Attractors for Partly Dissipative Reaction Diffusion Systems in R n

Journal of Mathematical Analysis and Applications 252, 790᎐803 Ž2000. doi:10.1006rjmaa.2000.7122, available online at http:rrwww.idealibrary.com on

Attractors for Partly Dissipative Reaction Diffusion Systems in ⺢ n Anibal Rodriguez-Bernal1 Departamento de Matematica ´ Aplicada, Uni¨ ersidad Complutense de Madrid, 28040 Madrid, Spain E-mail: [email protected]

and Bixiang Wang 2 Department of Applied Mathematics, Tsinghua Uni¨ ersity, Beijing 100084, People’s Republic of China; and Department of Mathematics, Brigham Young Uni¨ ersity, Pro¨ o, Utah 84602 Submitted by William F. Ames Received April 23, 1999

In this paper, we study the asymptotic behavior of solutions for the partly dissipative reaction diffusion equations in ⺢ n. We prove the asymptotic compactness of the solutions and then establish the existence of the global attractor in L2 Ž⺢ n . = L2 Ž⺢ n .. 䊚 2000 Academic Press Key Words: global attractor; asymptotic compactness; partly dissipativeness; reaction diffusion equation.

1. INTRODUCTION In this paper, we investigate the asymptotic behavior of solutions for the partly dissipative reaction diffusion equations which are concerned with 1 2

Partially supported by DGES PB96-0648. Partially supported by Postdoctorship of Ministerio de Educacion y Cultura ŽSpain.. 790

0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.

REACTION DIFFUSION SYSTEMS

791

two unknown functions u and ¨ defined in ⺢ n,

⭸u ⭸t

y ␯ ⌬ u q ␭u q h Ž u . q ␣ ¨ s f ,

⭸¨ ⭸t

q ␦¨ y ␤u s g,

Ž 1.1. Ž 1.2.

where ␯ , ␭, and ␦ are positive constants, ␣ , ␤ g ⺢, f and g are given functions, and h is a nonlinear function satisfying some growth conditions. System Ž1.1. ᎐ Ž1.2. describes the signal transmission across axons and is a model of FitzHugh᎐Nagumo equations in neurobiology; see w1᎐3x and the references therein. The long time behavior of solutions for problem Ž1.1. ᎐ Ž1.2. in a bounded domain has been studied by several authors; see, e.g., w4᎐6x. Since the dynamical system associated with problem Ž1.1. ᎐ Ž1.2. is not compact even in the case of a bounded domain, Marion w4x used a splitting technique and proved the existence of the global attractor in this case. More precisely, the author decomposed the semigroup into two parts such that one part asymptotically tends to zero and the other part is compact. Then the existence of the global attractor follows from a standard result Žsee, e.g., w7᎐9x.. However, to prove the compactness of one part of the decomposition, the author used the fact that Sobolev embeddings are compact when domains are bounded. For the case of unbounded domains, the lack of compactness of Sobolev embeddings introduces then some extra difficulties. In fact, even for the single reaction diffusion equation in an unbounded domain, the existence of the global attractor in usual Sobolev spaces is not easy to prove and does not follow from the same arguments as for the bounded domain case. To overcome this difficulty and regain some kind of compactness, there have been two different approaches. First, some authors have used weighted Sobolev spaces with weights of the form Ž1 q < x < 2 . ␣ for some either positive or negative ␣ , which forces the functions to decrease or grow sufficiently fast at infinity. For this approach the nonhomogeneous forcing terms are also assumed to be in a weighted space and existence of global attractors in L2 Ž⺢ n . is not achieved; see Remark 2.14 in w10x. On the other hand, some authors have used spaces of bounded continuous functions together with suitably smooth and fast decreasing super-solutions; see, e.g., w10᎐15x and the references therein. Recently, in w16x the existence of the global attractor was proved in L2 Ž⺢ n . for a single reaction diffusion equation in an unbounded domain. Here, using a similar idea, we shall study the existence of the global attractor for the partly dissipative reaction diffusion system Ž1.1. ᎐ Ž1.2. in

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RODRIGUEZ-BERNAL AND WANG

L2 Ž⺢ n . = L2 Ž⺢ n .. The key point to our method relies on the exact estimates on solutions which show that the solutions are uniformly small at infinity for large time; see Lemma 2.3 below. This combined with suitable estimates and the compactness of Sobolev embeddings in bounded domains allows us to prove the existence of the global attractor. This paper is organized as follows. In the next section, we derive some estimates on the solutions which will be used in the proof of existence of the global attractor. In Section 3, we present the proof of the asymptotic compactness and then conclude our result by an abstract theorem. Finally Section 4 contains some other situations in which our proofs work the same.

