Parabolic Systems in Unbounded Domains. II. Equations with Time Delays

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

225, 557᎐586 Ž1998.

AY986051

Parabolic Systems in Unbounded Domains. II. Equations 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 March 20, 1998 This paper extends the results of an earlier article concerning the existence, uniqueness, comparison, and dynamics problems for a coupled system of parabolic equations in unbounded domains, including the whole space ⺢ n. The present extension is for a system of functional parabolic-ordinary equations where the ‘‘reaction function’’ is allowed to depend on the unknown function with time delay, which may be discrete or continuous, finite or infinite. Applications are made to some model problems, with emphasis on the convergence fo the time-dependent solution to a steady-state solution and the decay property of the solution as < x < ª ⬁. 䊚 1998 Academic Press

1. INTRODUCTION This paper is a continuation of the investigation of an earlier article w10x concerning the existence, uniqueness, and dynamics of a coupled system of parabolic equations in unbounded domains, including the whole space ⺢ n. The purpose here is to extend the results in w10x to a system of parabolicordinary equations where the ‘‘reaction function’’ may depend on the unknown functions with time delays, which may be discrete or continuous, finite or infinite. The system of equations under consideration is given in the form

Ž u i . t y L i u i s f i Ž t , x, u, J )u. , Ž i s 1, . . . , nU . Ž u i . t s f i Ž t , x, u, J )u. Ž i s nU q 1, . . . , N . ui Ž t , x . s hi Ž t , x . Ž i s 1, . . . , nU . u i Ž t , x . s ␩i Ž t , x . Ž i s 1, . . . , N .

Ž t ) 0, x g ⍀ . Ž 1.1. Ž t ) 0, x g ⭸ ⍀ . Ž t g Ii , x g ⍀ . ,

557 0022-247Xr98 $25.00 Copyright 䊚 1998 by Academic Press All rights of reproduction in any form reserved.

558

C. V. PAO

where u ' Ž u1 , . . . , u N ., J )u s Ž J1 ) u1 , . . . , JN ) u N ., ⍀ is a general unbounded domain in ⺢ n with boundary ⭸ ⍀, and for each i s 1, . . . , nU , L i is a uniformly elliptic operator in the form n

Li u i s

Ý

n

aŽjki. Ž t , x . ⭸ 2 u ir⭸ x j ⭸ x k q

j, ks1

Ý bjŽ i. Ž t , x . ⭸ u ir⭸ x j . js1

The uniform ellipticity property of L i is in the sense that there exist positive constants ␮ i , ␯ i such that

␮i < ␰ < 2 G

n

Ý

aŽjki. ␰ j ␰ k G ␯ i < ␰ < 2 ,

Ž ␰ ' Ž ␰ 1 , . . . , ␰ n . g ⺢ n . . Ž 1.2.

j, ks1

The function Ji ) u i and the interval Ii are given by Ji ) u i s

t

Hy⬁ J Ž t y s, x . u Ž s, x . ds, i

i

Ii s Ž y⬁, 0 x

Ž 1.3.

when i s 1, . . . , mU , Ii s w yri , 0 x

Ji ) u i s u i Ž t y r i , x . ,

when i s mU q 1, . . . , N,

where mU F N is a nonnegative integer and ri ) 0 is a constant representing the discrete time delay. If the integral Ji ) u i is replaced by the finite integral Ji ) u i s

t

Hr J Ž t y s, x . u Ž s, x . ds i

i

Ž 1.4.

i

for some or all i s 1, . . . , mU , where 0 - ri - ⬁, then the interval Ii in Ž1.3. is replaced by wyri , 0x. The above consideration of Ji ) u i includes various combinations of discrete and continuous time delays, including a combination of finite continuous and discrete time delays. In particular, when mU s N, the delays in the system are all of the continuous type, whereas when mU s 0, they are all of the discrete type. On the other hand, if nU s N, then problem Ž1.1. becomes a system of parabolic equations with various possible combinations of continuous and discrete delays. Throughout this paper we assume that for each i s 1, . . . , mU , Ji Ž t, x . is piecewise continuous in t and Holder continuous in x Žuniformly ¨ . in t and possesses the property Ji Ž t , x . G 0,

⬁

H0 J Ž t , x . dt s 1 i

Ž t ) 0, x g ⍀ . .

Ž 1.5.

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PARABOLIC SYSTEMS IN UNBOUNDED DOMAINS

For finite continuous delays the above condition is replaced by Ji Ž t , x . G 0,

Ji Ž t , x . s 0

ri

for t ) ri ,

H0 J Ž t , x . ds s 1, i

and

Ž 1.6.

Ž x g ⍀. .

Parabolic systems with time delays are often formulated as evolution equations in Banach spaces, and the existence and dynamics problems are treated in the framework of semigroup theory and dynamical systems Že.g., see w2, 6᎐8, 16, 17x and the references therein .. However, because of the absence of the usual compactness property, as in bounded-domain problems, this approach is not suitable for system Ž1.1.. Recently the method of upper and lower solutions has been used successfully for the treatment of some parabolic systems in unbounded domains without time delays Žcf. w10x.. It has also bee used in w12᎐14x for bounded domain problems with time delays. In this paper we combine the techniques used in w10, 14x to extend this method to system Ž1.1. with ⍀ s ⺢ n as well as a general unbounded domain ⍀, including the exterior of a bounded domain and a half-space in ⺢ n. As in bounded-domain problems, the dynamics of Ž1.1. is closely related to the solutions or quasi-solutions of the elliptic system yL i u i s f i Ž x, u, u . 0 s f i Ž x, u, u . ui Ž x . s hi Ž x .

Ž i s 1, . . . , nU . Ž i s n* q 1, . . . , N . U

Ž i s 1, . . . , n .

Ž x g ⍀.

Ž 1.7.

Ž x g ⭸⍀. .

The plan of the paper is as follows. In Section 2 we investigate the existence and uniqueness of a solution to Ž1.1. in ⺢ n and in a general unbounded domain ⍀. The dynamics of Ž1.1. in ⍀ and the existence problem of Ž1.7. are discussed in Section 3. Section 4 contains some applications of the existence and dynamics results to some model problems arising from gas᎐liquid interaction and pattern formulation of certain chemical concentrations.

2. THE EXISTENCE-COMPARISON PROBLEM To investigate the existence and uniqueness of a solution to Ž1.1., we consider an arbitrary constant T ) 0 and define D T s Ž 0, T x = ⍀ , Q0Ž i. s Ii = ⍀ ,

ST s Ž 0, T x = ⭸ ⍀ , QTŽ i. s Ž Ii j w 0, T x . = ⍀

Q0 s Q0Ž1. = ⭈⭈⭈ = Q0Ž N . ,

D T s w 0, T x = ⍀

Ž i s 1, . . . , N .

QT s QTŽ1. = ⭈⭈⭈ = QTŽ N . ,

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C. V. PAO

mq ␣ Ž where ⍀ s ⍀ j ⭸ ⍀. Denote by C loc D T . the set of functions in C m Ž D T . that are locally Holder continuous in x, uniformly in t g Ž0, T x, and by ¨ C 1, 2 Ž D T . the set of functions that are once continuously differentiable in t and twice continuously differentiable in x for Ž t, x . g D T , where m is a nonnegative integer and ␣ g Ž0, 1.. The corresponding set of N-vector mq ␣ Ž functions is denoted by Cloc D T . and C 1, 1 Ž D T ., respectively. Similar function spaces with possibly different domains will be used throughout the paper. It is always assumed that ⭸ ⍀ is of class C 2q ␣ , and for each i s 1, . . . , nU and any T ) 0 the coefficients aŽjki., bjŽ i. of L i are in C 2q ␣ Ž D T . and C 1q ␣ Ž D T ., respectively; the functions f i Ž t, x, ⭈ ., h i Ž t, x ., and ␩i Ž t, x . belong to C ␣ Ž D T ., C 2q ␣ Ž ST ., and C 2q ␣ Ž Q0Ž i. ., respectively; and h i and ␩i satisfy the compatibility condition h i Ž0, x . s ␩i Ž0, x . on ⭸ ⍀. In addition to the above smoothness assumptions, we impose the following main conditions on the vector function:

f Ž ⭈, u, v . ' Ž f 1 Ž ⭈, u, v . , . . . , f N Ž ⭈, u, v . . . Ž H1 . Ži. fŽ⭈, u, v. is mixed quasi-monotone for u, v g S, and for each i s 1, . . . , N, there exists a constant K i G 0 such that < f i Ž t , x, u, v . y f i Ž t , x, uX , v X . < F K i Ž
Ž Ž t , x . g DT . ,

where S is a subset of ⺢ N. Žii. There exists w U g S and positive constants A i , ␤i with ␤i Ž4␯ i T .y1 such that < f i Ž t , x, w U , J )w U . < F A i exp Ž ␤i < x < 2 .

as < x < ª ⬁,

Ž t G 0.

where ␯ i ) 0 is the constant appeared in Ž1.2.. Recall that by writing u, v in the split form, u s Ž u i , wux a i , wux b i . ,

v s Ž wvx c i , wvx d i . ,

we say that fŽ⭈, u, v. is mixed quasi-monotone in S 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, c i q d i s N such that f i Ž⭈, u i , wux a i , wux b i , wvx c i , wvx d i . is nondecreasing with respect to the components of wux a i and wvx c i , and is nonincreasing with respect to the components of wux b i and wvx d i for all u, v in S. In particular, fŽ⭈, u, v. is said to be quasi-monotone nondecreasing in S if bi s d i s 0 for every i Žthat is, f i Ž⭈, u, v. is nondecreasing with respect to every component

561

PARABOLIC SYSTEMS IN UNBOUNDED DOMAINS

of u and v, except possibly the component u i .. The subset S in the hypothesis Ž H1 . is taken as the sector between a pair of coupled upper and lower solutions. For notational convenience we define L i ' 0 when i s nU q 1, . . . , N. u ' Žu DEFINITION 2.1. Two functions ˜ ˜1 , . . . , u˜N ., ˆu ' Ž uˆ1 , . . . , uˆN . in C ␣ Ž QT . l C 1, 2 Ž D T . are called coupled upper and lower solutions of Ž1.1. if ˜ uGˆ u, and if for each i s 1, . . . , N, u ˜i y L i u˜i G f i Ž t , x, u˜i , wˆux a i , wˆux b i , w J )u ˜ x c i , w J )u ˆx di .

in D T

u ˆi y L i uˆi F f i Ž t , x, uˆi , wˆux a i , w˜ux b i , w J )u ˆ x c i , w J )u ˜x di . u ˜i Ž t , x . G h i Ž t , x . G uˆi Ž t , x .

on ST

u ˜i Ž t , x . G ␩i Ž t , x . G uˆi Ž t , x .

in Q0Ž i.

