arbitrariness of T leads to wŽ t, x . G ws Ž x . in Rq= V. Letting t ª ` and

762

C. V. PAO

using the relation Ž3.11. gives w U Ž x . G ws Ž x .. However, since w U g ²w ˆs , w ˜s :, the maximal property of ws Ž x . implies that w U F ws . This proves U w s ws which shows that w U is the maximal solution of Ž3.2. in ²w ˆs , w ˜s :. A similar argument shows that w U s ws and is the minimal solution in ²w ˆs , w ˜s :. Finally, by the quasimonotone nondecreasing property of F Ž?, w, wt ., wŽ t, x . and wŽ t, x . are ordered upper and lower solutions of Ž3.1. when w ˆs F h Ž t, x . F w ˜s in Q0 . The relation Ž3.10. follows from Theorem 2.1. This proves the theorem.

4. ASYMPTOTIC BEHAVIOR AND GLOBAL ATTRACTOR To investigate the asymptotic behavior of the solution of Ž2.1. where the reaction function is mixed quasimonotone we first write the elliptic system Ž1.2. in the form yL i u i s f i Ž x, u i , w u s x a i , w u s x b i , w u s x c i , w u s x d i .

in V on ­ V

Bi u i s h i Ž x .

Ž i s 1, . . . , n . , Ž 4.1.

where u s ' Ž u1 , . . . , u n .. This leads to the following definition of upper and lower solutions. DEFINITION 4.1. A pair of functions ˜ u s ' Žu ˜1 , . . . , u˜n ., ˆu s ' Ž uˆ1 , . . . , uˆn . in C 2 Ž V . l C Ž V . are called coupled upper and lower solutions of Ž4.1. if ˜u s G ˆu s and if yL i u ˜i G f i Ž x, u˜i , ˜u s

ai ,

ˆu s

bi ,

˜u s

ci ,

y Li u ˆi F f i Ž x, uˆi , ˆu s

ai ,

˜u s

bi ,

ˆu s

ci ,

Bi u ˜i G h i Ž x . G Bi uˆi

ˆu s

. ˜u s d .

Ž i s 1, . . . , n . .

di

i

Ž 4.2.

Notice that the above definition is slightly different from the definition of upper and lower solutions for the usual coupled elliptic systems when wu s x d contains the component u i . However, if u i is not a component of i wu s x d , and in particular, if fŽ?, u, u. is quasimonotone nondecreasing Žthat i is, bi s d i s 0. then Definition 4.1 coincides with the usual definition of upper and lower solutions Žcf. w11x.. Moreover when bi s d i s 0 for all i, the pair ˜ u s and ˆ u s are not coupled and are again referred to as ordered upper and lower solutions. It is obvious from Ž4.2. that for any pair of coupled Žor ordered. upper and lower solutions ˜ u s, ˆ u s of Ž4.1. they are also

763

NONLINEAR PARABOLIC SYSTEMS

upper and lower solutions of Ž2.1. when ˆ us F h F ˜ u s in Q0 . Hence by Theorem 2.1, the sector ²ˆ u s, ˜ u s : given by Ž2.3. with C Ž D T . replaced by C Ž V . is an invariant set of the system Ž2.1.. It is easy to see as for the usual coupled elliptic systems that the sequences  uŽsm.4 '  u1Ž m., . . . , uŽnm.4 ,  uŽsm.4 '  u1Ž m., . . . , uŽnm.4 obtained from the linear iteration process yL i uŽi m. q K i uŽi m. s f iU x, uŽi my1. , uŽsmy1.

ž

uŽsmy1. y L i uŽi m. q K i uŽi m. s f iU x, uŽi my1. , uŽsmy1.

ž

uŽsmy1. Bi uŽi m. s Bi uŽi m. s h i Ž x .

ai , ci , ai , ci ,

uŽsmy1. uŽsmy1. uŽsmy1. uŽsmy1.

bi , di

/

bi , di

Ž i s 1, . . . , n .

/ Ž 4.3.

with uŽ0. u s and uŽ0. u s , where K i is the Lipschitz constant in Ž2.4. s s˜ s sˆ U and f i Ž x, u, u. s K i u i q f i Ž x, u, u., are uniquely determined and possess the monotone property

ˆu s F uŽsm. F uŽsmq1. F uŽsmq1. F uŽsm. F ˜u s

in V , m s 1, 2, . . . . Ž 4.4.

Furthermore, the limits lim uŽsm. s u s ,

lim uŽsm. s u s as m ª `

Ž 4.5.

exist and satisfy the equations

wu s x b i , u s c i , wu s x d i . yL i u i s f i Ž x, u i , w u s x a i , u s b i , w u s x c i , u s d i . yL i u i s f i Ž x, u i , u s

ai ,

Bi u i s Bi u i s h i Ž x .

on ­ V Ž i s 1, . . . , n .

in V

Ž 4.6.

Žcf. w11x.. The limits u s , u s , called quasisolutions of Ž4.1. in ²ˆ u s, ˜ u s :, are ordered but in general are not true solutions. However, if fŽ?, u, u. is quasimonotone nondecreasing in ²ˆ u s, ˜ u s :, then both u s and u s are true solutions of Ž4.1.. In fact, they are the respective maximal and minimal solutions in ²ˆ u s, ˜ u s : Žsee Theorems 8.10.1]8.10.3 in w11x.. Our goal is to show that for any initial function h in ²ˆ u s, ˜ u s : the corresponding solution u not only remains in ²ˆ u s, ˜ u s : but also enters the sector ²u s , u s : as t ª `, where ²u s , u s : '  u g C Ž V . ; u s F u F u s in V 4 .

Ž 4.7.

