Bulletin of the. Iranian Mathematical Society

Transcript

1 ISSN: X (Print) ISSN: (Online) Bulletin of the Iranian Mathematical Society Vol. 43 (2017), No. 2, pp Title: Pullback D-attractors for non-autonomous partly dissipative reaction-diffusion equations in unbounded domains Author(s): X. Li Published by Iranian Mathematical Society

2 Bull. Iranian Math. Soc. Vol. 43 (2017), No. 2, pp Online ISSN: PULLBACK D-ATTRACTORS FOR NON-AUTONOMOUS PARTLY DISSIPATIVE REACTION-DIFFUSION EQUATIONS IN UNBOUNDED DOMAINS X. LI (Communicated by Fatemeh Helen Ghane) Abstract. At present paper, we establish the existence of pullback D- attractor for the process associated with non-autonomous partly dissipative reaction-diffusion equation in L 2 (R n ) L 2 (R n ). In order to do this, by energy equation method we show that the process, which possesses a pullback D-absorbing set, is pullback D 0 -asymptotically compact. Keywords: Pullback attractors, partly dissipative, reaction-diffusion equations. MSC(2010): Primary: 37B55; Secondary: 37L05, 37L30, 35B40, 35B41, 35K Introduction Consider the following non-autonomous partly dissipative reaction-diffusion equation (1.1) u t ν u + λu + h(u) + αυ = f(x, t) in [, + ) Rn, (1.2) υ t + δυ βu = g(x, t) in [, + ) Rn, with the initial data (1.3) u(, x) = u (x), υ(, x) = υ (x) in R n, Article electronically published on 30 April, Received: 6 September 2014, Accepted: 4 December c 2017 Iranian Mathematical Society 515

3 Pullback D-attractors 516 where ν, λ, δ > 0 with λ < δ, f, g L loc (R, L 2 (R n ), α, β R with αβ > 0. Suppose that the nonlinear function h satisfies (1.4) (1.5) h(s)s C 0 s 2, h(0) = 0, h (s) C 1, s R, h (s) C 2 (1 + s r ), s R, with r 0 for n 2 and r min{ 4 n, 2 n 2 } for n 3, where C i, i = 0, 1, 2, are positive constants and 0 C 0 < λ. The long-time behavior of the solutions of non-autonomous evolution partial differential equations can be characterized in terms of uniform attractors, that is, the compact sets that uniformly (with respect to time symbol) attract every bounded subset of the phase space and are minimal in the sense of attracting property. By constructing the skew-product flow, one can reduce the non-autonomous system to a autonomous one in an extended phase space. In this situation, the structure of the uniform attractor can be described as the union of kernel sections. See details in [6]. At the same time, pullback (or cocycle) attractors have been introduced to non-autonomous systems to investigate the dynamics of them, see, e.g. [5, 7]. By definition, pullback attractor is a parameterized family of compact sets which attracts every bounded set of phase space from. It is noticed that in contrast to the existence of uniform attractors, the existence of pullback attractors can be obtained under the weak assumptions on the external forces. Thus, the development of the theory on pullback attractors plays an important role in understanding the dynamics of non-autonomous systems. Recently, authors in [4, 9, 12, 17] develop the theory of pullback attractors in the classical sense and study the pullback D-attractors for non-autonomous systems under the consideration of universes of initial data changing in time. In this case, the pullback attracting property of pullback D-attractors is about the families of sets depending on time which are not necessary to be bounded. The purpose of this paper is to investigate the existence of pullback D- attractors for system (1.1)-(1.3). For the evolution equations in unbounded domains, since the Sobolev embedding is not compact, there are some difficulties in obtaining the compact attracting sets for the semigroup (autonomous case) or process (non-autonomous case) corresponding to them. One way to overcome such difficulties is to investigate the problem in weighted spaces, see, e.g. [2, 8]. By using the method of tail estimates of the solutions, authors in [15] obtained that the semigroup associated with autonomous system (1.1)- (1.3) (f and g are independent of t) is asymptotically compact. Motivated by the energy equation method, see, e.g. [3,13,16,19 21], we show that the process associated with (1.1)-(1.3) is pullback D 0 -asymptotically compact, and obtain the existence of pullback D-attractor.