Ž i s 1, . . . , nU . Ž i s 1, . . . , N . ,

Ž 2.1.

and there exist positive constants A i and ␥ i with ␥ i - Ž4␯ i T .y1 such that
as < x < ª ⬁

Ž t ) 0. .

Ž 2.2.

For a given pair of coupled upper and lower solutions ˜ u, ˆ u, we define the sectors ²ˆ u, ˜ u: '  u g C Ž QT . ; ˆ uFuF˜ u4 ²u ˆi , u˜i : '  u i g C Ž QT . ; uˆi F u i F u˜i 4 .

Ž 2.3.

Define also Li u i ' Ž u i . t y L i u i q K i u i Fi Ž t , x, u, J )u . s K i u i q f i Ž t , x, u, J )u .

Ž i s 1, . . . , N . , Ž 2.4.

where L i u i ' 0 for i s nU q 1, . . . , N, and K i is the constant in Ž H1 .. Then problem Ž1.1. is equivalent to the system Li u i s Fi Ž t , x, u i , w u x a i , w u x b i , w J )u x c i , w J )u x d i . in D T

Ž i s 1, . . . , N .

ui Ž t , x . s hi Ž t , x .

on ST

Ž i s 1, . . . , nU .

u i Ž t , x . s ␩i Ž t , x .

in Q0Ž i.

Ž i s 1, . . . , N . .

Ž 2.5.

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C. V. PAO

It is obvious from Ž2.4. and hypothesis Ž H1 . that Fi Ž ⭈, u i , w u x a i , w v x b i , w J )u x c i , w J )v x d i . G Fi Ž ⭈, ¨ i , w v x a i , w u x b i , w J )v x c i , w J )u x d i . when ˜ uGuGv Gˆ u.

Ž 2.6.

Hence for the existence-uniqueness problem of Ž1.1. it suffices to show the existence of a unique solution to Ž2.5.. We first show this for the Cauchy problem Li u i s Fi Ž t , x, u, J )u .

in Ž 0, T x = ⺢ n

u i Ž t , x . s ␩i Ž t , x .

in Ii = ⺢ n

Ž i s 1, . . . , N . Ž 2.7.

by the method of monotone iterations. Specifically, by starting from the initial iterations uŽ0. s ˜ u and uŽ0. s ˆ u, we construct two sequences  uŽ k .4 ' Ž k . Ž k . Ž k . Ž k . Ž k .  u1 , . . . , u N 4 ,  u 4 '  u1 , . . . , u N 4 from the linear iteration process, Li uŽi k . s Fi Ž t , x, uŽi ky1. , w uŽ ky1. x a i , uŽ ky1.

bi ,

w J )uŽ ky1. x c i , J )uŽ ky1.

Li uŽi k . s Fi t , x, uŽi ky1. , uŽ ky1.

ai ,

w uŽ ky1. x b i , J )uŽ ky1.

ci ,

ž

di

/

in D T

Ž 2.8.

w J )uŽ ky1. x d i /

uŽi k . Ž t , x . s uŽi k . Ž t , x . s ␩i Ž t , x .

in Q0Ž i.

Ž i s 1, . . . , N . ,

where D T s Ž0, T x = ⺢ n, Q0Ž i. s Ii = ⺢ n. To ensure that these sequences are well defined and converge to a unique solution of Ž2.7., we used the integral representation for the solution of Ž2.8.. Define qiŽ k . Ž t , x . ' Fi t , x, uŽi k . , w uŽ k . x a i , uŽ k .

ž

qiŽ k . Ž t , x . ' Fi t , x, uŽi k . , uŽ k .

ai ,

ž

JiŽ0. Ž t , x . '

bi ,

w uŽ k . x b i , J )uŽ k .

H⺢ ⌫ Ž t , x ; 0, ␰ . ␩ Ž 0, ␰ . d ␰ , n

i

i

w J )uŽ k . x c i , J )uŽ k . ci ,

di

/

w J )uŽ k . x d i / Ž 2.9.

563

PARABOLIC SYSTEMS IN UNBOUNDED DOMAINS

where ⌫i Ž t, x; ␶ , ␰ . is the fundamental solution of the operator Li Žcf. w4, 11x.. Then the solutions uŽi k . and uŽi k . of Ž2.8. may be expressed as uŽi k . Ž t , x . s JiŽ0. Ž t , x . q

t

H0 d␶ H⺢ ⌫ Ž t , x ; ␶ , ␰ . q n

i

Ž ky1. i

Ž ␶ , ␰ . d␰ Ž i s 1, . . . , nU .

uŽi k . Ž t , x . s JiŽ0. Ž t , x . q

t

H0 d␶ H⺢ ⌫ Ž t , x ; ␶ , ␰ . q n

i

Ž ky1. i

Ž ␶ , ␰ . d␰ Ž 2.10.

uŽi k . Ž t , x . s eyK i t␩i Ž 0, x . q

t yK Ž ty ␶ . Ž ky1. i

H0 e

qi

Ž ␶ , x . d␶ Ž i s nU q 1, . . . , N .

uŽi k . Ž t , x . s eyK i t␩i Ž 0, x . q

t yK Ž ty ␶ . Ž ky1. i

H0 e

qi

Ž ␶ , x . d␶ .

In the following lemma we show that the sequences  uŽ k .4 ,  uŽ k .4 are uniquely determined and are monotonic. LEMMA 2.1. The sequences  uŽ k .4 '  u1Ž k ., . . . , uŽNk .4 and  uŽ k .4 '  u1Ž k ., . . . , uŽNk .4 gi¨ en by Ž2.8. are well defined and possess the monotone property

ˆu F uŽ k . F uŽ kq1. F uŽ kq1. F uŽ k . F ˜u

on w 0, T x = ⺢ n Ž 2.11.

for e¨ ery k. Moreo¨ er, for each k s 1, 2, . . . , the pair uŽ k . and uŽ k . are coupled upper and lower solutions of Ž2.7.. Proof. It is easy to see from Ž2.4., Ž2.9. and hypothesis Ž H1 . that both ␣ Ž qiŽ ky1. and qiŽ ky1. are in C loc D T . whenever uŽ ky1. and uŽ ky1. are in ␣ Ž Ž ky1. Cloc QT .. Moreover, if u and uŽ ky1. are in ²ˆ u, ˜ u:, then by Ž H1 . there X X exist positive constants A i , ␤i such that < qiŽ ky1. Ž t , x . < F AXi exp Ž ␤iX < x < 2 . < qiŽ ky1. Ž t , x . < F AXi exp Ž ␤iX < x < 2 .

as < x < ª ⬁

Ž i s 1, . . . , N .

Žcf. w10x.. This implies that for each i s 1, . . . , N, the solutions uŽi k ., uŽi k . of 2q ␣ Ž Ž2.8. exist, are in C loc D T ., and possess the growth property < uŽi k . < F AYi exp Ž ␤iY < x < 2 . ,

< uŽi k . < F AYi exp Ž ␤iY < x < 2 .

as < x < ª ⬁

Ž 2.12.

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C. V. PAO

for some constants AYi and ␤iY whenever uŽ ky1. and uŽ ky1. are in ²ˆ u, ˜ u: l ␣ Ž Cloc D T . Žcf. w4, 11x.. It is clear from uŽ0. s ˜ u and uŽ0. s ˆ u that uŽ1. and uŽ1. exist and satisfy Ž2.12. for k s 1. Moreover, by Ž2.8., Ž2.4., and Ž2.1., the Ž1. Ž0. Ž1. functions z iŽ0. s uŽ0. and z iŽ0. s uŽ1. ˜i y uŽ1. ˆi i y ui s u i i y ui s ui y u satisfy the relation Li z iŽ0. s Li u ˜i y Fi Ž u˜i , w˜ux a i , wˆux b i , w J )u ˜ x c i , w J )u ˆx di . G 0

in D T

Li z iŽ0. s Fi Ž u ˆi , wˆux a i , w˜ux b i , w J )u ˆ x c i , w J )u ˜ x d i . y Li uˆi G 0

z iŽ0. Ž t , x . s u ˜i Ž t , x . y ␩i Ž t , x . G 0, z iŽ0. Ž t , x . s ␩i Ž t , x . y uˆi Ž t , x . G 0 in Q0Ž i. . < and < z iŽ0. < F < uŽ1. < q
2

< x
Rª⬁

2

s 0, Ž 2.13.

Rª⬁

where A i s A i q AYi and z iŽ0. stands for either z iŽ0. or z iŽ0.. By the Phragman᎐Lindelof ¨ principle, z iŽ0. G 0 in DT Žcf. w11, 15x.. This shows that Ž1. Ž0. u F u and uŽ1. G uŽ0.. Ž1. Ž . Ž . Let z iŽ1. s uŽ1. i y u i , i s 1, . . . , N. By 2.8 and 2.6 , Ž0. Ž0. Li z iŽ1. s Fi uŽ0. i , wu x ai , u

ž

Ž0. yFi uŽ0. i , u

ž

ai ,

bi ,

w J )uŽ0. x c i , J )uŽ0.

w uŽ0. x b i , J )uŽ0.

ci ,

di

/

w J )uŽ0. x d i / G 0,

and z iŽ1. Ž t, x . s 0 in Q 0Ž i.. Since by Ž2.12. with k s 1, < z iŽ1. < F 2 AYi expŽ ␤iY < x < 2 . as < x < ª ⬁, a relation similar to that in Ž2.13. hold for z iŽ1.. It follows again from the Phragman᎐Lindelof ¨ principle that z iŽ1. G 0. The Ž1. Ž1. 2q ␣ Ž above conclusions show that u and u are in Cloc D T . and satisfy the Ž1. Ž1. relation ˆ uFu Fu F˜ u on D T . The monotone property Ž2.11. follows from an induction argument similar to that in w10x. To show that uŽ k . and uŽ k . are coupled upper and lower solutions of Ž2.7. for every k G 1, we observe from Ž2.8., Ž2.6., and Ž2.11. that

Ž uŽi k . . t y Li uŽi k . s K i Ž uŽi ky1. y uŽi k . . q f i uŽi ky1. , w uŽ ky1. x a i , uŽ ky1.

ž

G f i uŽi k . , w uŽ k . x a i , uŽ k .

ž

bi ,

bi ,

w J )uŽ ky1. x c i , J )uŽ ky1.

w J )uŽ k . x c i , J )uŽ k .

di

/

di

/

PARABOLIC SYSTEMS IN UNBOUNDED DOMAINS

565

Ž uŽi K . . t y Li uŽi k . s yK i Ž uŽi k . y uŽi ky1. . q f i uŽi ky1. , uŽ ky1.

ž

F f i uŽi k . , uŽ k .