764

C. V. PAO

To achieve the above goal we imbed the system Ž4.1. into the extended system: yL i u i s f i Ž x, u i , w u s x a i , w M y vs x b i , w u s x c i , w M y vs x d i . y L i ¨ i s yf i Ž x, Mi y ¨ i , w M y vs x a i , w u s x b i , w M y vs x c i , w u s x d i . Bi u i s h i Ž x . ,

Bi ¨ i s hUi Ž x . Ž i s 1, . . . , n . .

Ž 4.8.

It is obvious that the above system is the steady-state problem of Ž2.6., and by Lemma 2.1 the reaction function at the right-hand side of Ž4.8. is quasimonotone nondecreasing in ²ˆ u s, ˜ u s :. This property leads to the following analogous result as that in Lemma 2.2. LEMMA 4.1. The pair w ˜s ' Ž˜u s , M y ˆu s ., w ˆs ' Žˆu s , M y ˜u s . are ordered upper and lower solutions of Ž4.8. if and only if ˜ u s and ˆ u s are coupled upper and lower solutions of Ž4.1.. Proof. The proof is similar to that for Lemma 2.2 and is omitted. As a consequence of Lemma 4.1 and Theorem 8.10.1 of w11x the extended system Ž4.8. has a maximal solution ws ' Žu s , vs . and a minimal solution ws ' Žu s , vs . in ²w ˆs , w ˜s : when system Ž4.1. has a pair of coupled upper and lower solutions ˜ u s, ˆ u s . In the following lemma we establish an equivalence relation between the maximal and minimal solutions of Ž4.8. and the quasisolutions of Ž4.1.. LEMMA 4.2. Let ˜ u s, ˆ u s be a pair of coupled upper and lower solutions of Ž4.1., and let w ˜s ' Ž˜u s , M y ˆu s ., w ˆs s Žˆu s , M y ˜u s .. Then the limits u s , u s in Ž4.5. are the quasisolutions of Ž4.1. in ²ˆ u s, ˜ u s : if and only if Žu s , M y u s . and Žu s , M y u s . are the maximal and minimal solutions of Ž4.8. in ²w ˆs , w ˜s :. Proof. Suppose u s , u s are the quasisolutions of Ž4.1. in ²ˆ u s, ˜ u s :. Since by Lemma 4.1 the pair w ˜s and w ˆs are ordered upper and lower solutions of Ž4.8., the sequence  uŽ m., v Ž m.4 obtained from the linear iteration process yL i uŽi m. q K i uŽi m. s f iU uŽi my1. , uŽsmy1.

ž

uŽsmy1.

ai , ci ,

M y vsŽ my1. M y vsŽ my1.

y L i ¨ iŽ m. q K i ¨ iŽ m. s yf iU Mi y ¨ iŽ my1. , M y vsŽ my1.

ž

M y vsŽ my1. Bi uŽi m. s h i Ž x . ,

Bi ¨ iŽ m. s

hUi

ai , ci ,

bi , di

/

uŽsmy1. uŽsmy1.

Ž x . Ž i s 1, . . . , n .

bi , di

/

Ž 4.9. ŽuUs , vsU .

converges monotonically from above to the maximal solution in Ž0. Ž0. . ²w : Ž Ž . , w if u , v s u , M y u , and it converges monotonically from ˆs ˜s ˜s ˆs s s

765

NONLINEAR PARABOLIC SYSTEMS

Ž0. . below to the minimal solution ŽuUs , vsU . if ŽuŽ0. s Žˆ u s, M y ˜ u s . Žcf. s , vs Ž m. Ž m. w11x.. Denote these two sequences by  u e , ve 4 and  uŽem., veŽ m.4 , respectively. In view of uŽ0. u s , uŽ0. u s and the uniqueness property of the s s˜ s sˆ sequence in Ž4.9., a comparison between the iteration processes Ž4.3. and Ž4.9. shows that

uŽem. s uŽsm. ,

uŽem. s uŽsm. ,

veŽ m. s M y uŽsm. ,

veŽ m. s M y uŽsm.

Ž 4.10. for every m s 1, 2, . . . . This implies that the two iteration processes are equivalent. It follows from uŽsm. ª u s and uŽsm. ª u s as m ª ` that

Ž uUs , vsU . '

lim Ž uŽem. , veŽ m. . s Ž u s , M y u s .

mª`

Ž uUs , vsU . ' lim Ž uŽem. , veŽ m. . s Ž u s , M y u s . . mª`

Ž 4.11.

This shows that the maximal and minimal solutions of Ž4.8. are given by Žu s , M y u s . and Žu s , M y u s ., respectively. Conversely if Žu s , M y u s . and Žu s , M y u s . are the maximal and minimal solutions of Ž4.8., then they are the limits of the respective sequences  uŽem., veŽ m.4 and  uŽem., veŽ m.4 obtained from Ž4.9.. In view of Ž4.10., u s s lim uŽsm.

and

u s s lim uŽsm.

as m ª `.

Since  uŽsm.4 and  uŽsm.4 are the sequences governed by Ž4.3. with uŽ0. s ˜ us and uŽ0. s ˆ u s we conclude that u s and u s are the quasisolutions of Ž4.1. in ²ˆ u s, ˜ u s :. This proves the lemma. Based on Lemma 4.2 and Theorem 3.1 we have the following conclusion. THEOREM 4.1. Let ˜ u s, ˆ u s be a pair of coupled upper and lower solutions of Ž4.1., and let hypothesis ŽH. hold in L ' ²ˆ u s, ˜ u s :. Denote by u s ' Ž u1 , . . . , u n . and u s ' Ž u1 , . . . , u n . the quasisolutions of Ž4.1. which satisfy Ž4.6.. Then for any initial function h satisfying ˆ us F h F ˜ u s in Q0 the corresponding solution uU Ž t, x . of Ž2.1. possesses the property u s Ž x . F uU Ž t , x . F u s Ž x .

as t ª ` Ž x g V . .