4 517 Li For convenience, hereafter, let H s (R n ) be the standard Sobolev spaces. Denote by H = L 2 (R n ) with the norm and the scalar product,, respectively. For any 1 < p, let p be the norm of L p (R n ). Let X be a Banach space with distance d(, ), C be generic positive constant which may vary from line to line or even in the same line. This paper is organized as follows: in the next section, we give some definitions and recall some results which will be used in the following sections; in Section 3, we prove the existence of pullback D-attractor for the process associated with (1.1)-(1.3) in H H. 2. Preliminaries We first recall some basic definitions and abstract results on the existence of pullback D-attractors. Let U be a process acting in a Banach space X, i.e., a family {U(t, ) : < t < } of mappings U(t, ) : X X satisfying (1) U(, ) = Id (identity), (2) U(t, ) = U(t, r)u(r, ) for all r t. Let D be a nonempty class of parameterized sets D = {D(t) : t R} P(X), where P(X) denotes the family of all nonempty subsets of X. Definition 2.1. It is said that D 0 D is pullback D-absorbing for the process U(t, ) on X if for any t R and any D D, there exists a 0 (t, D) t such that U(t, )D() D 0 (t), for all 0 (t, D). Definition 2.2. A family A D = {A D (t) : t R} P(X) is said to be a pullback D-attractor for the process U(t, ) on X if (1) A D (t) is compact for every t R, (2) A D is pullback D-attracting, i.e., lim dist(u(t, )D(), A D(t)) = 0, for all D D, and all t R, (3) A D is invariant, i.e., U(t, )A D () = A D (t) for < t < +, where dist(a, B) is the Hausdorff semi-distance between A and B, defined by dist(a, B) = sup inf d(x, y), for A, B X. x A y B In order to illuminate the invariance of pullback D-attractor, the following property is needed. Definition 2.3. A process U(t, ) on X is said to be closed if for any t, and any sequence {x n } X with x n x X and U(t, )x n y, then U(t, )x = y. It is clear that if the process is closed, then it is norm-to-weak continuous, and if it is continuous or weak continuous, then it is norm-to-weak continuous.

5 Pullback D-attractors 518 Definition 2.4. It is said that a process U(t, ) on X is pullback D 0 -asymptotically compact if for any t R, any sequences { n } (, t] and {x n } X satisfying n and x n D 0 ( n ) for all n, the sequence {U(t, n )x n } is relatively compact in X. The family D is said to be inclusion-closed if D D, and D = {D (t) : t R} P(X) with D (t) D(t) for all t, then D D. Recall the results in [9]. Theorem 2.5. Consider a closed process U(t, ), t, t, R, on X, a universe D in P(X) which is inclusion-closed, and a family D 0 = {D 0 (t) : t R} D which is closed and pullback D-absorbing for U(t, ), and assume also that U(t, ) is pullback D 0 -asymptotically compact. Then there exists a minimal and unique pullback D-attractor A D = {A D (t) : t R} defined by A D (t) = s t s X U(t, )D 0 (), t R. 3. Pullback D attractor of system (1.1)-(1.3) We notice that if (u, υ) is a solution of (1.1)-(1.2) with the data (α, β, f, g), then (u, υ) is a solution of (1.1)-(1.2) with the data ( α, β, f, g). Since αβ > 0, we assume without loss of generality in this paper that α and β are both positive. By the standard Fatou-Galerkin methods (see, e.g. [1,10,14,18]), we can get the well-posedness of (1.1)-(1.3). Theorem 3.1. Assume that h satisfies (1.4)-(1.5) and f, g L loc (R, L 2 (R n )). Then for any R, any initial data (u, υ ) H H and any T >, there exists a unique solution (u, υ) = (u(t;, u, υ ), υ(t;, u, υ )) for problem (1.1)-(1.3) satisfying (u, υ) C([, T ]; H H), u L 2 (, T ; H 1 (R n )), and the mapping (u, υ ) (u(t), υ(t)) is continuous in H H. By Theorem3.1, we can define a continuous process {U(t, )} on H H by U(t, )(u, υ ) = (u(t), υ(t)) := (u(t;, u, υ ), υ(t;, u, υ )) for all t. Let D be the class of all families {D(t) : t R} of nonempty subsets of H H such that (3.1) where and lim t eσt [D(t)] + = 0, 0 < σ < min{λ C 0, δ, 2λ r + 1 } [D(t)] + = sup{ u 2 + v 2 : (u, v) D(t)}.