2. PRELIMINARIES In this section, we consider the following partly dissipative reaction diffusion system,

⭸u ⭸t

y ␯ ⌬ u q ␭u q h Ž u . q ␣ ¨ s f

⭸¨ ⭸t

q ␦¨ y ␤u s g

in ⺢ n = ⺢q,

in ⺢ n = ⺢q,

Ž 2.1. Ž 2.2.

with the initial data u Ž 0, x . s u 0 Ž x . ,

¨ Ž 0, x . s ¨ 0 Ž x . in ⺢ n .

Ž 2.3.

where ␯ , ␭, ␦ ) 0, f, g g L2 Ž⺢ n . given, ␣ and ␤ are numbers satisfying

␣␤ ) 0,

Ž 2.4.

while h is a nonlinear function satisfying the conditions h Ž s . s G yC0 s 2 ,

h Ž 0 . s 0,

h⬘ Ž s . G yC,

s g ⺢, Ž 2.5.

for some 0 F C0 - ␭, < h⬘ Ž s . < F C Ž 1 q < s < r . ,

s g ⺢,

Ž 2.6.

with r G 0 for n F 2 and r F minŽ n4 , n y2 2 . for n G 3, and C is a positive constant. Note that since 0 F C0 - ␭, adding a linear term to h and relabeling the constant ␭, we can assume that ␭ ) 0, C0 s 0. Note that conditions Ž2.5. and Ž2.6. are usual conditions for the study of the asymptotic behavior of solutions in an unbounded domain; see, e.g., w9, 10x.

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In view of Ž2.1. ᎐ Ž2.2., it is clear that if Ž u, ¨ . is a solution for the data Ž ␣ , ␤ , f, g ., then Ž u, y¨ . is a solution of Ž2.1. ᎐ Ž2.2. for the data Žy␣ , y␤ , f, yg .. Using this and condition Ž2.4. we will assume throughout this paper and without loss of generality that ␣ and ␤ are both positive. In the sequel, we denote by H s Ž⺢ n . the standard Sobolev spaces and H s L2 Ž⺢ n .. We also use 5 ⭈ 5 and Ž⭈, ⭈ . for the usual norm and inner product of L2 Ž⺢ n .. For any 1 F p F ⬁, we denote by 5 ⭈ 5 p the norm of L p Ž⺢ n . and in particular, for p s 2 we denote 5 ⭈ 5 2 s 5 ⭈ 5. In general, 5 ⭈ 5 X denotes the norm of any Banach space X. By standard methods, it is easy to prove that if Ž2.4. ᎐ Ž2.6. hold and f, g g H, then problem Ž2.1. ᎐ Ž2.3. is well-posed in H = H; that is, for any Ž u 0 , ¨ 0 . g H = H, there exists a unique solution Ž u, ¨ . g C Ž⺢q, H = H .. This establishes the existence of a dynamical system  S Ž t .4t G 0 which maps H = H into H = H such that SŽ t .Ž u 0 , ¨ 0 . s Ž uŽ t ., ¨ Ž t .., the solution of problem Ž2.1. ᎐ Ž2.3.; see w10x. In what follows, we shall formally derive a priori estimates on the solutions which hold for smooth functions and will become rigorous by a limiting process. Hereafter, we shall denote by C any positive constants which may change from line to line. We begin with the estimates in H = H. LEMMA 2.1. Assume that Ž2.4. ᎐ Ž2.6. hold and f, g g H. Then the solution Ž u, ¨ . of problem Ž2.1. ᎐ Ž2.3. satisfies 5 u Ž t . 5 q 5 ¨ Ž t . 5 F M,

t G T1 ,

tq1

t G T1 ,

Ht

5 ⵜu Ž ␶ . 5 2 d␶ F M,

Ž 2.7.

where M is a constant depending only on the data Ž ␯ , ␭, ␦ , ␣ , ␤ , f, g . and T1 depends on the data Ž ␯ , ␭, ␦ , ␣ , ␤ , f, g . and R when 5Ž u 0 , ¨ 0 .5 H= H F R. Proof. Taking the inner product of Ž2.1. with ␤ u in H, we find that 1 2

␤

d dt

5 u 5 2 q ␤␯ 5 ⵜu 5 2 q ␤␭ 5 u 5 2

q␤

H⺢ h Ž u . u q ␤␣H⺢ u¨ s ␤H⺢ fu. n

n

n

Ž 2.8.