Ž 4.12.

Moreo¨ er, if u s s u s ' uUs then uUs is the unique solution of Ž4.1. in ²ˆ u s, ˜ u s: and uU Ž t, x . ª uUs Ž x . as t ª `. Proof. It is easily seen by considering w s Žu, v. and using the relations Ž2.5. and Ž3.3. that problems Ž2.6. and Ž4.8. are special cases of the systems in Ž3.1. and Ž3.2., respectively. Since by Lemma 4.1, w ˜s ' Ž˜u s , M y ˆu s . and Ž . w ' u , M y u are ordered upper and lower solutions of Ž4.8., and ˆs ˆ s ˜s

766

C. V. PAO

since the function F Ž?, w, wt . given by Ž2.7. is quasimonotone nondecreasing in ²w ˆs , w ˜s :, the conditions in Theorem 3.1 are satisfied. Hence if w ' Žu, v. and w ' Žu, v. are the solutions of Ž2.6. with their respective initial function Žh , hU . ' Ž˜ u s, M y ˆ u s . and Žh , hU . ' Žˆ u s, M y ˜ u s . then the limits lim Ž u Ž t , x . , v Ž t , x . . s Ž u s Ž x . , vs Ž x . . ,

tª`

lim Ž u Ž t , x . , v Ž t , x . . s Ž u s Ž x . , vs Ž x . .

tª`

Ž 4.13.

exist and are the respective maximal and minimal solutions of Ž4.8. in ²w ˆs , w ˜s :. Moreover, for arbitrary Žh , hU . g ²w ˆs , w ˜s : which is equivalent to u ˆi F hi F u˜i ,

Mi y u ˜i F hiU F Mi y uˆi

in Q0Ž i. Ž i s 1, . . . , n . ,

Ž 4.14. the corresponding solution Žu, v. satisfies the relation

Ž u Ž t , x . , v Ž t , x . . F Ž u Ž t , x . , v Ž t , x . . F Ž u Ž t , x . , v Ž t , x . . . Ž 4.15. Now if hiU s Mi y hi then Ž4.14. holds for any h g ²ˆ u s, ˜ u s : and by Theorem 2.2, Žu, v. s ŽuU , M y uU ., where uU is the unique solution of Ž2.1.. This implies that Žu, v. F ŽuU , M y uU . F Žu, v. which yields the relation u Ž t , x . F uU Ž t , x . F u Ž t , x .

in Rq= V .

Ž 4.16.

It follows from Ž4.13. that relation Ž4.12. holds. Finally if u s s u s ' uUs , then by the maximal and minimal property of u s and u s , uUs is the unique solution of Ž4.1. in ²ˆ u s, ˜ u s :. The convergence of uU Ž t, x . to uUs Ž x . follows Ž . from 4.12 . This proves the theorem. It is seen from the above theorem that the sector ²u s , u s : between the two quasisolutions u s , u s is an attractor for the system Ž2.1. if ˆ us F h F ˜ us in Q0 . For arbitrary h we have the following result which is useful in the investigation of global attractors in some model problems. THEOREM 4.2. Let the conditions in Theorem 4.1 hold, and let u ' Ž u1 , . . . , u n . be the solution of Ž2.1. corresponding to an arbitrary initial function h Ž t, x .. If there exist tU G 0 such that u ˆi Ž x . F u i Ž t , x . F u˜i Ž x .

for all t g w tU y t i , tU x Ž i s 1, . . . , n . ,

Ž 4.17. then all the conclusions in Theorem 4.1 remain true.

767

NONLINEAR PARABOLIC SYSTEMS

Proof. Let t s t y tU , UŽt , x . s uŽt q tU , x ., and consider the system Ž2.1. with t replaced by t q tU . Since for each i s 1, . . . , n, u i Ž t y t i , x . s u i Žt q tU y t i , x . s Ui Žt y t i , x ., where U ' ŽU1 , . . . , Un ., we see that U satisfies the same equations in Ž2.1. with respect to Žt , x .. The initial condition for U is given by Ui Ž t , x . s u i Ž t y tU , x .

Ž yt i F t F 0, x g V . .

In view of Ž4.17., Ui Žt , x . satisfies the relation u ˆi Ž x . F Ui Ž t , x . F u˜i Ž x .

Ž yt i F t F 0, x g V . .

It follows from Theorem 4.1 that u s Ž x . F UŽt , x . F u s Ž x . as t ª `. This proves the relation Ž4.12. for uŽ t, x .. When u s s u s ' uUs the convergence of uŽ t, x . to uU Ž x . follows from Ž4.12.. For the parabolic-ordinary system Ž1.3. the requirements of upper and lower solutions ˜ u s ' Žu ˜1 , . . . , u˜n ., ˆu s ' Ž uˆ1 , . . . , uˆn . of the corresponding Ž steady-state problem 1.4. are given by yL i u ˜i G f i Ž x, u˜i , ˜u s

ai ,

ˆu s

bi ,

˜u s

ci ,

yL i u ˆi F f i Ž x, uˆi , ˆu s

ai ,

˜u s

bi ,

ˆu s

ci ,

0 G f i Ž x, u ˜i , ˜u s

ai ,

0 F f i Ž x, u ˆi , ˆu s

us ai , ˜

ˆu s

˜u s

ci ,

us bi , ˆ

ci ,

bi ,

Bi u ˜i G h i Ž x . G Bi uˆi

ˆu s

ˆu s

. ˜u s d . di

Ž i s 1, . . . , nU .

i

. ˜u s d . di

Ž i s nU q 1, . . . , n .

i

Ž i s 1, . . . , nU . .

Ž 4.18.