6 519 Li For the external terms, we suppose that (3.2) e σs f(s) 2 ds < +, e σs g(s) 2 ds < +, t R. Lemma 3.2. Assume that (1.4)-(1.5) hold and f, g L loc (R, H) satisfy (3.2). Then there exists a pullback D-absorbing set in H H. Proof. Taking the inner product of (1.1) with βu in H, we have (3.3) β d 2 dt u 2 +βν u 2 +βλ u 2 +β h(u)u + βα uυ = β f(x, t)u. R n R n R n Similarly, taking the inner product of (1.2) with αυ in H, we have α d (3.4) 2 dt υ 2 +αδ υ 2 βα uυ = α g(x, t)υ. R n R n Summing up (3.3) and (3.4), we have 1 d 2 dt (β u 2 +α υ 2 ) + βν u 2 +βλ u 2 +αδ υ 2 +β h(u)u R n (3.5) = β f(x, t)u + α g(x, t)υ. R n R n Note that the terms on the right-hand side of (3.5) can be estimated by (3.6) β f(x, t)u β f(x, t) u β(λ C 0) u 2 β + R 2 2(λ C n 0 ) f(x, t) 2, and (3.7) α g(x, t)υ α g(x, t) υ 1 R 2 αδ υ 2 + α 2δ g(x, t) 2. n By (3.6)-(3.7) and (1.4), we obtain from (3.5) that (3.8) d dt (β u 2 +α υ 2 ) + 2βν u 2 +β(λ C 0 ) u 2 +αδ υ 2 β f(x, t) 2 + α λ C 0 δ g(x, t) 2, which implies that (3.9) d dt (β u 2 +α υ 2 ) + 2βν u 2 +σ(β u 2 +α υ 2 ) β f(x, t) 2 + α λ C 0 δ g(x, t) 2.

7 Pullback D-attractors 520 Neglecting 2βν u 2 and utilizing Gronwall lemma, we get that (3.10) β u(t) 2 +α υ(t) 2 e σ(t ) (β u() 2 +α υ() 2 ) + + α δ g(x, s) 2 )ds e σ(t ) (β u() 2 +α υ() 2 ) + βe σt λ C 0 + αe σt δ e σs g(x, s) 2 ds. e σ(t s) β ( f(x, s) 2 λ C 0 e σs f(x, s) 2 ds By (3.1) we know that for any (u(), υ()) D(), there exists a 0 (t, D) such that u(t) 2 + υ(t) 2 2e σt β t ( e σs f(x, s) 2 ds + α e σs g(x, s) 2 ds) γ λ C 0 δ (3.11) = R(t) 2, 0 (t, D), where γ = min{α, β}. Note that (3.12) lim t eσt (R(t)) 2 = 0, we get that D 0 = {D 0 (t) : t R}, defined by (3.13) D 0 (t) = {(u(t), υ(t)) H H : u(t) 2 + υ(t) 2 R(t) 2 }, is a pullback D-absorbing set. Lemma 3.3. Let the assumptions of Lemma 3.2 hold. Then for any (u(), υ()) D() and any t R, there exists 1 (t, D) t such that for all 1 (t, D), the following inequality holds (3.14) u(t) 2 Ce σt ( e σs ( f(x, s) 2 + g(x, s) 2 )ds, where the positive constant C is independent of D() and t. Proof. Taking the inner product of (1.1) with u in H, we get that (3.15) 1 d 2 dt u 2 + ν u 2 + λ u 2 = h(u) u + α υ u f(t) u. R n R n R n We now estimate every term on the right-hand side of (3.15). By (1.4), we have (3.16) h(u) u = h (u) u 2 C 1 u 2. R n R n