ž

ai ,

ai ,

w uŽ ky1. x b i , J )uŽ ky1.

w uŽ k . x b i , J )uŽ k .

ci ,

ci ,

w J )uŽ ky1. x d i /

w J )uŽ k . x d i / ,

and uŽi k . Ž t, x . s uŽi k . Ž t, x . s ␩i Ž t, x . in Q0Ž i.. Since uŽ k . and uŽ k . are in ²ˆ u, ˜ u:, they satisfy the growth condition Ž2.2.. This shows that uŽ k . and uŽ k . are coupled upper and lower solutions of Ž2.7., which completes the proof of the lemma. In view of the monotone property Ž2.11. the pointwise limits lim uŽ k . Ž t , x . s u Ž t , x . , lim uŽ k . Ž t , x . s u Ž t , x .

kª⬁

kª⬁

Ž 2.14.

exist and satisfy the relation ˆ uFuFuF˜ u on D T . Letting k ª ⬁ in Ž2.8. and following the argument in the proof of Lemma 3.2 of w10x shows that u and u satisfy the relation

Ž u i . t y L i u i s f i Ž t , x, u i , w ux a , w ux b , w J )ux c , w J )ux d . Ž u i . t y L i u i s f i Ž t , x, u i , w ux a , w ux b , w J )ux c , w J )ux d . i

i

i

i

i

u i Ž t , x . s u i Ž t , x . s ␩i Ž t , x .

in D T

i

i

Ž 2.15.

i

in Q0Ž i. ,

Ž i s 1, . . . , N . .

In the following theorem we show that u s u, and this is the unique solution of Ž2.7.. THEOREM 2.1. Let ˜ u, ˆ u be a pair of coupled upper and lower solutions of Ž2.7., and let hypothesis Ž H1 . hold with S s ²ˆ u, ˜ u:. Then problem Ž2.7. has a 2q ␣ Ž unique solution u g C loc D T . l ²ˆ u, ˜ u:. Moreo¨ er, the sequences  uŽ k .4 ,  uŽ k .4 gi¨ en by Ž2.8. con¨ erge monotonically to u and satisfy the relation

ˆu F uŽ k . F uŽ kq1. F u F uŽ kq1. F uŽ k . F ˜u

on w 0, T x = ⺢ n . Ž 2.16.

Proof. The proof is along the same line as that in w10x, and we give a sketch as follows. Let B be an arbitrarily large ball in ⺢ n with surface ⭸ B, and let Gi Ž t, x; ␶ , ␰ . be the Green’s function of the operator Li in the bounded domain DUT ' Ž0, T x = B under the Dirichlet boundary condition

566

C. V. PAO

on STU ' Ž0, T x = ⭸ B. Define qi Ž x, t . ' Fi Ž t , x, u i , w u x a i , w u x b i , w J )u x c i , w J )u x d i . qi Ž x, t . ' Fi Ž t , x, u i , w u x a i , w u x b i , w J )u x c i , w J )u x d i . JiŽ1. Ž t , x . '

HBG Ž t , x ; 0, ␰ . ␩ Ž 0, ␰ . d ␰ i

i

Ž 2.17.

Ž i s 1, . . . , N . .

By considering Ž2.15. in the bounded domain DUT , we may write u i Ž t , x . s JiŽ1. Ž t , x . q

t

H0 d␶ HBG Ž t , x ; ␶ , ␰ . q Ž ␶ , ␰ . d ␰ i

⭸ ⌫i

t

H0 d␶ H⭸ B ⭸␯

q

␰

u i Ž t , x . s JiŽ1. Ž t , x . q

t

i

H0 d␶ H⭸ B ⭸␯

q

Ž t , x ; ␶ , ␰ . ␺i Ž ␶ , ␰ . d ␰

H0 d␶ HG Ž t , x ; ␶ , ␰ . q Ž ␶ , ␰ . d ␰ ⭸ ⌫i

t

i

␰

i

Ž t , x ; ␶ , ␰ . ␺i Ž ␶ , ␰ . t yK Ž ty ␶ . i

qi Ž ␶ , x . d␶

t yK Ž ty ␶ . i

qi Ž ␶ , x . d␶

u i Ž t , x . s eyK i t␩i Ž 0, x . q

H0 e

u i Ž t , x . s eyK i t␩i Ž 0, x . q

H0 e

Ž 2.18. U

Ž i s 1, . . . , n .

Ž i s nU q 1, . . . , N . ,

where ␺ i and ␺ i are the densities given by

␺i Ž t , x . s 2

t

␺i Ž t , x . s 2

t

H0 d␶ H⭸ B R Ž t , x ; ␶ , ␰ . u Ž ␶ , ␰ . d ␰ q 2 u Ž t , x . i

i

i

Ž 2.19.

H0 d␶ H⭸ B R Ž t , x ; ␶ , ␰ . u Ž ␶ , ␰ . d ␰ q 2 u Ž t , x . i

i

i

Žcf. w4, 11x.. The kernel R i Ž t, x; ␶ , ␰ . has a weak singularity at Ž t, x . s Ž␶ , ␰ . and has the estimate < R i Ž t , x ; ␶ , ␰ . < F Ci < t y ␶
Ž 1 y ␥r2 - ␮ - 1 . Ž 2.20.

PARABOLIC SYSTEMS IN UNBOUNDED DOMAINS

567

for some constant Ci . It is clear from Ž2.18. that the function z i ' u i y u i satisfies the relation zi Ž t , x . s

t

H0 d␶ HBG Ž t , x ; ␶ , ␰ . i

q

⭸ ⌫i

t

H0 d␶ H⭸ B ⭸␯

qi Ž ␶ , ␰ . y qi Ž ␶ , ␰ . d ␰

Ž t, x; ␶ , ␰ .

␰

Ž 2.21. Ž i s 1, . . . , nU .

= ␺i Ž ␶ , ␰ . y ␺i Ž ␶ , ␰ . d ␰ zi Ž t , x . s

t yK Ž ty ␶ . i

qi Ž ␶ , x . y qi Ž ␶ , x . d␶

H0 e

Ž i s nU q 1, . . . , N . .

For each fixed t ) 0, define 5 z i 5 t ' max  < z i Ž ␶ , x . < ; 0 F ␶ F t , x g B 4

Ž 2.22.

5z 5 t ' 5 z1 5 t q ⭈⭈⭈ q5 z N 5 t ,

where B s B j ⭸ B. It is clear that z i Ž t, x . s 0 in Q0Ž i., and by Ž1.3., Ž1.5. and Ž H1 ., < Ž Ji ) z i . Ž ␶ , ␰ . < F 5 z i 5 t < qi Ž ␶ , ␰ . y qi Ž ␶ , ␰ . < F K i < z i Ž ␶ , ␰ . < q
Ž 2.23.

Ž0 F ␶ F t , ␰ g B . .

Since by Ž2.19., Ž2.20., and Ž2.22., < ␺i Ž t , x . y ␺i Ž t , x . < t

H0 d␶ H⭸ B< R Ž t , x ; ␶ , ␰ . < < z Ž ␶ , ␰ . < d ␰ q 2 < z Ž t , x . <

F2

i

i

i

F CiX 5 z i 5 t for some constant CiX independent of t, we see from Ž2.21. and Ž2.23. that < zi Ž t , x . < F

t

žH

d␶

q

H0 d␶ H⭸ B

0

ž

HBG Ž t , x ; ␶ , ␰ . d ␰ i

t

⭸ ⌫i ⭸␯␰

/Ž

3 K i 5z 5 t .

Ž t ; x ; ␶ , ␰ . d ␰ Ž CiX 5 z i 5 t . .

/

568

C. V. PAO

The property of Gi and ⭸ ⌫ir⭸␯␰ Žwhich is similar to that in Ž2.20.. leads to the estimate < z i Ž t , x . < F K iU t 1y ␮ 5z 5 t q CiU t 1y ␮ 5 z i 5 t

Ž i s 1, . . . , nU . , Ž 2.24.

where K iU and CiU are some constants independent of t Žsee also w10x.. For i s nU q 1, . . . , N, the relations in Ž2.21. and Ž2.23. yield the estimate < z i Ž t , x . < F K iU t 5z 5 t

Ž i s nU q 1, . . . , N .

Ž 2.25.

for some constant K iU . Addition of the inequalities in Ž2.24. and Ž2.25. leads to
Ž 2.26.

for some constant K U . This implies that zŽ t, x . s 0 on w0, t 1 x = B for some t 1 ) 0. A ladder argument like that in w11x shows that zŽ t, x . s 0 on w0, T x = B. This proves that u s u, and this is the unique solution of Ž2.7. in w0, T x = B. The arbitrariness of B ensures that it is the unique solution of Ž2.7.. Finally, the monotone property Ž2.16. is a direct consequence of Lemma 2.1. This proves the theorem. We next show the existence and uniqueness of a solution to Ž1.1. for the general unbounded domain ⍀. Starting again from the initial iterations uŽ0. s ˜ u and uŽ0. s ˆ u, we construct two sequences  uŽ k .4 '  u1Ž k ., . . . , uŽNk .4 , Ž k . Ž k . Ž k  u 4 '  u1 , . . . , u N .4 from the linear iteration process Ž2.8., with D T ' Ž0, T x = ⍀ and with the boundary condition uŽi k . Ž t , x . s uŽi k . Ž t , x . s h i Ž t , x .

on ST

Ž i s 1, . . . , nU . . Ž 2.27.

To ensure the existence and the monotone property of these sequences we consider, for each k s 1, 2, . . . , the linear scalar parabolic boundary-value problem Li u i s qiŽ ky1. Ž t , x .

in D T

ui Ž t , x . s hi Ž t , x .

on ST

u i Ž t , x . s ␩i Ž t , x .

in Q0Ž i.

Ž i s 1, . . . , nU . ,

Ž 2.28.

where qiŽ ky1. stands for either qiŽ ky1. or qiŽ ky1., which are defined by Ž2.9.. The following lemma gives the existence-uniqueness result for problem Ž2.28. when k s 1. LEMMA 2.2. Let qiŽ0. stand for either qiŽ0. or qiŽ0., which are gi¨ en by Ž2.9. with k s 0. Then for k s 1 and i s 1, . . . , nU , problem Ž2.28. has a unique solution uŽ1. that coincides with uŽ1. if qiŽ0. s qiŽ0. and with uŽ1. if qiŽ0. s qiŽ0.. i i i Ž1. Ž1. Moreo¨ er, u i G u i on D T .

PARABOLIC SYSTEMS IN UNBOUNDED DOMAINS

569

Proof. Let  ⍀ m 4 be a sequence of increasing smooth bounded domains in ⺢ n such that ⬁

D ⍀m s ⍀,

⍀ m ; ⍀ mq1 ; ⍀ ,

⭸ ⍀ m s ⭸ ⍀U j ⭸ ⍀Xm , Ž 2.29.

ms1

and let Dm s Ž0, T x = ⍀ m , Sm s Ž0, T x = ⭸ ⍀Xm , where ⭸ ⍀U is the portion of ⭸ ⍀ contains in ⍀ m and ⭸ ⍀Xm is the boundary of ⍀ m lying in ⍀. Consider the linear problem Ž2.28. in the bounded domain Dm under the additional boundary condition ui Ž t , x . s u ˜i Ž t , x .

on Sm

if qiŽ0. s qiŽ0.

ui Ž t , x . s u ˆi Ž t , x .

on Sm

if qiŽ0. s qiŽ0. .