It is easy to show by the same reasoning as that for the elliptic system Ž4.1. that the sequence  uŽsm.4 ,  uŽsm.4 obtained from the iteration process Ž4.3. with uŽ0. u s , uŽ0. u s , and L i s 0 Žand without the boundary condition. s s˜ s sˆ U for i s n q 1, . . . , n, converges monotonically to a pair of quasisolutions u s ' Ž u1 , . . . , u n ., u s ' Ž u1 , . . . , u n . which satisfy the equations

wu s x b i , u s c i , wu s x d i . yL i u i s f i Ž x, u i , w u s x a i , u s b i , w u s x c i , u s d i . yL i u i s f i Ž x, u i , u s

ai ,

wu s x b i , u s c i , wu s x d i . 0 s f i Ž x, u i , w u s x a i , u s b i , w u s x c i , u s d i . 0 s f i Ž x, u i , u s

ai ,

Bi u i s Bi u i s h i Ž x .

Ž i s 1, . . . , nU . Ž i s nU q 1, . . . , n .

Ž i s 1, . . . , nU . .

Ž 4.19.

In terms of the above quasisolutions we have the following asymptotic behavior of the solution for the system Ž1.3..

768

C. V. PAO

THEOREM 4.3. Let ˜ u s, ˆ u s be a pair of coupled upper and lower solutions of Ž1.4., and let hypothesis ŽH. hold in L ' ²ˆ u s, ˜ u s :. Denote by u s ' Ž u1 , . . . , u n . and u s ' Ž u1 , . . . , u n . the quasisolutions of Ž1.4. which satisfy Ž4.19.. Then all the conclusions in Theorem 4.1 hold true for the system Ž1.3. when ˆ us F h F ˜ u s in Q0 . If there exists tU G 0 such that the solution uŽ t, x . of Ž1.3. satisfies Ž4.17., then the same conclusion in Theorem 4.2 holds for Ž1.3. with arbitrary initial function h Ž t, x .. Proof. The proof follows from the same argument as that for the system Ž2.1.. Details are omitted.

5. APPLICATIONS In this section we give some applications of the results given in the previous sections to three model problems arising from ecology and nuclear engineering. This includes a logistic-delayed equation, a nuclear reactor model, and a Volterra]Lotka competition model with n competing species. The dynamics of these model problems have been investigated by many researchers in the current literature Žcf. w4, 6, 8, 14, 15, 20, 21, 23, 24x.. ŽA. A Diffusion Logistic Equation with Time Delays In the traditional logistic equation with time delays if the effect of disposition is taken into consideration, then the equation for the population density u ' uŽ t, x . in a bounded habitat V is governed by the scalar parabolic initial boundary-value problem

­ ur­ t y Lu s u Ž a y bu y cut . Bu ' a­ ur­n q b Ž x . u s 0 uŽ t , x . s h Ž t , x .

Ž t ) 0, x g V . Ž t ) 0, x g ­ V .

Ž 5.1.

Ž yt F t F 0, x g V . ,

where ut ' uŽ t y t , x . for some t ) 0, Lu s = ? Ž D=u., and D ' DŽ x ., a ' aŽ x ., b ' bŽ x ., and c ' cŽ x . are all positive smooth functions in V. The boundary coefficients are given either by a s 0, b Ž x . ' 1 or by a s 1, b Ž x . G 0. It is clear that problem Ž5.1. is a special case of Ž2.1. with n s 1, and the reaction function f Ž x, u, ut . ' u Ž a Ž x . y b Ž x . u y c Ž x . ut . is mixed quasimonotone in L s Rq with a1 s b1 s c1 s 0 and d1 s 1. The above problem has been given considerable attention in recent years, and most of the discussions are concerned with the large time behavior of

NONLINEAR PARABOLIC SYSTEMS

769

the solution in relation to the positive solution of the corresponding steady-state problem yLu s u Ž a y bu y cu .

Ž x g V. Ž x g ­V.

Bu s 0

Ž 5.2.

Žcf. w20, 21, 23, 24x.. Here we apply the results given in Sections 3 and 4 to study the asymptotic behavior of the solution by constructing some suitable upper and lower solutions of Ž5.2.. In view of Definition 4.1, the requirements of upper and lower solutions u ˜s , uˆs for Ž5.2. become yLu ˜s G u˜s Ž a y bu˜s y cuˆs . in V ,

Bu ˜s G 0 on ­ V

y Lu ˆs F uˆs Ž a y buˆs y cu˜s . in V ,

Bu ˆs F 0 on ­ V .

Ž 5.3.

Unlike the usual scalar boundary-value problem, the pair u ˜s , uˆs in Ž5.3. are coupled. Our construction of upper and lower solutions is based on the smallest eigenvalue l ' lŽ p . of the eigenvalue problem Lf y p Ž x . f q l a Ž x . f s 0 in V ,

Bf s 0 on ­ V ,

Ž 5.4.

where p s pŽ x . is a given continuous nonnegative function in V. It is well known that lŽ p . is nonnegative and increasing with respect to p G 0, and if pŽ x . and b Ž x . are not both identically zero then lŽ p . is strictly positive Žcf. w17x.. In either case, the corresponding eigenfunction f Ž x . of lŽ p . can be chosen positive with max f Ž x . s 1 in V. It is also known that problem Ž5.2. has only the trivial solution u s s 0 if lŽ0. G 1 and it has a unique positive solution uUs Ž x . if lŽ0. - 1, where lŽ0. is the smallest eigenvalue of Ž5.4. corresponding to pŽ x . ' 0 Žcf. w11x.. Consider first the case lŽ0. G 1. It is easy to verify that for any constant M G aŽ x .rbŽ x . in V the pair u ˜s s M and uˆs s 0 satisfy all the inequalities in Ž5.3.. Since for problem Ž5.2. the monotone iteration process Ž4.3. becomes yLuŽ m. q KuŽ m. s KuŽ my1. q uŽ my1. Ž a y buŽ my1. y cuŽ my1. . y LuŽ m. q KuŽ m. s KuŽ my1. q uŽ my1. Ž a y buŽ my1. y cuŽ my1. . Ž 5.5. BuŽ m. s BuŽ m. s 0,

m s 1, 2, . . . ,

we see that if the coupled initial iterations are uŽ0. s M and uŽ0. s 0 then uŽ m. s 0 for every m and uŽ m. ª 0 as m ª ` Žcf. w11x.. It follows from Theorem 4.1 that u s s u s s 0 and the time-dependent solution uŽ t, x . of Ž5.1. converges to 0 as t ª ` whenever 0 F h Ž t, x . F M in wyt , 0x = V.