8 521 Li Also, we find that (3.17) f(t) u f u 1 R 2 ν u ν f 2, n and (3.18) α It follows from (3.15)-(3.18) that (3.19) Rn υ u α υ u 12 ν u 2 + α2 2ν υ 2. d dt u 2 2C 1 u ν f 2 + α2 ν υ 2. Integrating (3.9) with respect to t and considering (3.10), we get that +1 t u(s) 2 ds + α2 ν +1 C(β u(t) 2 +α υ(t) 2 ) + C t υ(s) 2 ds +1 t ( f(x, s) 2 + g(x, s) 2 )ds +1 C(β u(t) 2 +α υ(t) 2 ) + Ce σt e σs ( f(x, s) 2 + g(x, s) 2 )ds (3.20) +1 Ce σ(t ) (β u() 2 +α υ() 2 ) + Ce σt e σs ( f(x, s) 2 + g(x, s) 2 )ds. By (3.20) and using uniform Gronwall lemma (see, e.g. [18]), we obtain from (3.19) that (3.21) u(t + 1) 2 Ce σ(t ) (β u() 2 +α υ() 2 ) +1 + Ce σt e σs ( f(x, s) 2 + g(x, s) 2 )ds, where the constant C is independent of u(), υ() and t. The estimate (3.21) implies the desired estimate (3.14). In order to get the pullback D 0 -asymptotical compactness for the process, we endow external terms with the additional assumptions: (A) e σs f(x, s) 2 dxds = 0, lim k lim k e σs g(x, s) 2 dxds = 0, t R. Lemma 3.4. Let the assumptions of Lemma 3.2 and (A) hold, and D 0 be defined by (3.13). Then for every ε > 0, any t R, there exists k = k(ε, t) 0

9 Pullback D-attractors 522 such that (3.22) lim sup e 2λ(t s) U(s, )(u, υ ) 2 dxds < ε, k k, (u, υ ) D 0 (). Proof. Let θ be a smooth function satisfying 0 θ(s) 1 for s R +, and θ(s) = 0 for 0 s 1 and θ(s) = 1 for s 2. Then there exists a constant C such that θ (s) C for s R +. Multiplying (1.1) by βθ( x 2 /k 2 )u and integrating in R n, we get that (3.23) 1 2 β d Rn θ( x 2 Rn dt k 2 ) u 2 βν θ( x 2 Rn )u u + βλ θ( x 2 k2 k 2 ) u 2 Rn = β θ( x 2 Rn )h(u)u βα θ( x 2 Rn k2 k 2 )uυ + β θ( x 2 k 2 )fu. Similarly, multiplying (1.2) by αθ( x 2 /k 2 )υ and integrating in R n, we get that (3.24) α d 2 dt Rn θ( x 2 k 2 ) υ 2 + αδ Rn θ( x 2 Rn k 2 ) υ 2 = βα θ( x 2 Rn )uυ +α θ( x 2 k2 k 2 )gυ. Summing up (3.23) and (3.24), we have (3.25) 1 d Rn Rn Rn θ( x 2 2 dt k 2 )(β u 2 + α υ 2 ) + βλ θ( x 2 k 2 ) u 2 + αδ θ( x 2 k 2 ) υ 2 = βν Rn θ( x 2 k 2 )u u β Rn Rn Rn θ( x 2 k )h(u)u + β θ( x 2 2 k )fu + α θ( x 2 2 k )gυ. 2 We now estimate every term on the right-hand side of (3.25). First, (3.26) Rn βν θ( x 2 )u u = βν k2 Rn θ( x 2 2x k 2 ) u 2 βν R k 2 θ ( x 2 k 2 )u u. n