Ž 2.30.

It is easy to see that u ˜i and uˆi are upper and lower solutions of Ž2.28., Ž2.30. in Dm for either qiŽ0. s qiŽ0. or qiŽ0. s qiŽ0.. By Theorem 2.2 of w14x Žsee also w11x. for the time-delayed problem in bounded domains, a unique solution Ž u i . m exists and is in ² u ˆi , u˜i :. Consider the case qiŽ0. s qiŽ0. and denote the corresponding solution by Ž u i . m . Since Ž u i . mq1 satisfies Ž2.28. in Dm and Ž u i . mq1 F u ˜i s Ž u i . m on Sm , the standard comparison theorem ensures that Ž u i . mq1 F Ž u i . m on Dm . Similarly, when qiŽ0. s qiŽ0., the corresponding solution Ž u i . m of Ž2.28., Ž2.30. exists and satisfies Ž u i . m F Ž u i . mq1 on Dm . Moreover, the relation Ž u i . mq1 s u ˜i G uˆi s Ž u i . mq1 on Smq1 implies that Ž u i . mq1 G Ž u i . mq1 on Dmq1 , and therefore Ž u i . mq1 G Ž u i . mq1 on Dm . This leads to the relation Ž u i . m F Ž u i . mq1 F Ž u i . mq1 F Ž u i . m on Dm . Define extensions ŽUi . m and ŽUi . m to D T by

Ž Ui . m s

½

Ž ui . m

on Dm

u ˜i

on D T _ Dm ,

Ž Ui . m s

½

Ž ui . m

on Dm

u ˆi

on D T _ Dm .

Ž 2.31. Then u ˆi F Ž Ui . m F Ž Ui . mq1 F Ž Ui . mq1 F Ž Ui . m F u˜i

on D T .

This implies that the pointwise limits lim Ž Ui . m ' u i ,

mª⬁

lim Ž Ui . m ' u i

mª⬁

exist and satisfy u ˆi F u i F u i F u˜i on DT . The justification that u i and u i are the solutions of Ž2.28. corresponding to qiŽ0. s qiŽ0. and qiŽ0. s qiŽ0.

570

C. V. PAO

follows from the same argument as that in w10x. Finally, by the definition of Ž1. qiŽ0., qiŽ0. and the uniqueness property of the solution, u i s uŽ1. i and u i s u i . This proves the lemma. In view of the above lemma, we have the following existence and monotone property of the sequences  uŽ k .4 ,  uŽ k .4 . LEMMA 2.3. The sequences  uŽ k .4 ,  uŽ k .4 , gi¨ en by Ž2.8., Ž2.27., are well defined and possess the monotone property Ž2.11. on D T s w0, T x = ⍀. Moreo¨ er, for each k s 1, 2, . . . , uŽ k . and uŽ k . are coupled upper and lower solutions of Ž1.1.. Proof. In view of Lemma 2.2 and the iteration process Ž2.8. for i s nU q 1, . . . , N, the first iterations uŽ1. ' Ž u1 , . . . , u N ., uŽ1. ' Ž u1 , . . . , u N . exist and satisfy ˆ u F uŽ1. F uŽ1. F ˜ u on D T . Moreover, from the relation Ž1. Ž0. Ž1. Ž uŽ1. i . t y Li u i s K i Ž u i y u i . Ž0. Ž0. q f i uŽ0. i , wu x ai , u

ž

Ž1. Ž1. G f i uŽ1. i , wu x ai , u

ž

bi ,

bi ,

w J )uŽ0. x c i , J )uŽ0.

w J )uŽ1. x c i , J )uŽ1.

di

/

Ž1. Ž0. Ž1. Ž uŽ1. i . t y Li u i s K i Ž u i y u i . Ž0. q f i uŽ0. i , u

ž

Ž1. F f i uŽ1. i , u

ž

ai ,

ai ,

w uŽ0. x b i , J )uŽ0.

w uŽ1. x b i , J )uŽ1.

ci ,

ci ,

di

/

Ž 2.32.

w J )uŽ0. x d i /

w J )uŽ1. x d i /

and the boundary condition Ž2.27., uŽ1. and uŽ1. are coupled upper and lower solutions of Ž1.1.. The same relation implies that for each i s 1, . . . , nU , uŽ1. and uŽ1. are upper and lower solutions of Ž2.28. when k s 1 i i Ž0. Ž0. and qi is either qi or qiŽ0.. Using uŽ1. and uŽ1. instead of ˜ u and ˆ u as the coupled upper and lower solutions, an induction argument shows that uŽ k . and uŽ k . exist and satisfy Ž2.11. for every k s 1, 2, . . . . The proof that uŽ k . and uŽ k . are coupled upper and lower solutions of Ž1.1. follows from a relation similar to that in Ž2.32.. As a consequence of the above lemma, we have the following existenceuniqueness result. u, ˆ u be a pair of coupled upper and lower solutions of THEOREM 2.2. Let ˜ Ž1.1., and let hypothesis Ž H1 . hold with S ' ²ˆ u, ˜ u:. Then problem Ž1.1. has 2q ␣ Ž a unique solution u g Cloc D T . l ²ˆ u, ˜ u:. Moreo¨ er, the sequences  uŽ k .4 ,  uŽ k .4 gi¨ en by Ž2.8., Ž2.27. con¨ erge monotonically to u and satisfy the relation Ž2.16. on D T .

PARABOLIC SYSTEMS IN UNBOUNDED DOMAINS

571

Proof. By Lemma 2.3 the sequences  uŽ k .4 ,  uŽ k .4 given by Ž2.8., Ž2.27. converge pointwise to some limits u and u, respectively, and ˆ u-uFuF˜ u on D T . To prove the existence and uniqueness problem, it suffices to show u s u and u satisfies Ž1.1. in every bounded subdomain Dm of D T . This follows from an argument similar to that in w10x. The details are omitted.

3. THE DYNAMICS PROBLEM The existence-comparison results given in Section 2 are fundamental to our study of the dynamics problem, which is closely related to the solutions of the elliptic system Ž1.7.. In analogy to the parabolic system, we write the elliptic system 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 . ui Ž x . s hi Ž x .

in ⍀

Ž i s 1, . . . , nU .

on ⭸ ⍀

Ž i s 1, . . . , N . Ž 3.1.

and give the following definition. DEFINITION 3.1. Two functions ˜ u s ' Žu ˜1 , . . . , u˜N ., ˆu s ' Ž uˆ1 , . . . , uˆN . in C 2 Ž ⍀ . l C Ž ⍀ . are called coupled upper and lower solutions of Ž3.1. 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 ,

yL i u ˆi F f i Ž x, uˆi , uˆs

ai ,

˜u s

bi ,

ˆu s

ci ,

u ˜i Ž x . G h i Ž x . G uˆi Ž x .

ˆu s

. ˜u s d . di

in ⍀ Ž i s 1, . . . , N .

i

on ⭸ ⍀ Ž i s 1, . . . , nU . .

Ž 3.2.

Recall that L i s 0 when i s nU q 1, . . . , N. It is seen from the above definition that the pair ˜ u s and ˆ u s are coupled upper and lower solutions of Ž1.1. when ␩ g ²ˆ u s, ˜ u s : in Q0 . Moreover, if fŽ x, u, v. is quasi-monotone nondecreasing Žthat is, bi s d i s 0 for all i ., then the pair ˜ u s and ˆ u s are not coupled and are referred to as ordered upper and lower solutions. Throughout this section we assumed that a pair of coupled Žor ordered. upper and lower solutions ˜ u s, ˆ u s exist and the following additional hypothesis holds. Ž H2 . For each i, the coefficients of L i and the functions f i ' f i Ž x, u, v. and h i ' h i Ž x . are all independent of t, and the vector function fŽ x, u, v. 1q ␣ ' Ž f 1Ž x, u, v., . . . , f N Ž x, u, v.. is a bounded Cloc -function of Ž x, u, v. for x g ⍀ and u, v g ²ˆ u s, ˜ u s :. Consider the elliptic system Ž3.1. in the domain ⍀ m with a suitable boundary condition on ⭸ ⍀Xm , where ⍀ m is a any one of the bounded domains given in Ž2.29.. As in Ž2.8., we use uŽ0. u s and uŽ0. u s as initial s s˜ s sˆ

572

C. V. PAO

iterations and construct two sequences  UsŽ m.4 ' U1Ž m., . . . , UNŽ m.4 ,  UsŽ m.4 ' U1Ž m., . . . , UNŽ m.4 from the linear iteration process yL i uŽi m. q K i uŽi m. s Fi x, Ui Ž my1. , UsŽ my1.

ž

UsŽ my1. y L i uŽi m. q K i uŽi m. s Fi x, Ui Ž my1. , UsŽ my1.

ž

UsŽ my1. in ⍀ m uŽi m. Ž x . s uŽi m. Ž x . s h i Ž x .

on ⭸ ⍀U

uŽi m. Ž x . s u ˜i Ž x . , uŽi m. Ž x . s uˆi Ž x .

on ⭸ ⍀Xm

ai

, UsŽ my1.

ci

, UsŽ my1.

ai ,

UsŽ my1.

ci ,

bi ,

di

bi

UsŽ my1.

/

,

di

/

Ž 3.3.

Ž i s 1, . . . , N . Ž i s 1, . . . , nU . ,

where Ui Ž0. s u ˜i , Ui Ž0. s uˆi , and Ui Ž m. and Ui Ž m. are the respective extensions of uŽi m., uŽi m. given by Ui

Ž m.

Ž x. '

Ui Ž m. Ž x . s

½ ½

uŽi m. Ž x .

when x g ⍀ m

u ˜i Ž x .

when x g ⍀ _ ⍀ m ,

uŽi m. Ž x .

when x g ⍀ m

u ˆi Ž x .

when x g ⍀ _ ⍀ m

Ž 3.4. Ž i s 1, . . . , N . .

Without any loss of generality, we may assume that K i ) 0 for i s nU q 1, . . . , N. It is clear from Ž H1 . and Ž H2 . that the sequences  UsŽ m.4 ,  UsŽ m.4 are well defined. In the following theorem we show that these sequences are monotonic and converge to some functions u s ' Ž u1 , . . . , u N ., u s ' Ž u1 , . . . , u N ., called quasi-solutions of Ž3.1., which satisfy the relation

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 ,

ui Ž x . s ui Ž x . s hi Ž x .

in ⍀ Ž i s 1, . . . , N . on ⭸ ⍀ , Ž i s 1, . . . , nU . .

Ž 3.5.