770

C. V. PAO

The arbitrariness of M ensures that uŽ t, x . ª 0 as t ª ` for any h Ž t, x . G 0. We next consider the more interesting case lŽ0. - 1. In this situation the scalar boundary-value problem yLUs s Us Ž a y bUs . in V ,

BUs s 0 on ­ V

Ž 5.6.

has a unique positive solution Us and Us - arb when b Ž x . k 0. We seek a pair of positive upper and lower solutions of Ž5.2. in the form u ˜s s r Us and u ˆs s d Us , where r and d are some positive constants with r ) d . Indeed, by Ž5.3. and Ž5.6., this pair are coupled upper and lower solutions of Ž5.2. if

r Us Ž a y bUs . G r Us Ž a y br Us y c d Us . d Us Ž a y bUs . F d Us Ž a y bd Us y c r Us . . The above inequalities are clearly satisfied by a sufficiently small d ) 0 if 1 F r - brc in V. In view of Ž5.5. and Theorem 8.10.1 of w11x, problem Ž5.2. has a pair of quasisolutions u s , u s which are in ² d Us , r Us : and satisfy the equations yLu s s u s Ž a y bu s y cu s . in V ,

Bu s s 0 on ­ V

y Lu s s u s Ž a y bu s y cu s . in V ,

Bu s s 0 on ­ V .

Since Lu s = ? Ž D=u. and Bu s a­ ur­n q b u, Green’s theorem implies that 0s

HV Ž u Lu s

s

y u s Lu s . dx s

HV Ž c y b . u u Ž u s

s

s

y u s . dx

Žcf. w11x.. It follows from c - b and the positive property of u s and u s that u s s u s . The uniqueness property of uUs ensures that u s s u s s uUs . By Theorem 4.1 the solution uŽ t, x . of Ž5.1. converges to uUs Ž x . as t ª ` when d Us Ž x . F h Ž t, x . F r Us Ž x . in wyt , 0x = V. For arbitrary h Ž t, x . G 0 with h Ž0, x . k 0, a comparison between the solution uŽ t, x . of Ž5.1. and the solution UŽ t, x . of the usual parabolic boundary-value problem

­ Ur­ t y LU s U Ž a y bU . ,

BU s 0,

U Ž 0, x . s h Ž 0, x . Ž 5.7.

shows that uŽ t, x . - UŽ t, x . in Ž0, `. = V. Since UŽ t, x . ª Us Ž x . as t ª ` Žcf. w11x. there exists a finite T1 ) 0 such that uŽ t, x . F r Us Ž x . for all

771

NONLINEAR PARABOLIC SYSTEMS

t G T1. Furthermore, by the maximum principle, the solution w Ž t, x . of the problem

­ wr­ t y Lw q Ž cut . w s w Ž a y bw . ,

Bw s 0,

w Ž 0, x . s h Ž 0, x . is positive in Ž0, `. when h Ž0, x . k 0. Since uŽ t, x . is also a solution of the above problem the uniqueness property of the solution ensures that uŽ t, x . s w Ž t, x .. Hence there exist constants d ) 0 and tU G T1 such that d Us F uŽ t, x . F r Us for all t g w tU y t , tU x. As a consequence of Theorems 4.1 and 4.2 we have the following conclusion: THEOREM 5.1. Let lŽ0. be the smallest eigen¨ alue of Ž5.4. corresponding to pŽ x . s 0, and let uŽ t, x . be the solution of Ž5.1. with arbitrary initial function h Ž t, x . G 0 and h Ž0, x . k 0. Then uŽ t, x . ª 0 as t ª ` if lŽ0. G 1, and uŽ t, x . ª uUs Ž x . as t ª ` if lŽ0. - 1 and bŽ x . ) cŽ x . in V, where uUs Ž x . is the unique positi¨ e solution of Ž5.2.. Remark 5.1. Theorem 5.1 gives a sufficient condition on the positive functions a, b, and c for the global stability of the trivial solution and the positive solution uUs Ž x .. Similar conditions are also given in w20, 21, 23, 24x when a, b, c, and D are all constants and the boundary condition is of either Neumann type or Dirichlet type. ŽB. A Nuclear Reactor Model In some nuclear reactor feedback systems where the reactor is considered spatially homogeneous and the temperature effect to the reactor power has a time delay, a mathematical model for the reactor power P and the temperature differential T is given by the ordinary-parabolic system Žcf. w6, 14x.

­ Pr­ t s P yaŽ P y Pe . y bTt ­ Tr­ t y Lu s c Ž P y Pe . y T BT s h Ž x . P Ž 0, x . s P0 Ž x . ,

Ž t ) 0, x g V .

Ž t ) 0, x g ­ V .

T Ž t , x . s T0 Ž t , x .