10 523 Li Note that (3.27) 2x βν R k 2 θ ( x 2 k 2 )u u n x C k x 2k k 2 u u C k k x u u 2k C u u k R n C k u u. For the second term on the right-hand side of (3.25), by (1.4), we have (3.28) β Rn θ( x 2 Rn k 2 )h(u)u βc 0 θ( x 2 k 2 ) u 2. For the third term on the right-hand side of (3.25), we have (3.29) Rn β θ( x 2 )fu β k2 β( Similarly, we can obtain (3.30) α β 2(λ C 0 ) From (3.25)-(3.30), we get that θ( x 2 k 2 )fu f 2 ) 1 2 ( R n θ( x 2 k 2 ) u 2 ) 1 2 (0 θ 1) f 2 + β(λ C 0) 2 Rn θ( x 2 k 2 ) u 2. Rn θ( x 2 k 2 )gυ α g 2 + αδ Rn θ( x 2 2δ 2 k 2 ) υ 2. d Rn θ( x 2 Rn dt k 2 )(β u 2 + α υ 2 ) + β(λ C 0 ) θ( x 2 Rn k 2 ) u 2 + αδ θ( x 2 k 2 ) υ 2 (3.31) C u u + β f 2 + α g 2. k λ C 0 δ

11 Pullback D-attractors 524 Since 0 < σ < min{λ C 0, δ}, we can find ε 0 > 0 such that 0 < σ + ε 0 < min{λ C 0, δ}. It follows from (3.31) that d Rn dr (e(σ+ε 0)r θ( x 2 k 2 )(β u 2 + α υ 2 )) = (σ + ε 0 )e (σ+ε 0)r Rn θ( x 2 k 2 )(β u 2 + α υ 2 ) + e (σ+ε 0)r d dr (3.32) C k e(σ+ε 0)r u u + βe(σ+ε 0)r λ C 0 f 2 + αe(σ+ε 0)r δ Integrating (3.32) on [, s] with s t, we get that (3.33) e (σ+ε0)s θ( x 2 R k 2 )(β u(s) 2 + α υ(s) 2 ) n Rn e (σ+ε 0) θ( x 2 k 2 )(β u 2 + α υ 2 ) + C k + β s e (σ+ε 0)r f 2 dr + α λ C 0 δ Multiplying (3.33) with e (2λ σ ε0)s 2λt, we have e 2λ(t s) θ( x 2 R k 2 )(β u(s) 2 + α υ(s) 2 ) n e 2λt e (2λ σ ε0)s e (σ+ε0) s Rn θ( x 2 k 2 )(β u 2 + α υ 2 ) s e (σ+ε 0)r g 2. e (σ+ε0)r u u dr g 2 dr. θ( x 2 R k 2 )(β u 2 + α υ 2 ) + Ce 2λt e (2λ σ ε0)s k n e (σ+ε 0)r f 2 dr e (σ+ε 0)r u u dr + βe 2λt e (2λ σ ε 0)s + αe 2λt e (2λ σ ε 0)s δ e (σ+ε 0)r λ C 0 g 2 dr. Integrating the above inequality with respect to s over [, t], we obtain (3.34) e 2λ(t s) R n θ( x 2 k 2 )(β u(s) 2 + α υ(s) 2 )ds e (σ+ε 0)(t ) (β u 2 +α υ 2 ) + Ce (σ+ε0)t 2λ σ ε 0 (2λ σ ε 0 )k βe (σ+ε 0)t t + e (σ+ε 0)s f(s) 2 ds (2λ σ ε 0 )(λ C 0 ) + αe (σ+ε 0)t (2λ σ ε 0 )δ e (σ+ε 0)s g(s) 2 ds. e (σ+ε 0)s u u ds