Similarly, taking the inner product of Ž2.2. with ␣ ¨ in H, we find 1 2

␣

d dt

5 ¨ 5 2 q ␣␦ 5 ¨ 5 2 y ␣␤

H⺢ u¨ s ␣H⺢ g¨ . n

n

Ž 2.9.

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RODRIGUEZ-BERNAL AND WANG

Summing up Ž2.8. and Ž2.9., we have 1 d 2

␤ 5 u 5 2 q ␣ 5 ¨ 5 2 . q ␤␯ 5 ⵜu 5 2 q ␤␭ 5 u 5 2 q ␣␦ 5 ¨ 5 2 Ž dt q␤

H⺢ h Ž u . u s ␤H⺢ fu q ␣H⺢ g¨ . n

n

Ž 2.10.

n

We now majorize the right-hand side of Ž2.10. as follows. First, we have

H⺢ fu

F ␤ 5 f 5 5 u5 F

H⺢ g¨

F ␣ 5 g 5 5¨ 5 F

␤

n

1 2

␤␭ 5 u 5 2 q

␤ 2␭

5 f 52,

Ž 2.11.

5 g 52.

Ž 2.12.

and

␣

n

1 2

␣␦ 5 ¨ 5 2 q

␣ 2␦

By Ž2.10. ᎐ Ž2.12. and Ž2.5., we get d

␤ 5 u 5 2 q ␣ 5 ¨ 5 2 . q 2 ␤␯ 5 ⵜu 5 2 q ␤␭ 5 u 5 2 Ž dt q ␣␦ 5 ¨ 5 2 F

␤ ␭

␣

5 f 52 q

␦

5 g 52.

Ž 2.13.

Let ␴ s min ␭, ␦ 4 . Then we find d

Ž ␤ 5 u5 2 q ␣ 5 ¨ 5 2 . q ␴ Ž ␤ 5 u5 2 q ␣ 5 ¨ 5 2 . F dt

␤ ␭

5 f 52 q

␣ ␦

5 g 52.

Thus, the Gronwall lemma gives, when 5Ž u 0 , ¨ 0 .5 H= H F R,

␤ 5 uŽ t . 5 2 q ␣ 5 ¨ Ž t . 5 2 F ey ␴ t Ž ␤ 5 u Ž 0 . 5 2 q ␣ 5 ¨ Ž 0 . 5 2 . q F Ž ␣ q ␤ . R 2 ey ␴ t q

␤ ␴␭

5 f 52 q

␤ ␴␭

␣ ␴␦

5 f 52 q

5 g52 F

␣ ␴␦

2␤

␴␭

5 g52

5 f 52 q

2␣

␴␦

5 g 52,

Ž 2.14. for t G T1 [ ␴1 lnŽ ␴␭␦ Ž ␣ q ␤ . R 2r␤␦ 5 f 5 2 q ␣␭ 5 g 5 2 .. Integrating Ž2.13. between t and t q 1, by Ž2.14. we get for t G T1 , 2 ␤␯

tq1

Ht

5 ⵜu Ž ␶ . 5 d␶ F

␤ ␭

5 f 52 q

␣ ␦

5 g 5 2 q ␤ 5 u Ž t . 5 2 q ␣ 5 ¨ Ž t . 5 2 F C.

Ž 2.15. Then, Ž2.14. and Ž2.15. conclude the proof of Lemma 2.1.

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REACTION DIFFUSION SYSTEMS

Note that from Ž2.14. ᎐ Ž2.15. if f s 0 s g then all solutions converge to the trivial solution Ž u, ¨ . s Ž0, 0. as time goes to infinity. Hence in the lemma above one can take M any positive number and T1 depends on Ž ␯ , ␭, ␦ , ␣ , ␤ . and on M. Since this is not a very interesting case, for the remainder of this section we will assume that f and g are not simultaneously zero. In what follows, we denote by B the ball, B s  Ž u, ¨ . g H = H : 5 Ž u, ¨ . 5 H= H F M 4 ,

Ž 2.16.

where M is the constant in Ž2.7.. Then it follows from Lemma 2.1 that B is an absorbing set for SŽ t . in H = H. By the boundedness of B and Lemma 2.1 again, we know that there exists a constant T Ž B . depending only on the data Ž ␯ , ␭, ␦ , ␣ , ␤ , f, g . and B such that S Ž t . B ; B,

t G T Ž B. .