Ž yt F t F 0, x g V . , Ž 5.8.

where L and B are the same as that in Ž5.1., Tt ' T Ž t y t , x . for some t G 0, Pe ' Pe Ž x . is a positive function representing the equilibrium power,

772

C. V. PAO

and a, b, and c are positive rate constants. The above system is a special case of Ž1.3. with Ž u1 , u 2 . s ŽT, P . and with f 1 Ž x, T , P . s c Ž P y Pe . y T f 2 Ž x, Tt , P . s P yaŽ P y Pe . y bTt .

Ž 5.9.

Clearly the function f ' Ž f 1 , f 2 . is mixed quasimonotone in R = Rq. Since the power P is always nonnegative we may define, if necessary, f 2 Ž?, P, T . s 0 whenever P - 0. With this modification, f is mixed quasimonotone in the whole space R 2 . Notice that since T is the differential between the reactor temperature and surrounding temperature, it is not necessarily nonnegative. For this reason the boundary and initial functions h and T0 are allowed to have positive as well as negative values in their respective domains. Consider the steady-state problem of Ž5.8. which is given by Ps yaŽ Ps y Pe . y bTs s 0

in V

yLTs s c Ž Ps y Pe . y Ts

Ž 5.10.

on ­ V .

BTs s h Ž x .

It is easy to verify that the above problem has exactly two solutions in the form Ž0, TsŽ0. . and Ž PsU , TsU ., where TsŽ0. and TsU are the respective solutions of the linear boundary-value problems BTsŽ0. s h on ­ V

yLTsŽ0. q cTsŽ0. s ycPe in V , yLTsU q c Ž 1 q bra. TsU s 0 in V ,

BTsU s h on ­ V ,

Ž 5.11.

and PsU s Pe y Ž bra.TsU Ž PsU Ž x . ' 0 at those points x where Pe Ž x . Ž bra.TsU Ž x ... When hŽ x . ' 0, the solution Ž PsU , TsU . is reduced to Ž Pe , 0. while Ž0, TsŽ0. . is nonpositive. To investigate the stability and instability of these solutions we construct a suitable pair of upper and lower solutions of Ž5.10. which are denoted by Ž P˜s , T˜s . and Ž Pˆs , Tˆs .. In view of the mixed quasimonotone property of Ž f 1 , f 2 ., the requirements on this pair are given by P˜s ya Ž P˜s y Pe . y bTˆs F 0, yLT˜s G c yLTˆs F c

Pˆs ya Ž Pˆs y Pe . y bT˜s G 0

Ž P˜s y Pe . y T˜s Ž Pˆs y Pe . y Tˆs

,

BT˜s G h Ž x .

,

BTˆs F h Ž x . .

Ž 5.12.

773

NONLINEAR PARABOLIC SYSTEMS

Assume that the rate constants in Ž5.8. satisfy the condition c Ž 1 y bra. q l 0 ) 0,

Ž 5.13.

where l 0 G 0 is the smallest eigenvalue of Ž5.4. corresponding to pŽ x . ' 0 and aŽ x . ' 1. Then for any nontrivial function hU Ž x . G 0 the linear boundary value problem yLwU q c Ž 1 y bra. wU s 0 in V ,

BwU s hU Ž x . on ­ V Ž 5.14.

has a unique positive solution wU s wU Ž x .. We choose hU Ž x . such that wU - Ž arb. Pe in V, and seek a pair of upper and lower solutions in the form U

U

ž P˜ , T˜ / s Ž P q r Ž bra. w , r w . , ž Pˆ , Tˆ / s Ž P y r Ž bra. w , yr w . , s

s

e

U

s

s

U

e

Ž 5.15.

where r is any constant satisfying 1 F r F aPer Ž bwU .

r k aPer Ž bwU . .

and

Ž 5.16.

The above condition on r ensures that Pˆs G 0 and Pˆs k 0. It is easy to verify that the pair in Ž5.15. satisfy the first two inequalities in Ž5.12. while the remaining inequalities are reduced to

r Ž yLwU q cwU . G c r Ž bra. wU , y r Ž yLwU q cwU . F yc r Ž bra. wU ,

r BwU G h Ž x . yr BwU F h Ž x . .

By the relation Ž5.14. the above inequalities hold for any hŽ x . satisfying < hŽ x .< F r hU Ž x .. Under this condition, the pair Ž P˜s , T˜s ., Ž Pˆs , Tˆs . given by Ž5.15. are coupled upper and lower solutions of Ž5.10.. Hence by using this pair as the coupled initial iterations in the iteration process Ž4.3. Žwith Ž u1 , u 2 . s ŽT, P ., Ž L1 , L2 . s Ž L, 0. and Ž f 1 , f 2 . given by Ž5.9.. we can construct two monotone sequences which converge to a pair of quasisolutions Ž Ps , Ts ., Ž Ps , Ts . such that Ž Ps , Ts . G Ž Ps , Ts . and Ps ya Ž Ps y Pe . y bTs s 0,

Ps yaŽ Ps y Pe . y bTs s 0

yLTs q cTs s c Ž Ps y Pe . ,

BTs s h Ž x .

yLTs q cTs s c Ž Ps y Pe . ,

BTs s h Ž x . .

We show that

Ž Ps , Ts . s Ž Ps , Ts . s Ž PsU , TsU . .

Ž 5.17.

774

C. V. PAO

Since Pˆs G 0 and Pˆs k 0 the first two equations in Ž5.17. give Ps y Pe s y Ž bra. Ts ,

Ps y Pe s y Ž bra. Ts .

Ž 5.18.

This leads to the relation yLTs q cTs s y Ž bcra. Ts ,

BTs s h Ž x .

y LTs q cTs s y Ž bcra. Ts ,

BTs s h Ž x . .