12 525 Li Now we bound each term on the right-hand side of (3.34). (u, υ ) D 0 (), by (3.12), for any t R we have First, since (3.35) Note that (3.36) e (σ+ε 0)(t ) 2λ σ ε 0 (β u 2 +α υ 2 ) 0, as. e (σ+ε 0)s u u ds 1 2 By (3.10), we get that (3.37) e (σ+ε 0)s u 2 ds e σs u(s) 2 1 β e σ(s ) (β u 2 +α υ 2 ) + λ C 0 Then, we have (3.38) + αe σs βδ e (σ+ε 0)s u(s) 2 ds s e σν g(ν) 2 dν, 1 β (β u 2 +α υ 2 )e σ e ε0s 1 ds + λ C 0 + α e ε0s ds e σν g(ν) 2 dν βδ eε 0t βε 0 e σ (β u 2 +α υ 2 ) + + αeε 0t e σν g(ν) 2 dν βδε 0 <, t, s [, t]. s e (σ+ε 0)s u 2 ds. e σν f(ν) 2 dν e ε0s ds e σν f(ν) 2 dν e ε 0t t e σν f(ν) 2 dν ε 0(λ C 0) which along with (3.12), for any t R and any ε > 0, there exists k 1 N such that (3.39) Ce (σ+ε0)t lim sup( 2(2λ σ ε 0)k e (σ+ε 0)s u(s) 2 ds) < ε, Reasoning as above, from (3.14), there exists k 2 N such that (3.40) Ce (σ+ε0)t lim sup( 2(2λ σ ε 0 )k k k 1, (u, υ ) D 0(). e (σ+ε 0)s u(s) 2 ds) < ε, k k 2, (u, υ ) D 0 ().

13 Pullback D-attractors 526 For the third term and the last term on the right-hand side of (3.4), it follows from assumption (A) that there exists k 3 N such that (3.41) e σs f(s) 2 ds < ε, e σs g(s) 2 ds < ε, k k 3. Combining (3.34)-(3.41), letting k > max{k 1, k 2, k 3 }, and taking into account that e 2λ(t s) ( u(s) 2 + υ(s) 2 )ds e 2λ(t s) θ( x 2 R n k 2 )( u(s) 2 + υ(s) 2 )ds, x 2k we get the result. As the proof of Lemma 4 in [19], we have the following result. Lemma 3.5. Assume that (1.4)-(1.5) hold and f, g L loc (R, H). If (u n, υ n ) (u, υ ) weakly in H H, then there exist subsequences {u nj } of {u n } and {υ nj } of {υ n } such that U(t, )(u nj, υ nj ) U(t, )(u, υ ) weakly in H H, for all t. And for all T, U(, )(u nj, υ nj ) U(, )(u, υ ) weakly in L 2 (, T ; H H), P 1 U(, )u nj P 1 U(, )u weakly in L 2 (, T ; H 1 (R n )), P1U(, )u nj P1U(, )u weakly in L 2 (, T ; H 1 ), where P 1 is the canonical projection from Hilbert space E F to E. Theorem 3.6. Assume that (1.4)-(1.5) hold and f, g L loc (R, H) satisfy (3.2) with 0 < σ < min{λ C 0, δ, 2λ r+1 } and assumption (A). Then the process U(t, ) corresponding to problem (1.1)-(1.3) possesses a unique pullback D-attractor A D = {A(t) : t R} in H H. Proof. By the proof of Lemma 3.2, we obtain that D 0 = {D 0 (t)} t R defined by (3.13) is a pullback D-absorbing set for the process U(t, ). To prove the result, by Theorem 2.5, we only need to prove that for any t R, any sequences n, and all (u n, υ n ) D 0 ( n ), the sequence {U(t, )(u n, υ n )} is precompact in H H. From (3.13) we know that for any t R, there exists D0 (t) t such that U(t, )D 0 () D 0 (t), D0 (t). Then for every k Z +, there exists D0 (k, t) t k such that (3.42) U(t k, )D 0 () D 0 (t k), D0 (k, t).