Ž 2.17.

We now establish the estimates in the space H ⺢ . 1Ž

n.

LEMMA 2.2. Assume that Ž2.4. ᎐ Ž2.6. hold and f, g g H. Then any solution Ž u, ¨ . of problem Ž2.1. ᎐ Ž2.3. satisfies 5 u Ž t . 5 H 1 F C,

t G T1 q 1,

Ž 2.18.

where C depends only on the data Ž ␯ , ␭, ␦ , ␣ , ␤ , f, g . and T1 is the constant in Lemma 2.1. Proof. Taking the inner product of Ž2.1. with y⌬u in H, we get 1 d 2 dt

5 ⵜu 5 2 q ␯ 5 ⌬ u 5 2 q ␭ 5 ⵜu 5 2 s

H⺢ h Ž u . ⌬ u q ␣H⺢ ¨ ⌬ u y H⺢ f⌬ u. n

n

n

Ž 2.19.

We now handle each term on the right-hand side of Ž2.19.. By Ž2.5. we have

H⺢ h Ž u . ⌬ u s yH⺢ h⬘ Ž u . < ⵜu < n

n

2

F C 5 ⵜu 5 2 .

Ž 2.20.

Also, we find that

H⺢ f⌬ u n

F 5 f 5 5 ⌬ u5 F

1 4

␯ 5 ⌬ u5 2 q

1

5 f 52,

Ž 2.21.

␯ 5 ⌬ u 5 2 q C.

Ž 2.22.

␯

and, by Ž2.7., we get for t G T1 ,

␣

H⺢ ¨ ⌬ u n

F ␣ 5 ¨ 5 5 ⌬ u5 F C 5 ⌬ u5 F

1 4

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RODRIGUEZ-BERNAL AND WANG

It follows from Ž2.19. ᎐ Ž2.22. that for t G T1 , d dt

5 ⵜu 5 2 q ␯ 5 ⌬ u 5 2 q 2 ␭ 5 ⵜu 5 2 F C Ž 1 q 5 ⵜu 5 2 . ,

which implies d dt

5 ⵜu 5 2 F C Ž 1 q 5 ⵜu 5 2 . .

Ž 2.23.

By Ž2.23., Lemma 2.1, and the uniform Gronwall lemma in w9x, we get that 5 ⵜu Ž t . 5 2 F C,

t G T1 q 1,

Ž 2.24.

which concludes the proof of Lemma 2.2. We are now in a position to derive the estimates on solutions for large time and space variables. These estimates will be used when we prove the asymptotic compactness of the dynamical system SŽ t .. In order to overcome the non-compactness of Sobolev embeddings in ⺢ n, we decompose the whole space ⺢ n into a bounded ball and its complement. Then the asymptotic compactness of SŽ t . will follow from the compact Sobolev embeddings in the bounded ball and the estimates in its complement. The following lemma will play a crucial role in the proof of our result. LEMMA 2.3. Assume that Ž2.4. ᎐ Ž2.6. hold, f, g g H, and Ž u 0 , ¨ 0 . g B, the bounded absorbing set in Ž2.16.. Then for e¨ ery ␧ ) 0, there exist T Ž ␧ . and k Ž ␧ . depending on ␧ such that the following holds, for t G T Ž ␧ . and k G k Ž ␧ .,

H< x
2

q < ¨ Ž t . < 2 . dx F ␧ .

Ž 2.25.

Proof. Choose a smooth function ␪ such that 0 F ␪ Ž s . F 1 for s g ⺢q, and

␪ Ž s . s 0 for 0 F s F 1;

␪ Ž s . s 1 for s G 2,

and there exists a constant C such that < ␪ ⬘Ž s .< F C for s g ⺢q. Multiplying Ž2.1. by ␤␪ Ž< x < 2rk 2 . u, and then integrating the resulting identity, we find 1 2

␤

d dt

H⺢ ␪ n

< x<2

< u < 2 y ␤␯

ž / H ž /

s y␤

k2

␪

⺢n

< x<2 k2

H⺢ ␪ n

< x<2

ž / k2

h Ž u . u y ␤␣

u⌬ u q ␤␭

H⺢ ␪ n

H⺢ ␪

< x<2

ž / k2

n

< x<2

< u< 2

ž / H ž /

u¨ q ␤

k2

␪

⺢n

< x<2 k2

uf .