THEOREM 3.1. Let ˜ u s, ˆ u s be a pair of coupled upper and lower solutions of Ž3.1., and let hypotheses Ž H1 . and Ž H2 . hold with S ' ²ˆ u s, ˜ u s :. Then the sequence  UsŽ m.4 ,  UsŽ m.4 gi¨ en by Ž3.3., Ž3.4. con¨ erge monotonically to a pair of quasi-solutions u s , u s , which satisfy Ž3.5. and the relation

ˆu s F UsŽ m. F UsŽ mq1. F u s F u s F UsŽ mq1. F UsŽ m. F ˜u s

on ⍀ . Ž 3.6.

PARABOLIC SYSTEMS IN UNBOUNDED DOMAINS

573

Moreo¨ er, if u s s u s Ž' uUs ., then uUs is the unique solution of Ž3.1. in ²ˆ u s, ˜ u s :. Proof. It is easy to see by Ž3.3., Ž3.2., and Ž2.6. for k s 1 that the first Ž1. . Ž1. . Ž Ž1. Ž Ž1. iterations uŽ1. and uŽ1. satisfy the relation s ' u1 , . . . , u N s ' u1 , . . . , u N Ž1. Ž1. ˆu s F u s F u s F ˜u s on ⍀ 1. In view of Ž3.4., ˆu s F UsŽ1. F UsŽ1. F ˜u s on ⍀. An induction argument shows that UsŽ m. F UsŽ mq1. F UsŽ my1. F UsŽ m. on ⍀ for every m. This implies that the pointwise limits lim UsŽ m. Ž x . s u s Ž x . ,

mª⬁

lim UsŽ m. Ž x . s u s Ž x .

mª⬁

Ž x g ⍀ . Ž 3.7.

exist and satisfy the relation Ž3.6.. The remaining proof of the theorem follows from the same argument as in w10x and is omitted. An immediate consequence of Theorem 3.1 is the following result for the case bi s d i s 0. COROLLARY 3.1. Let the hypotheses in Theorem 3.1 hold. If fŽ x, u, v. is quasi-monotone nondecreasing in ²ˆ u s, ˜ u s :, then the sequence  UsŽ m.4 con¨ erges monotonically from abo¨ e to a maximal solution u s in ²ˆ u s, ˜ u s :, and the sequence  UsŽ m.4 con¨ erges monotonically from below to a minimal solution u s . Moreo¨ er, relation Ž3.6. holds. Proof. This follows from Theorem 3.1 and the fact that every solution uUs g ²ˆ u s, ˜ u s : satisfies u s F uUs F u s Žcf. w10x.. To investigate the asymptotic behavior of the solution of Ž1.1. in relation to the quasi-solutions u s and u s , we need to establish the nonnegative property of a function z ' Ž z1 , . . . , z N . that satisfies the relation N

Ž z i . t y Li z i G

Ý

ci j z j q d i j Ž Jj ) z j .

js1

in D T zi Ž t , x . G 0

on ST

zi Ž t , x . G 0

in Q0Ž i.

Ž i s 1, . . . , N . Ž i s 1, . . . , nU . Ž i s 1, . . . , N . .

Ž 3.8.

␣ Ž LEMMA 3.1. Let c i j , d i j g C loc D T . be bounded on D T such that c i j G 0 2q ␣ Ž for j / i and d i j G 0 for all i, j. If z ' Ž z1 , . . . , z n . g Cloc D T . l C Ž QT . is bounded on Q0 and satisfies Ž3.8., then z G 0 and is bounded on QT .

574

C. V. PAO

Proof. In view of Ž3.8. there exists nonnegative functions qi , hUi , ␩iU ␣ Ž with qi g C loc D T . such that N

Ž z i . t y Li z i s

c i j z j q d i j Ž J j ) z j . q qi

Ý js1

in D T zi Ž t , x . s zi Ž t , x . s

hUi ␩iU

Ž t, x. Ž t, x.

on ST in Q0Ž i.

Ž i s 1, . . . , N . Ž i s 1, . . . , nU . Ž i s 1, . . . , N . .

Ž 3.9.

By the assumption of c i j and d i j , the function fŽ⭈, z, v. ' Ž f 1Ž⭈, z, v., . . . , f N Ž⭈, z, v.. with N

f i Ž ⭈, z, v . '

Ý

c i j z j q d i j ¨ j q qi

Ž i s 1, . . . , N .

js1

is quasi-monotone nondecreasing for all z, v g ⺢ N and satisfies hypothesis Ž H1 . with S ' ⺢ N , w U s 0, and ␤i s 0. It is easy to see from the nonnegative property of qi , hUi , and ␩iU that for any constants ␳ ' Ž ␳ , . . . , ␳ . ) 0 and ␣ ) 0, the pair ˜ z s ␳ e ␣ t and ˆ u s 0 are upper and U lower solutions of Ž3.9. if ␳ G h i , ␳ G ␩iU , and

␣␳ e␣t G

N

c i j Ž ␳ e ␣ t . q d i j J j ) Ž ␳ e ␣ t . q qi

Ý

Ž i s 1, . . . , N . .

js1

In view of Jj ) Ž ␳ e ␣ t . s ␳

⬁

H0 J Ž s, x . e j

␣ Ž tys.

J j ) Ž ␳ e ␣ t . s ␳ e ␣ Ž tyr . F ␳ e ␣ t

ds F ␳ e ␣ t

Ž i s 1, . . . , mU . Ž i s mU q 1, . . . , N . ,

the above requirements are satisfied by some sufficiently large ␳ and ␣ . Since ˜ z satisfies a growth condition similar to that in Ž2.2. with ␥ i s 0, it is a bounded upper solution. It follows from Theorem 2.2 that a unique solution zU to Ž3.9. exists, and 0 F zU F ␳ e ␣ t on D T . The uniqueness property of the solution ensures that z s zU G 0, which proves the lemma. Based on Lemma 3.1, we show the monotone convergence of the solution of Ž1.1. when fŽ x, u, v. is quasi-monotone nondecreasing and either ␩ s ˜ u s or ␩ s ˆ u s . Since ˜ u s and ˆ u s are also coupled upper and lower solutions of Ž1.1. when ␩ g ²ˆ u s, ˜ u s : in Q0 , Theorem 2.2 and the

575

PARABOLIC SYSTEMS IN UNBOUNDED DOMAINS

arbitrariness of T imply that a unique global solution uŽ t, x . to Ž1.1. exists and satisfies the relation

ˆu s Ž x . F u Ž t , x . F ˜u s Ž x .

for t G 0,

x g ⍀.

Ž 3.10.

Denote the solution by uŽ t, x . when ␩ Ž t, x . s ˜ u s Ž x . on Q0 and by uŽ t, x . when ␩ Ž t, x . s ˆ u s Ž x .. The following theorem gives the monotone convergence of these solutions as t ª ⬁. u s, ˆ u s be coupled upper and lower solutions of Ž3.1., THEOREM 3.2. Let ˜ and assume that fŽ⭈, u, v. is quasi-monotone nondecreasing for u, v in ²ˆ u s, ˜ u s: and that hypotheses Ž H1 ., Ž H2 . hold with S ' ²ˆ u s, ˜ u s :. Then as t ª ⬁, uŽ t, x . con¨ erges monotonically from abo¨ e to the maximal solution u s Ž x . of Ž3.1., and uŽ t, x . con¨ erges monotonically from below to the minimal solution u s Ž x .. Moreo¨ er, uŽ t, x . G uŽ t, x ., and if uU Ž t, x . is the solution of Ž1.1. corresponding to any ␩ g ²ˆ u s, ˜ u s :, then u Ž t , x . F uU Ž t , x . F u Ž t , x .

for t G 0,

x g ⍀.

Ž 3.11.

Proof. The proof is based on Lemma 3.1 and is a slight modification of the argument given in w10x. We give a sketch as follows. Let zŽ t, x . s uŽ t q ␦ , x . y uŽ t, x . for an arbitrary constant ␦ ) 0. By Ž1.1., Ž3.10., and the Mean Value Theorem,

Ž z i . t y L i z i s f i Ž x, u i Ž t q ␦ , x . , u Ž t q ␦ , x . y f i Ž x, u i Ž t , x . , u Ž t , x . N

s

Ý js1

ž

⭸ fi ⭸ uj

/

J )u Ž t q ␦ , x .

J )u Ž t , x .

ai ,

Ž x, ␰ . z j Ž t , x . q

ai ,

ž

⭸ fi ⭸ ¨j

in D T

ci

ci

.

.

Ž x, ␰X . Ž J j ) z j . Ž t , x .

/

Ž i s 1, . . . , N .

Ž 3.12.

zi Ž t , x . s hi Ž x . y hi Ž x . s 0

on ST

Ž i s 1, . . . , nU .

zi Ž t , x . s ui Ž t q ␦ , x . y u ˆi Ž x . G 0

in Q0

Ž i s 1, . . . , N . ,

where ␰ ' ␰ Ž t, x . and ␰X s ␰X Ž t, x . are some intermediate values between uŽ t, x . and uŽ t q ␦ , x .. Since by Ž H1 ., Ž H2 . and the quasi-monotone nondecreasing property of fŽ⭈, u, v. the functions c i j ' Ž ⭸ f ir⭸ u j .Ž x, ␰ . and d i j ' Ž ⭸ f ir⭸ ¨ j .Ž x, ␰X . are in C ␣ Ž D T . and c i j G 0 for j / i and d i j G 0 for all i, j, we conclude from Lemma 3.1 and the arbitrariness of T that z G 0 in w0, ⬁. = ⍀. This proves the monotone nondecreasing property of uŽ t, x . in t. A similar argument shows that uŽ t, x . is monotone nonincreasing in t and uŽ t, x . G uŽ t, x .. This implies that the pointwise limits lim u Ž t , x . ' u s Ž x . ,

tª⬁

lim u Ž t , x . ' u s Ž x .

tª⬁

576

C. V. PAO

exist and satisfy ˆ us F us F us F ˜ u s on ⍀. The proof that u s and u s are the respective maximal and minimal solutions of Ž3.1. follows from the same reasoning as in w10x. We now consider the general case where the function fŽ x, u, J )u. is mixed quasi-monotone in ²ˆ u s, ˜ u s :. Let v s M y u with a positive constant vector M ' Ž M1 , . . . , MN . G ˜ u s , and consider the extended system of 2 N parabolic-ordinary equations,

Ž u i . t y L i u i s Hi Ž x, u, v, J )u, J )v . Ž ¨ i . t y L i¨ i s Gi Ž x, u, v, J )u, J )v . Ž i s 1, . . . , N . u i Ž t , x . s h i Ž t , x . , ¨ i Ž t , x . s Mi y h i Ž t , x . on ST Ž i s 1, . . . , nU . u i Ž t , x . s ␩i Ž t , x . , ¨ i Ž t , x . s ␯ i Ž t , x . in Q0Ž i. Ž i s 1, . . . , N . , in D T

Ž 3.13.

and its corresponding elliptic system, yL i u i s Hi Ž x, u, v, u, v . yL i ¨ i s Gi Ž x, u, v, u, v .

in ⍀

Ž i s 1, . . . , N . Ž 3.14.

u i Ž x . s h i Ž x . , ¨ i Ž x . s Mi y h i Ž x . on ⭸ ⍀

Ž i s 1, . . . , nU . ,

where u s Ž u1 , . . . , u N ., v s Ž ¨ 1 , . . . , ¨ N ., and Hi Ž x, u, v, J )u, J )v . ' Fi Ž x, u i , w u x a i , w M y v x b i , w J )u x c i , J ) Ž M y v .

di

.