A subtraction of the above two equations yields yLT U q c Ž 1 y bra. T U s 0,

BT U s 0,

where T U s Ts y Ts . In view of Ž5.13. and Ts G Ts we have T U s 0 which is equivalent to Ts s Ts . It follows from Ž5.18. and Theorem 4.1 that ŽTs , Ps . s ŽTs , Ps . and is the unique solution of Ž5.10. between Ž Pˆs , Tˆs . and Ž P˜s , T˜s .. Since Ž PsU , TsU . is the only solution of Ž5.10. with Ps G 0 and Ps k 0 we conclude that Ž Ps , Ts . s Ž Ps , Ts . s Ž PsU , TsU .. As a consequence of Theorem 4.3 and the fact that Pˆs Ž x . can be made arbitrarily close to 0 we have the following conclusion. THEOREM 5.2. Let wU Ž x . be the positi¨ e solution of Ž5.14. for some h Ž x . G 0 such that wU - Ž arb. Pe , and let a, b, and c satisfy condition Ž5.13.. Then for any initial and boundary functions Ž P0 , T0 . and hŽ x . such that U

< P0 y Pe < F r Ž bra. wU in V ,

T0 Ž t , x . F r wU in w yt , 0 . = V ,

h Ž x . F r hU Ž x .

on ­ V ,

Ž 5.19.

where r is any constant satisfying Ž5.16., problem Ž5.8. has a unique global solution Ž P, T . such that lim Ž P Ž t , x . , T Ž t , x . . s Ž PsU Ž x . , TsU Ž x . .

as t ª `.

Ž 5.20.

Moreo¨ er, the solution Ž0, TsŽ0. Ž x .. is unstable. When hŽ x . ' 0, the steady-state solutions of Ž5.10. are reduced to Ž Pe , 0. and Ž0, TsŽ0. .. In this situation we have COROLLARY 5.1. Under the conditions in Theorem 5.2 and hŽ x . ' 0 the global solution Ž P Ž t, x ., T Ž t, x .. of Ž5.8. con¨ erges to Ž Pe , 0. as t ª `, and the solution Ž0, TsŽ0. . is unstable. ŽC. A Competition Model with n-Competing Species In the Volterra]Lotka competition model with n-competing species where the effect of disposition and time delay are both taken into consid-

775

NONLINEAR PARABOLIC SYSTEMS

eration, the equations governing the population densities u i ' u i Ž t, x . of the competition species are given by n

ž

­ u ir­ t y L i u i s u i a i y

n

Ý bi j u j y Ý c i j Ž u j . t

js1

js1

/

Bi u i s 0 u i Ž t , x . s hi Ž t , x .

Ž i s 1, . . . , n . ,

Ž 5.21.

where Ž u j .t ' u j Ž t y t j , x ., L i and Bi are the same as that in Ž1.1., and a i ' a i Ž x ., bi j ' bi j Ž x . and c i j ' c i j Ž x . are all nonnegative functions in V. We assume that a i , bi j , and c i j are in C a Ž V ., and a i ) 0, bi i ) 0 in V. This implies that the reaction function at the right-hand side of Ž5.21. is quasimonotone nonincreasing and satisfies the hypothesis ŽH. with L s ŽRq. n. Hence the requirements of upper and lower solutions for the steady-state problem of Ž5.21. become n

ž ž

yL i u ˜i G u˜i ai y bi i u˜i y y Li u ˆi F uˆi ai y bi i uˆi y Bi u ˜i G 0 G Bi uˆi

n

Ý bi j uˆj y Ý c i j uˆj j/i

js1

n

n

Ý bi j u˜j y Ý c i j u˜j j/i

js1

/ /

Ž i s 1, . . . , n . .

Ž 5.22.

It is easy to verify that for any constants Mi satisfying Mi G Ž a irbii . in V, i s 1, . . . , n, the pair ˜ u s Ž M1 , . . . , Mn . and ˆ u s Ž0, . . . , 0. satisfy all the Ž . inequalities in 5.22 . By Theorem 2.1, the system Ž5.21. has a unique global solution uŽ t, x . such that 0 F uŽ t, x . F M in Rq= V whenever 0 F h Ž t, x . F M in Q0 . Moreover, a comparison between the components u i of u and the solution Ui of the scalar boundary-value problem

­ Ur i ­ t y L i Ui s Ui Ž a i y bii U . ,

Bi Ui s 0,

Ui Ž 0, x . s hi Ž 0, x .

Ž 5.23. shows that u i F Ui in Rq= V, i s 1, . . . , n, Let l i Ž pi . and f i be the smallest eigenvalue and its corresponding Žnormalized. positive eigenfunction of the eigenvalue problem L i f i y pi f i q l i a i f i s 0 in V ,

Bi f i s 0 on ­ V .

Ž 5.24.

776

C. V. PAO

Since for any nontrivial hi Ž0, x . G 0 the solution Ui Ž t, x . of Ž5.23. converges to 0 as t ª ` when l i Ž0. G 1, and it converges to the unique positive solution UiU Ž x . of the boundary-value problem Bi Ui s 0 on ­ V ,

yL i Ui s Ui Ž a i y bii Ui . in V ,

Ž 5.25.

when l i Ž0. - 1, the relation 0 - u i - Ui implies that u i ª 0 as t ª ` if l i Ž0. G 1. To ensure the coexistence of the competing species it is necessary that l i Ž0. - 1 for all i s 1, . . . , n. This requirement ensures that a unique positive solution UiU Ž x . to Ž5.25. exists and can be used to construct a pair of upper and lower solutions for the steady-state problem of Ž5.21.. Specifically, we seek such a pair in the form ˜ u s s ŽU1U , . . . , UnU ., ˆu s s Ž d 1 f 1 , . . . , dn fn ., where for each i, di is a small positive constant. Indeed, this pair satisfy all the inequalities in Ž5.22. if

yL i UiU G UiU a i y bi i UiU y

ž

n

n

Ý bi j Ž d j f j . y Ý c i j Ž d j f j . j/i

js1 n

ž

yd i L i f i F d i f i a i y bii d i f i y

/

n

Ý bi j UjU y

Ý c i j UjU

j/i

js1

/

.