Ž 2.26.

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REACTION DIFFUSION SYSTEMS

Similarly, multiplying Ž2.2. by ␣␪ Ž< x < 2rk 2 . ¨ , and then integrating the resulting identity, we get 1 2

␣

d dt

< x<2

H⺢ ␪

< ¨ < 2 q ␣␦

H⺢ ␪

ž / H ž /

n

k2

s ␣␤

< x<2

␪

<¨ <2

ž / H ž / k2

n

u¨ q ␣

k2

⺢n

< x<2

< x<2

␪

k2

⺢n

¨g .

Ž 2.27.

Summing up Ž2.26. and Ž2.27. we find 1 d 2 dt

H⺢ ␪ n

< x<2

␤ < u< 2 q ␣ < ¨ < 2 .

ž /Ž H ž / H ž / H ž / k2

q ␤␭

< x<2

␪

k2

⺢n

s ␤␯

< x<2

␪

k2

⺢n

q␤

< u < 2 q ␣␦

< x<2

␪

k2

⺢n

< x<2

␪

<¨ <2

ž / H ž / H ž /

H⺢

u⌬ u y ␤ uf q ␣

k2

n

< x<2

␪

hŽ u. u

k2

⺢n

< x<2

␪

k2

⺢n

¨g .

Ž 2.28.

We now majorize each term on the right-hand side of Ž2.28.. We start with

␤␯

H⺢ ␪ n

< x<2

ž / k2

u⌬ u s y␤␯

H⺢ ␪

< x<2

< ⵜu < 2

ž / H ž / ž n

y ␤␯

␪⬘

⺢n

k2

< x<2 k

u

2

2x k2

⭈ ⵜu .

/

Ž 2.29.

Note that the first term above has a negative sign so it can be neglected, so we just need to bound the second term. By Ž2.17. and Ž2.24. we have y␤␯

H⺢

␪⬘

n

< x<2

ž /ž k

2

u

2x k

2

⭈ ⵜu

/

FC F F

< x<

HkF< x
2

< u < < ⵜu <

C

C

< u < < ⵜu < F H < u < < ⵜu < H k kF < x
C k

5 u 5 5 ⵜu 5 F

C k

,

t G T Ž B . q 1,

Ž 2.30.

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RODRIGUEZ-BERNAL AND WANG

when C is independent of k. So, given ␧ ) 0, setting k t Ž ␧ . s k G k 1 , by Ž2.29. and Ž2.30. we get

␤␯

H⺢

␪

n

< x<2

ž / k

2

␧

u⌬ u F

2

2C ␧

, then for

t G T Ž B . q 1.

,

Ž 2.31.

Note that the second term in the right hand side of Ž2.28. is negative and can be then neglected. For the third term we have

␤

H⺢ ␪ n

< x<2

ž / k2

uf F ␤

H< x
F␤

F␤

F

1 2

žH žH

< x
< x
␤␭

< x<2

ž /

< f <2

< f <2

H⺢ ␪ n

uf

k2

< x<2

1 2

␪2

1 2

< u< 2

/ žH ž / / / žH ž / / < x
␪

k

2

< u< 2

k2

⺢n

ž /

1 2

< x<2

1 2

< x<2

k2

< u< 2 q

␤ 2␭

H< x
Ž using 0 F ␪ F 1 . 2

.

Ž 2.32.

Similarly, we can also deduce that

␣

H⺢

␪

n

< x<2

ž / k

2

¨g F

1 2

␣␦

H⺢

␪

n

< x<2

ž / k

2

<¨ <2 q

␣ 2␦

H< x
2

.

Ž 2.33.

By Ž2.28. and Ž2.31. ᎐ Ž2.33., using Ž2.5. we get for ␧ ) 0, d dt

< x<2

H⺢ ␪

ž / k2

n

F␧q

Ž ␤ < u < 2 q ␣ < ¨ < 2 . q ␤␭

␤

H⺢ ␪

H
2

q

␣ ␦

H< x
n

2

< x<2

ž / k2

< u < 2 q ␣␦

.

H⺢ ␪ n

< x<2

ž / k2

<¨ <2

Ž 2.34.