Gi Ž x, u, v, J )u, J )v .

Ž 3.15.

' yFi Ž x, Mi y ¨ i , w M y v x a i , w u x b i , J ) Ž M y v .

ci ,

wux d i .

Ž i s 1, . . . , N . . It is clear from Ž H1 . that the 2 N-vector function F Ž u, v, w, z. ' Ž H1 Ž u, v, w, z. , . . . , HN Ž u, v, w, z. , G1 Ž u, v, w, z. , . . . , GN Ž u, v, w, z. .

Ž 3.16. ˆs, U ˜ s :, where U ˆ s ' Žˆu s , M y ˜u s ., U ˜s is quasi-monotone nondecreasing in ²U ' Ž˜ u s, M y ˆ u s ., and

ˆs , U ˜ s : s  Ž u, v . ; ˆu s F u F ˜u s , M y ˜u s F v F M y ˆu s 4 . ²U

577

PARABOLIC SYSTEMS IN UNBOUNDED DOMAINS

Moreover, ˜ u s and ˆ u s are coupled upper and lower solutions of Ž3.1. if and ˜ s and U ˆ s are ordered upper and lower solutions of Ž3.14. Žsee w10x.. only if U Hence by Corollary 3.1 the elliptic system Ž3.14. has a maximal solution ˆs, U ˜ s :. In the following lemma we show an and a minimal solution in ²U equivalence relation between these solutions and the quasi-solutions of Ž3.1.. LEMMA 3.2. The pair u s and u s are the quasi-solutions of Ž3.1. in ²ˆ u s, ˜ u s : if and only if Žu s , vs . ' Žu s , M y u s . and Žu s , vs . ' Žu s , M y u s . ˆs, U ˜ s :. are the maximal and minimal solutions of Ž3.14. in ²U Proof. Consider the quasi-solutions u s , u s , which are the respective limits of the sequences  uŽsm.4 ,  uŽsm.4 governed by Ž3.3. with uŽ0. u s and s s˜ Ž0. Ž0. . Ž Ž . and uŽ0. s u . It is easy to verify that when u , v s u , M y u ˆ ˜ ˆ s s s s s s Ž0. . Ž m. Ž m. 4 Ž m. Ž m. 4 ŽuŽ0. Ž .   , v s u , M y u , the sequences u , v ' u , M y u and ˆ ˜ s s s s s s s s  uŽsm., vsŽ m.4 '  uŽsm., M y uŽsm.4 satisfy the relation yL i uŽi m. q K i uŽi m. s Fi Ui Ž my1. , UsŽ my1.

ž

ai

, M y VsŽ my1.

UsŽ my1.

ci

bi

,

, M y VsŽ my1.

di

/

yL i ¨ iŽ m. q K i ¨ iŽ m. s yFi Mi y Vi Ž my1. , M y VsŽ my1.

ai

M y VsŽ my1.

ci

ž

yL i uŽi m.

q

, UsŽ my1.

bi

, UsŽ my1.

di

,

/

K i uŽi m.

s Fi Ui Ž my1. , UsŽ my1.

ž

M y VsŽ my1.

ai ,

UsŽ my1.

ci ,

bi ,

M y VsŽ my1.

di

Ž 3.17.

/

yL i ¨ iŽ m. q K i ¨ iŽ m. s yFi Mi y Vi Ž my1. , M y VsŽ my1.

ai ,

UsŽ my1.

bi ,

M y VsŽ my1.

ci ,

UsŽ my1.

di

ž

in ⍀ m

Ž i s 1, . . . , N .

uŽi m. s uŽi m. s h i Ž x . , ¨ iŽ m. s ¨ iŽ m. s Mi y h i Ž x . on ⭸ ⍀U uŽi m. s u ˜i , uŽi m. s uˆi , ¨ iŽ m. s Mi y uˆi , ¨ iŽ m. s Mi y u˜i on ⭸ ⍀Xm

Ž i s 1, . . . , nU . ,

/

578

C. V. PAO

for every m s 1, 2, . . . , where VsŽ m. ' Ž V1Ž m., . . . , VNŽ m. . and VsŽ m. ' Ž V1Ž m., . . . , VNŽ m. . are defined by Vi Ž m. Ž x . '

Vi Ž m. '

½ ½

¨ iŽ m. Ž x .

when x g ⍀ m

Mi y u ˆi Ž x .

when x g ⍀ _ ⍀ m ,

¨ iŽ m. Ž x .

when x g ⍀ m

Mi y u ˜i Ž x .

when x g ⍀ _ ⍀ m

Ž 3.18. Ž i s 1, . . . , N . .

Since vsŽ0. s M y ˆ u s , vsŽ0. s M y ˜ u s , and Ž˜ u s, M y ˆ u s . and Žˆ u s, M y ˜ u s. are ordered upper and lower solutions of Ž3.14., Corollary 3.1 implies that the pointwise limits lim Ž UsŽ m. , VsŽ m. . ' Ž Us , Vs . , lim Ž UsŽ m. , VsŽ m. . s Ž Us , Vs .

mª⬁

mª⬁

exist and are the respective maximal and minimal solutions of Ž3.14.. In view of VsŽ m. s M y UsŽ m., VsŽ m. s M y UsŽ m., and ŽUsŽ m., UsŽ m. . ª Žu s , u s . as m ª ⬁, the above limits lead to ŽUs , Vs . s Žu s , M y u s . and ŽUs , Vs . s Žu s , M y u s .. This shows that if u s and u s are the quasi-solutions of Ž3.1., then Žu s , M y u s . and Žu s , M y u s . are the maximal and the minimal solutions of Ž3.14.. The converse follows from the same argument as that in w10x. In view of Lemma 3.2 and Theorem 2.2, if ˜ u s and ˆ u s are bounded coupled upper and lower solutions of Ž3.1., then each of the problems Ž1.1. ˆs, U ˜ s :, respecand Ž3.13. has a unique bounded solution in ²ˆ u s, ˜ u s : and ²U tively, when ␩ g ²ˆ u s, ˜ u s : and ␯ s M y ␩. Moreover, u is the solution of Ž1.1. in ²ˆ u s, ˜ u s : if and only if Žu, v. ' Žu, M y u. is the solution of Ž3.13. in ˆs, U ˜ s : Žcf. w10x.. This observation leads to the following dynamic property ²U of Ž1.1. with mixed quasi-monotone functions. THEOREM 3.3. Let ˜ u s, ˆ u s be bounded coupled upper and lower solutions of Ž3.1., and let u s , u s be the corresponding quasi-solutions that satisfy Ž3.5.. Assume that hypotheses Ž H1 ., Ž H2 . hold with S ' ²ˆ u s, ˜ u s :. Then for any ␩ g ²ˆ u s, ˜ u s : in Q0 , the corresponding solution uŽ t, x . of Ž1.1. satisfies the relation u s Ž x . F lim inf u Ž t , x . F lim sup u Ž t , x . F u s Ž x . tª⬁

on ⍀ . Ž 3.19.

tª⬁

Moreo¨ er, if u s s u s Ž' uUs ., then uUs is the unique solution of Ž3.1. in ²ˆ u s, ˜ u s : and lim u Ž t , x . s uUs Ž x .

tª⬁

on ⍀ .

Ž 3.20.

PARABOLIC SYSTEMS IN UNBOUNDED DOMAINS

579

Proof. Let ŽuŽ t, x ., vŽ t, x .. and ŽuŽ t, x ., vŽ t, x .. be the solutions of Ž3.13. corresponding to Ž ␩, ␯ . s Ž˜ u s, M y ˆ u s . and Ž ␩, ␯ . s Žˆ u s, M y ˜ u s ., respectively. Since the function F Ž x, u, v, w, z. given by Ž3.16. is quasi-monoˆs, U ˜ s :, and U ˜ s ' Ž˜u s , M y ˆu s . and U ˆ s ' Žˆu s , M y tone nondecreasing in ²U ˜u s . are ordered upper and lower solutions of Ž3.14., an application of Theorem 3.2 to the system Ž3.13. shows that as t ª ⬁, ŽuŽ t, x ., vŽ t, x .. converges to the maximal solution Žu s , vs . of Ž3.14. and ŽuŽ t, x ., vŽ t, x .. converges to the minimal solution Žu s , vs ., where vs s M y u s and vs s M y u s . For arbitrary ␩ g ²ˆ u s, ˜ u s : and ␯ s M y ␩, the corresponding solution ŽuŽ t, x ., vŽ t, x .. satisfies

ŽuŽ t , x . , v Ž t , x . . F Ž uŽ t , x . , v Ž t , x . . F Ž uŽ t , x . , v Ž t , x . . . This leads to the relation Ž3.19.. It is clear from this relation that if u s s u s Ž' uUs ., then uŽ t, x . ª uUs as t ª ⬁. The arbitrariness of ␩ ensures that uUs is the unique solution of Ž3.1. in ²ˆ u s, ˜ u s : and satisfies Ž3.20.. This proves the theorem. Remarks. Ža. It is seen from Theorems 2.1 and 2.2 that the sector ²ˆ u s, ˜ u s : given by Ž2.3. is an invariant set of the parabolic system Ž1.1.. Since by Definition 3.1 and the relation in Ž3.5. the pair of quasi-solutions u s , u s are also coupled upper and lower solutions of Ž3.1., Theorems 3.1 and 3.2 imply that the sector ²u s , u s : is the smallest invariant set as well as an attractor of Ž1.1. Žwith respect to the pair of upper and lower solutions ˜u s and ˆu s .. Žb. If f i ' f i Ž x, u. is independent of J )u for every i, then the definition of upper and lower solutions for Ž1.1. and Ž3.1. and the quasi-monotone property of fŽ x, u. are reduced to that given in w10x. In this situation all of the results obtained in Sections 2 and 3 coincide with that in w10x and are applicable to a coupled system of parabolic-ordinary differential systems.