14 527 Li Since for any t R, k 0, D 0 (t k) is a bounded subset in H H, by a diagonal procedure, we can select { n, (u n, υ n )} { n, (u n, υ n )}, which n is a decreasing sequence, such that for every k 0, there exists a sequence ω k = (u k, υ k ) H H such that (3.43) U(t k, n )(u n, υ n ) ω k weakly in H H. Thus, it follows from Lemma 3.5 that ω 0 = (u 0, υ 0 ) = lim (H H) n ω U(t, n )(u n, υ n ) = lim (H H) n ω U(t, t k)u(t k, n )(u n, υ n ) = U(t, t k) lim (H H) n ω U(t k, n )(u n, υ n ) (3.44) = U(t, t k)ω k, for all k 0, where lim (H H)ω implies that denotes the weak limit in H H. The equality (3.44) also (β 1 2 P1U(t, n )(u n, υ n ), α 1 2 P2U(t, n )(u n, υ n )) (β 1 2 u0, α 1 2 υ0) weakly in H H, where P i (i = 1, 2) is the canonical projection. By the lower semi-continuity of the norm, we get that (β 1 2 u0, α 1 2 υ0 ) H H (3.45) lim inf (β 1 2 P1 U(t, n )(u n n, υ n ), α 1 2 P2 U(t, n )(u n, υ n )) H H. If we can also prove (3.46) lim n sup (β 1 2 P1 U(t, n )(u n, υ n ), α 1 2 P2 U(t, n )(u n, υ n )) H H (β 1 2 u0, α 1 2 υ0 ) H H, then we can get that (β 1 2 P1 U(t, n )(u n, υ n ), α 1 2 P2 U(t, n )(u n, υ n )) (β 1 2 u0, α 1 2 υ0 ) strongly in H H, which implies that and the result will be proved. U(t, n )(u n, υ n ) (u 0, υ 0 ) strongly in H H,

15 Pullback D-attractors 528 By (3.5), for all t and all (u, υ ) H H, we have (3.47) β P 1 U(t, )(u, υ ) 2 +α P 2 U(t, )(u, υ ) 2 = e 2λ(t ) (β u 2 +α υ 2 ) + 2α(λ δ) 2βν +2α e 2λ(t s) P 1 U(s, )(u, υ ) 2 ds 2β P 1U(s, )(u, υ ) ds + 2β e 2λ(t s) g, P 2 U(s, )(u, υ ) ds. Then for all k 0, n t k, we obtain (3.48) e 2λ(t s) P 2 U(s, )(u, υ ) 2 ds β P 1 U(t, n )(u n, υ n ) 2 +α P 2 U(t, n )(u n, υ n ) 2 e 2λ(t s) h(p 1 U(s, )(u, υ )), e 2λ(t s) f, P 1U(s, )(u, υ ) ds = β P 1 U(t, t k)u(, n )(u n, υ n ) 2 +α P 2 U(t, t k)u(t k, n )(u n, υ n ) 2 = e 2λk (β P 1 U(t k, n )(u n, υ n ) 2 +α P 2 U(t k, n )(u n, υ n ) ) 2 + 2α(λ δ) 2βν 2β + 2β + 2α e 2λ(t s) P 2 U(s, t k)u(t k, n )(u n, υ n ) 2 ds e 2λ(t s) P 1U(s, t k)u(t k, n )(u n, υ n ) 2 ds e 2λ(t s) h(p 1 U(s, t k)u(t k, n )(u n, υ n )), P 1 U(s, t k)u(t k, n )(u n, υ n ) ds e 2λ(t s) f, P 1 U(s, t k)u(t k, n )(u n, υ n ) ds e 2λ(t s) g, P 2 U(s, t k)u(t k, )(u, υ ) ds = I 1 + I 2 + I 3 + I 4 + I 5 + I 6. We now estimate I 1 I 6 one by one. First, from (3.42), we have (3.49) Then we have lim sup I1 n U(t k, n )(u n, υ n ) D 0 (t k) for any n D0(k,t), k 0. = lim n (e 2λk (β P 1U(t k, n )(u n, υ n ) 2 + α P 2U(t k, n )(u n, υ n ) 2 )) (3.50) e 2λk δ 1 (R(t k)) 2, k 0,