Since f and g belong to H, for given ␧ ) 0, there exists k 2 Ž ␧ . such that for k G k 2 Ž ␧ . ␤ ␣ < f <2 q < g<2 F ␧ . Ž 2.35. ␭ < x
H

H

Letting ␴ s min ␭, ␦ 4 and k Ž ␧ . s max k 1Ž ␧ ., k 2 Ž ␧ .4 , then from Ž2.34. and Ž2.35. we find for t G T Ž B . q 1 and k G k Ž ␧ ., d dt

H⺢

␪

n

< x<2

ž / k2

Ž ␤ < u < 2 q ␣ < ¨ < 2 . q ␴H

␪

⺢n

< x<2

ž /Ž k2

␤ < u< 2 q ␣ < ¨ < 2 . F 2 ␧ .

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REACTION DIFFUSION SYSTEMS

By Ž2.17., it follows from the Gronwall lemma that

H⺢

␪

n

< x<2

ž /Ž k2

␤ < uŽ t . < 2 q ␣ < ¨ < 2 .

F ey␴ Ž tyT Ž B .y1.

H⺢

␪

n

< x<2

ž / k2

= Ž ␤ < u Ž T Ž B . q 1. < 2 q ␣ < ¨ Ž T Ž B . q 1. < 2 . q F Ž ␣ q ␤ . M 2 ey ␴ Ž tyT Ž B .y1. q

2␧

␴

2␧

␴

.

Setting T Ž ␧ . s ␴1 lnŽ ␴ Ž ␭ q ␤ . M 2r␧ . q T Ž B . q 1, then for t G T Ž ␧ . and k G k Ž ␧ ., we have

H< x
Ž ␤ < uŽ t . < 2 q ␣ < ¨ Ž t . < 2 . F

H⺢

␪

n

< x<2

ž / k

2

Ž ␤ < uŽ t . < 2 q ␣ < ¨ Ž t . < 2 . F

3␧

␴

,

which concludes the proof of Lemma 2.3.

3. GLOBAL ATTRACTORS In this section, we present our main result, that is, the existence of the global attractor for S Ž t . in H = H. To this end, we first need to establish the asymptotic compactness for the dynamical system SŽ t .. Once this kind of compactness is obtained, the existence of the global attractor follows from a standard result which can be stated as follows Žsee w7᎐9, 17x for similar results .. PROPOSITION 3.1. Assume that X is a metric space and  SŽ t .4t G 0 is a semigroup of continuous operators in X. If  S Ž t .4t G 0 has a bounded absorbing set and is asymptotically compact, then  SŽ t .4t G 0 possesses a global attractor which is a compact in¨ ariant set and attracts e¨ ery bounded set in X. Below, we present a general result which concerns the asymptotic compactness for the functions defined in unbounded domains. For every k G 0, we denote by ⍀ k s  x g ⺢ n : < x < F k 4 and for a function w defined on ⺢ n, we denote by w N ⍀ k the restriction of w to ⍀ k . Then we have the following theorem. THEOREM 3.1. Let Z be a set in H r Ž⺢ n . = H s Ž⺢ n . with r, s G 0. Then Z is precompact in H r Ž⺢ n . = H s Ž⺢ n . if the following conditions are satisfied:

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Ži. For e¨ ery ␧ ) 0, there exists a constant K Ž ␧ . depending on ␧ such that for e¨ ery w g Z, 5 w 5 H r Ž⺢ n _ ⍀ K Ž ␧ . .=H s Ž⺢ n _ ⍀ K Ž ␧ . . - ␧ . Žii. For e¨ ery k G 0, the set of restrictions Zk s  w N ⍀ , w g Z 4 is k precompact in H r Ž ⍀ k . = H s Ž ⍀ k .. Proof. We prove that for every ␧ ) 0 there exists a finite covering of Z of balls of radii at most 3 ␧ . Let ␧ ) 0 and K Ž ␧ . be as in Ži.. Since ZK Ž ␧ . is m precompact in H r Ž ⍀ K Ž ␧ . . = H s Ž ⍀ K Ž ␧ . ., then ZK Ž ␧ . ; D is1 B Ž z i , ␧ i . where z i g ZK Ž ␧ . and ␧ i F ␧ . m Now we claim that Z ; D is1 B Ž z i , 3 ␧ .. In fact, if z g Z then for some r 5 5 i s 1, . . . , m, z y z i H Ž ⍀ K Ž ␧ . .=H s Ž ⍀ K Ž ␧ . . F ␧ i and then 5 z y z i 5 H r Ž⺢ n .=H s Ž⺢ n . F 5 z y z i 5 H r Ž⺢ n _ ⍀ K Ž ␧ .=H s Ž⺢ n _ ⍀ K Ž ␧ . . q 5 z y z i 5 H r Ž ⍀ K Ž ␧ . .=H s Ž ⍀ K Ž ␧ . . which is less than 2 ␧ q ␧ i F 3 ␧ . We now prove the asymptotic compactness for the dynamical system SŽ t . generated by problem Ž2.1. ᎐ Ž2.3.. THEOREM 3.2. Assume that Ž2.4. ᎐ Ž2.6. hold and f, g g H. Then the semigroup SŽ t . is asymptotically compact, that is, if  u m , ¨ m 4 is bounded in H = H and t m ª q⬁, then  SŽ t m .Ž u m , ¨ m .4 is precompact in H = H. Proof. Let Z s  S Ž t m .Ž u m , ¨ m . : m s 1, 2, . . . 4 and we check below that Z satisfies the conditions in Theorem 3.1 with r s s s 0. First, since Ž u m , ¨ m .4 is bounded in H = H, we can assume that there exists R such that 5Ž u m , ¨ m .5 H= H F R. Then by Lemma 2.1 we see that there exists a constant T1Ž R . depending on R such that

Ž u m Ž t . , ¨ m Ž t . . s S Ž t . Ž u m , ¨ m . ; B,

t G T1 Ž R . ,

Ž 3.1.

for every m g ⺞, where B is the absorbing set given in Ž2.16.. Since t m ª q⬁, there exists M1Ž R . such that if m G M1Ž R ., then t m G T1Ž R ., and hence S Ž t m . Ž u m , ¨ m . s S Ž t m y T1 . Ž S Ž T1 . Ž u m , ¨ m . . .

Ž 3.2.

Using Ž3.1. and Lemma 2.3, we find that for every ␧ ) 0, there exist T Ž ␧ . and K Ž ␧ . such that 5 S Ž t . Ž S Ž T1 . Ž u m , ¨ m . . 5 L2 Ž⺢ n _ ⍀ K Ž ␧ . .=L2 Ž⺢ n _ ⍀ K Ž ␧ . . - ␧ ,

t G T Ž ␧ . . Ž 3.3.

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Again, since t m ª q⬁, there exists M2 Ž ␧ . such that if m G M2 Ž ␧ ., then t m y T1 G T Ž ␧ .. Therefore, by Ž3.2. and Ž3.3. we get for m G max M1Ž ␧ ., M2 Ž ␧ .4 , S Ž tm . Ž um , ¨ m .

L2 Ž ⺢ n _⍀ K Ž ␧ . .=L2 Ž ⺢ n _⍀ K Ž ␧ . .

s S Ž t m y T1 . Ž S Ž T1 . Ž u m , ¨ m . .

L2 Ž ⺢ n _⍀ K Ž ␧ . .=L2 Ž ⺢ n _⍀ K Ž ␧ . .

- ␧ , Ž 3.4.

which verifies the condition Ži. in Theorem 3.1. On the other hand, assume k ) 0 is given. Using now Lemma 2.2 we have, like in Ž3.2.,

Ž u m Ž t m . , ¨ m Ž t m . . s S Ž t m . Ž u m , ¨ m . s S Ž t m y T2 . Ž S Ž T2 . Ž u m , ¨ m . . Ž 3.5. with T2 s T1 q 1, for all sufficiently large m. Since the embedding H 1 Ž ⍀ k . ; L2 Ž ⍀ k . is compact, from the estimate Ž2.18. it is clear that  u m Ž t m .4 lies in a compact set in L2 Ž ⍀ k .. On the other hand, for the solutions of Ž2.1. ᎐ Ž2.3., one can decompose Ž ¨ t . s ¨ 1Ž t . q ¨ 2 Ž t . where ¨ i Ž t . solves, respectively,

⭸ c1 ⭸t

q ␦ ¨ 1 s 0, ¨ 1 Ž 0 . s ¨ 0 ,

⭸ ¨2 ⭸t

q ␦ ¨ 2 y ␤ u s g , ¨ 2 Ž 0. s 0

that is, ¨ 1 Ž x, t . s ¨ 0 Ž x . ey␦ t , ¨ 2 Ž x, t . s ␤

t y␦ Ž tys.