4. APPLICATIONS In this section we give some applications of the existence and dynamics results obtained in the previous sections to two model problems arising from gas᎐liquid interaction and pattern formulation of chemical concentrations. Our first model involves a dissolved gas and a dissolved reactant that interact in ⍀ e , where ⍀ e is the exterior of a ball of radius r 0 in ⺢ n with n G 3. The equations governing the concentrations of the dissolved

580

C. V. PAO

gas u ' uŽ t, x . and the reactant ¨ s ¨ Ž t, x . are given by u t y D 1 ⵜ 2 u s y␴ 1 uJ2 ) ¨ q q1 Ž x . ¨ t y D 2 ⵜ 2 ¨ s y␴ 2 ¨ J 1 ) u q q 2 Ž x .

Ž t ) 0, x g ⍀ e .

u Ž t , x . s h1 Ž x . , ¨ Ž t , x . s h 2 Ž x . u Ž t , x . s ␩1 Ž t , x . , ¨ Ž t , x . s ␩ 2 Ž t , x .

Ž t ) 0, x g ⭸ ⍀ . Ž t g Ii , x g ⍀ e . ,

Ž 4.1.

where for each i s 1, 2, Di and ␴i are positive constants, and qi , h i , and ␩i are smooth functions Žcf. w1, 3x.. The consideration of J1 ) u and J 2 ) ¨ is for possible time delays in the reaction process, including the case J1 ) u s u, J 2 ) ¨ s ¨ , without any time delay. We assume that qi , h i , and ␩i are all nonnegative in their respective domains and qi possesses the decay property qi Ž x . F ␳ i < x
Ž x g ⍀e.

Ž i s 1, 2 .

Ž 4.2.

for some positive constants ␳ i and ␦ i . It is easily seen that by considering Ž u1 , u 2 . s Ž u, ¨ . and fŽ x, u, ¨ , J1 ) u, J 2 ) ¨ . s Ž f 1Ž x, u, J 2) ¨ ., f 2 Ž x, J1) u, ¨ .., where f 1 Ž x, u, J 2 ) ¨ . s y␴ 1 uJ2 ) ¨ q q1 Ž x . , f 2 Ž x, J1 ) u, ¨ . s y␴ 2¨ J1 ) u q q2 Ž x . ,

Ž 4.3.

2 all of the conditions in Ž H1 . and Ž H2 . are satisfied with S ' ⺢q . Hence for the existence and dynamics of problem Ž4.1., it suffices to construct a suitable pair of upper and lower solutions of the corresponding steady-state problem. Since the function fŽ x, u, ¨ , J1 ) u, J 2 ) ¨ . is quasi-monotone non2 increasing in ⺢q , the requirement of these functions in Ž3.2., denoted by Žu ˜s , ¨˜s . and Ž uˆs , ¨ˆs ., becomes

yD 1 ⵜ 2 u ˜s G y␴ 1 u˜s¨ˆs q q1 Ž x . , y D1 ⵜ 2 u ˆs F y ␴ 1 uˆs¨˜s q q1 Ž x . y D 2 ⵜ 2 ¨˜s G y␴ 2 u ˆs¨˜s q q2 Ž x . , y D 2 ⵜ 2 ¨ˆs F y ␴ 2 u ˜s¨ˆs q q2 Ž x . u ˜s Ž x . G h1 Ž x . G uˆs Ž x .

¨˜s Ž x . G h 2 Ž x . G ¨ˆs Ž x . .

Ž 4.4.

PARABOLIC SYSTEMS IN UNBOUNDED DOMAINS

581

Let qi Ž r . s ␳ i ryŽ2 q ␦ i ., h i Ž r . ' h i Ž< x <. G h i Ž x ., and let Us Ž r . and Vs Ž r . be any positive functions satisfying the relation X

y Ž D 1rr ny1 .Ž r ny1UsX . G q1 Ž r . , X

y Ž D 2rr ny1 .Ž r ny1 VsX . G q2 Ž r . ,

Us Ž r 0 . G h1 Ž r 0 . Vs Ž r 0 . G h 2 Ž r 0 .

Ž r ) r0 . ,

Ž 4.5.

where UsX ' dUsrdr and VsX ' dVsrdr. It is easy to verify from ⵜ 2 w s Ž1rr ny1 .Ž r ny1 wX .X for w ' w Ž r . and qi Ž r . G qi Ž x . G 0 that the pair Žu ˜s , ¨˜s . s ŽUs , Vs . and Ž uˆs , ¨ˆs . s Ž0, 0. satisfy all of the inequalities in Ž4.4.. By Theorem 3.1 the steady-state problem of Ž4.1. has a pair of quasi-solutions Ž u s , ¨ s ., Ž u s , ¨ s . that satisfy the relation

Ž 0, 0 . F Ž u s , ¨ s . F Ž u s , ¨ s . F Ž Us , Vs .

Ž 4.6.

and the equations yD 1 ⵜ 2 u s s y␴ 1 u s¨ s q q1 Ž x . , y D 1 ⵜ 2 u s s y␴ 1 u s¨ s q q1 Ž x . , y D 2 ⵜ 2 ¨ s s y␴ 2 u s¨ s q q2 Ž x . , y D 2 ⵜ 2 ¨ s s y␴ 2 u s¨ s q q2 Ž x . ,

Ž 4.7. Ž x g ⍀.

u s Ž x . s u s Ž x . s h1 Ž x . ,

Ž x g ⭸⍀. .

¨ s Ž x . s ¨ s Ž x . s h2 Ž x .

In fact, the above equations imply that Ž u s , ¨ s . and Ž u s , ¨ s . are both true steady-state solutions. To ensure the decay property of these solutions, we choose Us Ž r ., Vs Ž r . in the form Us Ž r . s Vs Ž r . s Ary␦

Ž r G r0 .

Ž 4.8.

for some positive constants A, ␦ with ␦ - min 1, ␦ 1 , ␦ 2 4 . Indeed, direct computation shows that this pair satisfies Ž4.5. when A G max  h i r 0␦ , ␳ irDi ␦ Ž n y 2 y ␦ . 4

Ž i s 1, 2 . .

It follows from Ž4.6. and Ž4.7. that the solutions Ž u s , ¨ s . and Ž u s , ¨ s . decay to Ž0, 0. as < x < ª ⬁. Note that these solutions are positive in ⍀ when either qi Ž x . k 0 in ⍀ or h i Ž x . k 0 on ⭸ ⍀, and they are identically zero when qi Ž x . ' h i Ž x . ' 0, i s 1, 2. We next show that Ž u s , ¨ s . s Ž u s , ¨ s ., and this is the unique positive steady-state solution of Ž4.1. when q1Ž x . k 0 or h i Ž x . k 0 for i s 1, 2. Let

582

C. V. PAO

w 1 s u s y u s G 0, w 2 s ¨ s y ¨ s G 0. Then a subtraction of the corresponding equations in Ž4.7. gives ⵜ 2 w 1 s Ž ␴ 1rD 1 . Ž u s¨ s y u s¨ s .

in ⍀ ,

w 1 s 0, on ⭸ ⍀ ,

ⵜ 2 w 2 s Ž ␴ 2rD 2 . Ž u s¨ s y u s¨ s .

in ⍀ ,

w 2 s 0 on ⭸ ⍀ .

This implies that W ' Ž ␴ 2rD 2 . w 1 q Ž ␴ 1rD 1 . w 2 satisfies ⵜ 2 W s 0 in ⍀

W s 0 on ⭸ ⍀ .

Since W Ž x . ª 0 as < x < ª ⬁, the Phragman᎐Lindelof ¨ principle ensures that W Ž x . F 0 Žcf. w11, 15x.. This leads to W Ž x . s 0, which proves Ž u s , ¨ s . s Ž u s , ¨ s .. Since qi Ž r . can be made arbitrarily large, Ž u s , ¨ s . is the unique positive steady-state solution. As a consequence of Theorems 3.1, 2.2, and 3.3, we have the following conclusion. THEOREM 4.1. Let ⍀ e be the exterior of a ball of ⺢ n with n G 3, and let qi Ž x . and h(x ., i s 1, 2, be nonnegati¨ e and qi Ž x . satisfy condition Ž4.2.. Then Ži. problem Ž4.1. has a unique nonnegati¨ e steady-state solution Ž u s Ž x ., ¨ s Ž x .., and this solution is positi¨ e in ⍀ e if either qi Ž x . k 0 or h i Ž x . k 0, i s 1, 2. Žii. For any Ž␩1 , ␩ 2 . G Ž0, 0. in Q0 , problem Ž4.1. has a unique nonnegati¨ e solution Ž uŽ t, x ., ¨ Ž t, x .. that is bounded on w0, ⬁. = ⍀ e , and Žiii. Ž u s Ž x ., ¨ s Ž x .. ª Ž0, 0. as < x < ª ⬁ and

Ž uŽ t , x . , ¨ Ž t , x . . ª Ž us Ž x . , ¨ s Ž x . .

as t ª ⬁

Ž x g ⍀e..

Our second model arises from the pattern formation of some activation᎐inhibition system by a reaction᎐diffusion mechanism in ⺢ n. A mathematical model for this system is given by u t y D 1 ⵜ 2 u s f Ž x, J 2 ) ¨ , J4 ) z . yc1 u q q1 Ž x . ¨ t y D 2 ⵜ ¨ s g Ž x, J1 ) u, J 3 ) w . 2

yc2¨ q q2 Ž x . wt y D 3 ⵜ w s c 3 Ž J 1 ) u y w . 2

Ž t ) 0, x g ⺢ n .

Ž 4.9.

z t y D4 ⵜ 2 z s c 4 Ž J 2 ) ¨ y z . u Ž t , x . s ␩1 Ž t , x . , ¨ Ž t , x . s ␩ 2 Ž t , x . , w Ž t , x . s ␩3 Ž t , x . ,

z Ž t , x . s ␩4 Ž t , x .

Ž t g Ii , x g ⺢ n . ,

where u, ¨ , w, and z represent various concentrations of the activators and inhibitors, q1 and q2 are some source functions, Di and c i Ž i s 1, . . . , 4.

PARABOLIC SYSTEMS IN UNBOUNDED DOMAINS

583

are positive constants, and f Ž x, J 2 ) ¨ , J4 ) z . s

g Ž x, J1 ) u, J 3 ) w . s

␴ 1 Ž x . Ž J4 ) z .

2

2

M Ž J4 ) z . q Ž J 2 ) ¨ .

␴ 2 Ž x . Ž J3 ) w . 2

2

Ž 4.10.

2

M Ž J 3 ) w . q Ž J1 ) u .