In view of Ž5.24. and Ž5.25., the first inequality is trivially satisfied while the second inequality is reduced to

l i a i y pi F a i y bii d i f i y

n

n

j/i

js1

Ý bi j UjU y Ý c i j UjU .

Hence if we choose n

pi Ž x . '

n

Ý bi j UjU q

Ý c i j UjU

j/i

js1

Ž 5.26.

then the above inequality holds for a sufficiently small d i ) 0 provided that l i Ž pi . - 1. Under this condition the pair ˜ u s, ˆ u s are coupled upper and lower solutions. It follows from the discussion in Section 3 that the monotone sequences  uŽ m.4 ,  uŽ m.4 obtained from Ž4.3. with uŽ0. ' ŽU1U , . . . , UnU . and uŽ0. ' Ž d 1 f 1 , . . . , dn fn . converge monotonically to a pair

777

NONLINEAR PARABOLIC SYSTEMS

of positive quasisolutions u s ' Ž u1 , . . . , u n ., u s ' Ž u1 , . . . , u n . which satisfy the equations n

ž ž

yL i u i s u i a i y bi i u i y y L i u i s u i a i y bi i u i y

n

Ý bi j u j y

Ý ci j u j

j/i

js1

n

n

Ý bi j u j y Ý c i j u j j/i

js1

/ /

Ž 5.27.

Bi u i s Bi u i s 0. By Theorem 4.1, the solution u of Ž5.21. satisfies the relation u s F u F u s as t ª ` if ˆ us F h F ˜ u s in Q0 . Since for arbitrary h Ž t, x . with hi Ž0, x . k 0 the components u i of u are strictly positive in V for every t ) 0, and since Ui ª UiU as t ª `, the relation 0 - u i - Ui implies that for some positive constants tU and d , u i satisfies the relation d i f i F u i F UiU for tU y t i F t F tU . By an application of Theorems 4.1 and 4.2 we have the following conclusion: THEOREM 5.3. Let l i Ž0., l i Ž pi . be the smallest eigen¨ alues of Ž5.24. corresponding to pi s 0 and pi gi¨ en by Ž5.26., respecti¨ ely, and let u s ' Ž u1 , . . . , u n ., u s ' Ž u1 , . . . , u n . be the positi¨ e quasisolutions which satisfy Ž5.27.. Then for any initial function h ' Žh1 , . . . , hn . with hi Ž0, x . k 0, i s 1, . . . , n, a global solution u ' Ž u1 , . . . , u n . to Ž5.21. exists and possesses the property unique u s Ž x . F uŽ t , x . F u s Ž x .

as t ª `

Ž 5.28.

if l i Ž pi . - 1 for all i, and lim u Ž t , x . s 0

as t ª `

Ž 5.29.

if l i Ž0. G 1 for all i. Moreo¨ er, uŽ t, x . ª u s Ž x . as t ª ` when l i Ž pi . - 1 and u s s u s . Remark 5.2. Ža. Theorem 5.3 implies that under the condition l i Ž pi . - 1 the solution u of Ž5.21. enters the sector ²u s , u s : as t ª `. Since this is true for any nontrivial nonnegative initial function h , the trivial solution and all forms of semitrivial solutions Žthat is, a steady-state solution which has at least one zero component and one positive component. are unstable. In terms of ecological dynamics this property implies that system Ž5.21. is uniformly persistent and permanent Žcf. w1, 2x..

778

C. V. PAO

Žb. It is easily seen that if a i , bi j , and c i j are constants then by the relation Ui - a irbii and the increasing property of l i Ž pi . in pi , the condition l i Ž pi . - 1 is satisfied if n

mi q

n

Ý a j Ž bi jrbj j . q Ý a j Ž c i jrbj j . - ai j/i

Ž i s 1, . . . , n . , Ž 5.30.

js1

where m i is the smallest eigenvalue of Ž5.24. corresponding to pi s 0 and a i s 1. A similar condition for the permanence of the system Ž5.21. is given in w4, 8x for n s 3. In the special case c i j s 0 for all i, j the above relation is reduced the condition in w15x for the competition model without time delays.

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17. M. H. Protter and H. F. Weinberger, ‘‘Maximal Principles in Differential Equations,’’ Prentice Hall, Englewood Cliffs, NJ, 1967. 18. S. M. Rankin, Existence and asymptotic behavior of a functional differential equation in Banach space, J. Math. Anal. Appl. 88 Ž1982., 531]542. 19. R. Redlinger, Existence theorems for semilinear parabolic systems with functionals, Nonlinear Anal. 8 Ž1984., 667]682. 20. A. Schiaffino, On a diffusion Volterra equation, Nonlinear Anal. 3 Ž1970., 595]600. 21. A. Schiaffino and A. Tesei, Monotone methods and attractivity results for Volterra integro-partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 89 Ž1981., 135]142. 22. C. C. Travis and G. F. Webb, Existence, stability and compactness in the a-norm for partial differential equations, Trans. Amer. Math. Soc. 240 Ž1978., 129]143. 23. Y. Yamada, Asymptotic stability for systems of semilinear Volterra diffusion equations, J. Differential Equations 52 Ž1984., 295]326. 24. Y. Yamada, Asymptotic behavior of solutions for semilinear Volterra diffusion equations, Nonlinear Anal. 21 Ž1993., 227]239. 25. K. Yosida, The Hopf bifurcation and its stability for semilinear diffusion equations with time delay arising in ecology, Hiroshima Math. J. 12 Ž1982., 321]348.