H0 e

u Ž x, s . ds q g Ž x .

Ž 1 y ey␦ t . ␦

.

Using Ž3.5., the estimate Ž2.18., and the above with ¨ Ž t . s ¨ m Ž t q T2 . s q T2 . q ¨ 2m Ž t q T2 ., uŽ x, s . s u m Ž x, s q T2 ., t s t m y T2 , and ¨ 0 s ¨ T2 ., we obtain that the H 1 Ž ⍀ k . norm of the first term in ¨ 2m Ž t m . is uniformly bounded in m. On the other hand, the second term lies in the compact set in L2 Ž ⍀ k .,  ␪ g Ž x ., ␪ g w0, 1␦ x4 . Therefore, ¨ 2m Ž t m . lies in a compact set in L2 Ž ⍀ k .. Since ¨ 1m Ž t m . decays to zero as t m ª ⬁ we get that ¨ m Ž t m . also lies in a compact set of L2 Ž ⍀ k .. Hence, we find that the set  SŽ t m .Ž u m , ¨ m . N ⍀ k : m s 1, 2 ⭈⭈⭈ 4 is precompact in L2 Ž ⍀ k . = L2 Ž ⍀ k . which verifies the hypothesis Žii. in Theorem 3.1. Then the proof is complete. ¨ 1m Ž t mŽ

We are now in a position to state our main result. THEOREM 3.3. Assume that Ž2.4. ᎐ Ž2.6. hold and f, g g H. Then problem Ž2.1. ᎐ Ž2.3. has a global attractor which is a compact in¨ ariant set and attracts e¨ ery bounded set in H = H.

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Proof. We note that we have established the existence of a bounded absorbing set for SŽ t . in Lemma 2.1 and the asymptotic compactness in Theorem 3.2, so the existence of the global attractor follows from Proposition 3.1. As observed after Lemma 2.1, if the forcing terms f and g are both zero, then all solutions converge to Ž0, 0.. In this trivial case the global attractor reduces to the single point Ž0, 0..

4. FINAL REMARKS The arguments in the previous sections can be applied to more general partly dissipative reaction diffusion systems of the form

⭸u ⭸t

in ⺢ n = ⺢q,

y ␯ ⌬ u q ␭ u q h Ž u, ¨ . s f

⭸¨ ⭸t

q ␦ ¨ q j Ž u, ¨ . s g

in ⺢ n = ⺢q,

Ž 4.1. Ž 4.2.

with initial data u Ž 0, x . s u 0 Ž x . ,

¨ Ž 0, x . s ¨ 0 Ž x . in ⺢ n ,

Ž 4.3.

where ␯ , ␭, ␦ ) 0 and f, g g L ⺢ are given. On the nonlinear terms we assume that they satisfy some natural growth assumptions such that Ž4.1. ᎐ Ž4.3. is well posed in L2 Ž⺢ n . = L2 Ž⺢ n .. In order for Lemma 2.1 to remain true, observe that it is enough that for some ␣ , ␤ ) 0 one has 2Ž

n.

␤ h Ž u, ¨ . u q ␣ j Ž u, ¨ . ¨ G yC0 < u < 2 y C1 < ¨ < 2

Ž 4.4.

for some 0 F C0 - ␭ and 0 F C1 - ␦ . With this the estimate derived from Ž2.10. can be carried over. On the other hand, the proof of Lemma 2.2 can be repeated with no major changes provided for example that for some positive C h Ž u, ¨ . s h 0 Ž u . q h1 Ž u, ¨ . , h 0 Ž 0 . s 0, < h1 Ž u, ¨ . < F C Ž < u < q < ¨ < . .

hX0 Ž u . G yC,

Ž 4.5.

On the other hand, condition Ž4.4. allows us to repeat the proof of Lemma 2.3 along the same lines as above, since one can derive an estimate very similar to Ž2.28.. Finally, the proof of Theorem 3.2 can be carried out provided for example that j s jŽ u. ,

< j⬘ Ž u . < F C Ž 1 q < u < r .

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803

with r G 0 for n F 2 and r - n y2 2 for n G 3, and C a positive constant since in this case, for any bounded set ⍀, the mapping j : H 1 Ž ⍀ . ª L2 Ž ⍀ . is compact. Therefore, the argument using the splitting ¨ Ž t . s ¨ 1Ž t . q ¨ 2 Ž t . runs as before.

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