2

Žcf. w5, 9x.. It is assumed that qi and ␴i Ž i s 1, 2. are smooth bounded nonnegative functions in ⺢ n with qi Ž x . k 0 and M is a positive constant. This implies that the vector function f Ž x, u, J )u . s Ž f 1 Ž x, u, J )u . , . . . , f 4 Ž x, u, J )u . . , where u ' Ž u, ¨ , w, z . and f 1 Ž x, u, J )u . s f Ž x, J 2 ) ¨ , J4 ) z . y c1 u q q1 Ž x . f 2 Ž x, u, J )u . s g Ž x, J1 ) u, J 3 ) w . y c 2¨ q q2 Ž x . f 3 Ž x, u, J )u . s c 3 Ž J1 ) u y w . f 4 Ž x, u, J )u . s c 4 Ž J 2 ) ¨ y z . 4 is mixed quasi-monotone in ⺢q and satisfies all of the conditions in Ž H1 . and Ž H2 .. To show the existence of a unique bounded solution of Ž4.9., we first construct a pair of constant upper and lower solutions of the corresponding steady-state problem. By the mixed quasi-monotone property of the reaction function fŽ x, u, J )u., the requirement of steady-state upper and lower solutions becomes

yD1 ⵜ 2 u ˜ q c1 u˜ G f Ž x, ¨ˆ, ˜z . q q1Ž x . , y D1 ⵜ 2 u ˆ q c1 uˆ F f Ž x, ¨˜, ˆz . q q1Ž x . y D 2 ⵜ 2 ¨˜ q c 2¨˜ G g Ž x, u, ˆ w ˜ . q q2 Ž x . , y D 2 ⵜ 2 ¨ˆ q c 2¨ˆ F g Ž x, u, ˜ w ˆ . q q2 Ž x . y D3 ⵜ 2 w ˜ q c3 w ˜ G c3 u˜

yD 3 ⵜ 2 w ˆ q c3 w ˆ F c3 uˆ

y D4 ⵜ 2 ˜ z q c4 ˜ z G c 4¨˜

yD4 ⵜ 2 ˆ z q c4 ˆ z F c 4¨ˆ.

Ž 4.11.

It is easily seen that for any constant K G Ž ␴irM . q qi , i s 1, 2, the pair Ž u, ˜ ¨˜, w, ˜ ˜z . s Ž Krc1 , Krc2 , Krc3 , Krc4 . and Ž u, ˆ ¨ˆ, w, ˆ ˆz . s Ž0, 0, 0, 0. satisfy all of the inequalities in Ž4.11.. By an application of Theorem 3.1, the

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C. V. PAO

extensions  U Ž k .4 ,  U Ž k .4 of the sequences  uŽ k .4 '  uŽ k ., ¨ Ž k ., w Ž k ., z Ž k .4 and  uŽ k .4 '  uŽ k ., ¨ Ž k ., w Ž k ., z Ž k .4 given by Ž3.3., Ž3.4. converge monotonically to a pair of positive quasi-solutions u s ' Ž u s , ¨ s , ws , z s . and u s ' Ž u s , ¨ s , ws , z s ., which satisfy the relation yD 1 ⵜ 2 u s q c1 u s s f Ž x, ¨ s , z s . q q1 , y D 1 ⵜ 2 u s q c1 u s s f Ž x, ¨ s , z s . q q1 y D 2 ⵜ 2 ¨ s q c 2¨ s s g Ž x, u s , ws . q q2 ,

Ž 4.12.

y D 2 ⵜ 2 ¨ s q c 2¨ s s g Ž x, u s , ws . q q2 y D 3 ⵜ 2 ws q c 3 ws s c 3 u s , y D4 ⵜ 2 z s q c 4 z s s c 4¨ s ,

yD 3 ⵜ 2 ws q c 3 ws s c 3 u s , yD4 ⵜ 2 z s q c 4 z s s c 4¨ s .

Moreover, for any nonnegative initial function, ␩ Ž t , x . ' Ž ␩1 Ž t , x . . , . . . , ␩4 Ž t , x . . F Ž Krc1 , . . . , Krc4 .

in Q0 ,

Theorems 2.1 and 3.3 guarantee that a unique global solution u s Ž u, ¨ , w, z . to Ž4.9. exists and enters the sector ²u s , u s : as t ª ⬁. Since the constant K can be made arbitrarily large, the existence of the solution u is ensured for any nonnegative ␩ Ž t, x .. To obtain a unique positive steady-state solution that decays to zero as < x < ª ⬁, we assume that n G 3 and ␴i and qi satisfy the decay property

␴i Ž x . F ␳ i < x
␳ i Ž x . F ␳ i < x
Ž i s 1, 2 . Ž 4.13.

for some positive constants ␳ i and ␦ i . Let ␳ Ž r . s ␳ ryŽ2 q ␦ . with ␳ G ␳ i and 0 - ␦ F ␦ i , and let Us Ž r ., Vs Ž r . be any positive functions satisfying Ž4.5., with qi Ž r . replaced by Ž1 q My1 . ␳ Ž r . and without the condition at r 0 . Then it is easy to verify that the pair ˜ uUs ' ŽUs , Vs , Us , Vs . and ˆ us ' o satisfy all of the inequalities in Ž4.11.. It follows from Theorem 3.1 that the steady-state problem of Ž4.9. has a pair of quasi-solutions u s and u s such that o F u s F u s F ˜ uUs , and they satisfy the relation Ž4.12.. Since, as in the case of problem Ž4.1., Us s Vs s Ary␦ satisfies relation Ž4.5. and decays to zero as r ª ⬁, we see that lim u s Ž x . s lim u s Ž x . s o

as < x < ª ⬁.

Ž 4.14.

To ensure u s s u s , we let w ' Ž w 1 , w 2 , w 3 , w4 ., with w1 s u s y u s ,

w2 s ¨ s y ¨ s ,

w 3 s ws y ws ,

w4 s z s y z s .

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PARABOLIC SYSTEMS IN UNBOUNDED DOMAINS

Then w G o, and by Ž4.12. and the Mean Value Theorem, yⵜ 2 w 1 s Dy1 yf¨ Ž ␰ 1 , ␰ 2 . w 2 q f z Ž ␰ 1 , ␰ 2 . w4 y c1w 1 1 y ⵜ 2 w 2 s Dy1 yg u Ž ␰ 1X , ␰ 2X . w 1 q g w Ž ␰ 1X , ␰ 2X . w 3 y c 2 w 2 2 y ⵜ 2 w 3 s Ž c3rD 3 . Ž w 1 y w 3 .

Ž 4.15.

y ⵜ 2 w4 s Ž c4rD4 . Ž w 2 y w4 . , where Ž ␰ 1 , ␰ 2 . and Ž ␰ 1X , ␰ 2X . are some intermediate values between o and ˜uU . Let Mi , i s 1, . . . , 4, be the maximum values of < f¨ Ž ␰ 1 , ␰ 2 .<, < f z Ž ␰ 1 , ␰ 2 .<, < g uŽ ␰ 1X , ␰ 2X .<, and < g w Ž ␰ 1X , ␰ 2X .< in ²o, ˜ uU :, respectively. Then the addition of the equations in Ž4.15. gives yⵜ 2 W U F Ž M3rD 2 q c 3rD 3 y c1rD 1 . w 1 q Ž M1rD 1 q c 4rD4 y c 2rD 2 . w 2 q Ž M4rD 2 y c 3rD 3 . w 3 q Ž M2rD 1 y c 4rD4 . w4 , where W U s w 1 q w 2 q w 3 q w4 . Hence if M3rD 2 q c 3rD 3 F c1rD 1 ,

M1rD 1 q c 4rD4 F c 2rD 2 ,

M4rD 2 F c 3rD 3 ,

M2rD 1 F c 4rD4 ,

Ž 4.16.

then yⵜ 2 W U F 0. Since by Ž4.14., W U Ž x . ª 0 as < x < ª ⬁, the Phragman᎐Lindalof ¨ principle implies that W U Ž x . F 0. This leads to W U s 0, which proves u s s u s . As a consequence of Theorems 3.1, 2.1, and 3.3, we have the following conclusions. THEOREM 4.2. Ži. The steady-state problem of Ž4.9. has a pair of quasisolutions u s ' Ž u s , ¨ s , ws , z s ., u s ' Ž u s , ¨ s , ws , z s . that satisfy Ž4.12.. Žii. For any nonnegati¨ e initial function ␩ ' Ž␩1 , . . . , ␩4 ., problem Ž4.9. has a unique global solution u ' Ž u, ¨ , w, z . that enters the sector ²u s , u s : as t ª ⬁. Žiii. If n G 3 and conditions Ž4.13. and Ž4.16. hold, then the steady-state problem has a unique positi¨ e solution u s Ž x . and u s Ž x . ª 0 as < x < ª ⬁. Moreo¨ er, for any initial function ␩ 0 G 0, the corresponding solution Ž u, ¨ , w, z . of Ž4.9. con¨ erges to u s Ž x . as t ª ⬁. REFERENCES 1. L. E. Bobisud, A nonlinear perturbation problem arising in chemical engineering, Nonlinear Analysis 3 Ž1979., 337᎐345. 2. J. M. Cushing, Integrodifferential equations and delay models in population dynamics, Lecture Notes in Biomath. 20 Ž1977..

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3. P. V. Danckwets, ‘‘Gas-Liquid Reactions,’’ McGraw-Hill, New York, 1970. 4. A. Friedman, ‘‘Partial Differential Equations of Parabolic Type,’’ Prentice Hall, Englewood Cliffs, NJ, 1964. 5. A. Gmira, Generation of biological pattern and form: Some physical, mathematical and logical aspects, Progr. Biophys. Mol. Biol. 37 Ž1981., 1᎐47. 6. J. Hale, ‘‘Theory of Functional Differential Equations,’’ Springer-Verlag, New York, 1977. 7. Y. Kuang, ‘‘Delay Differential Equations with Applications in Population Dynamics,’’ Academic Press, New York, 1993. 8. R. H. Martin and H. L. Smith, Reaction-diffusion systems with time delays: Monotonicity, invariance, comparison and convergence, J. Reine Angew. Math. 413 Ž1991., 1᎐35. 9. H. Meinhardt, Tailoring and coupling of reaction-diffusion systems to obtain reproducible complex pattern formation during development of the high organisms, Appl. Math. Comput. 32 Ž1989., 103᎐135. 10. C. V. Pao, Parabolic systems in unbounded domains. I. Existence and dynamics, J. Math. Anal. Appl. 217 Ž1998., 129᎐160. 11. C. V. Pao, ‘‘Nonlinear Parabolic and Elliptic Equations,’’ Plenum Press, New York, 1992. 12. C. V. Pao, Coupled nonlinear parabolic systems with time delays, J. Math. Anal. Appl. 196 Ž1995., 237᎐265. 13. C. V. Pao, Dynamics of nonlinear parabolic systems with time delays, J. Math. Anal. Appl. 198 Ž1996., 751᎐779. 14. C. V. Pao, Systems of parabolic equations with continuous and discrete delays, J. Math. Anal. Appl. 205 Ž1997., 157᎐185. 15. M. H. Protter and H. F. Weinberger, ‘‘Maximum Principles in Differential Equations,’’ Prentice-Hall, Englewood Cliffs, NJ, 1967. 16. C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc. 200 Ž1974., 395᎐418. 17. J. Wu, ‘‘Theory and Applications of Partial Differential Equations,’’ Springer-Verlag, New York, 1996.