16 529 Li where δ 1 = max{α, β}. For I 2, since U(t k, n )(u n, υ n ) ω k weakly in H H, and it follows, from Lemma 3.5, that P 2 U(s, )U(t k, n )(u n, υ n ) P 2 U(s, t k)ω k weakly in L 2 (t k, t; H). Then we can deduce that e 2λ(t s) P 2 U(s, t k)ω k 2 ds lim inf n Thus, by the fact that λ < δ we have (3.51) e 2λ(t s) P 2 U(s, t k)u(t k, n )(u n, υ n ) 2 ds. lim sup I n 2 2α(λ δ) e 2λ(t s) P 2 U(s, t k)ω k 2 ds. Similarly, reasoning as above, we have P 1 U(s, )U(, n )(u n, υ n ) P 1 U(s, t k)ω k weakly in L 2 (t k, t; H). Then, Thus, (3.52) e 2λ(t s) P 1 U(s, t k)ω k 2 ds lim inf n lim sup I n 3 2βν e 2λ(t s) P 1 U(s, t k)u(t k, n )(u n, υ n ) 2 ds. e 2λ(t s) P 1 U(s, t k)ω k 2 ds. Now, we estimate I 4 by decomposing R n into a bounded domain and its complement to overcome the lack of compactness of Sobolev imbeddings. Note that 2β e 2λ(t s) h(p 1 U(s, t k)u(t k, n )(u n, υ n )), P 1 U(s, t k) U(t k, n )(u n, υ n ) ds = 2β e 2λ(t s) h(p 1 U(s, t k)u(t k, n )(u n, υ n ))P 1 U(s, t k) x m U(t k, n )(u n, υ n )dxds 2β e 2λ(t s) h(p 1U(s, t k)u(t k, n )(u n, υ n ))P 1U(s, t k) x m U(t k, n )(u n, υ n )dxds. By Lemma 3.4, for any ε 0, there exists t, m N and ñ N such that e 2λ(t s) P 1 U(s, n )(u n, υ n ) 2 dxds < ε, n x m

17 Pullback D-attractors 530 for n min{, t k}, m m, n > ñ and (u 0nj, υ 0nj ) D 0 ( n ). Without loss of generality, we can choose ñ large enough such that n min{, t k} for all n ñ. Combining (1.4) and (1.5), we have h(u) C( u + u r+1 ), and then we can deduce that for n ñ, m m, and (u n, υ n ) D 0 ( n ), there holds e 2λ(t s) C n x m h(p 1U(s, t k)u(t k, n )(u n, υ n ))P 1U(s, t k) U(t k, n )(u n, υ n )ds e 2λ(t s) P 1U(s, n )(u n, υ n ) 2 dxds + C x m εc + C( x m e 2λ(t s) P 1 U(s, t k)u(t k, n )(u n, υ n ) r+1 P 1 U(s, n )(u n, υ n ) dxds ( εc + εc( εc + εc( n e 2λ(t s) e 2λ(t s) e 2λ(t s) x m x m x m P 1U(s, t k)u(t k, n )(u n, υ n ) 2(r+1) dxds) 1 2 P 1U(s, n )(u n, υ n ) 2 dxds) 1 2 P 1 U(s, t k)u(t k, n )(u n, υ n ) 2(r+1) dxds) 1 2 e 2λ(t s) P 1 U(s, t k)u(t k, n )(u n, υ n ) 2(r+1) H 1 ds) 1 2. Let Ω m = {x R n x m}, and note that U(t k, n )(u n, υ n ) ω k weakly in H H. From Lemma 3.5, we get that P 1 U(, t k)u(t k, n )(u n, υ n ) P 1 U(, t k)ω k weakly in L 2 (t k, t; H 1 (Ω m )), P1U(, t k)u(t k, n )u n P1U(, t k)ω k weakly in L 2 (t k, t; H 1 (Ω m )). By the compactness result in [11], we have P 1U(, t k)u(t k, n )(u n, υ n ) P 1U(, t k)ω k strongly in L 2 (t k, t; L 2 (Ω m). Thus, for m m we have (3.53) lim sup I n 4 2β +ε lim n sup( e 2λ(t s) x m h(p 1 U(s, t k)ω k )P 1 U(s, t k)ω k ds e 2λ(t s) P 1 U(s, n )(u n, υ n ) 2(r+1) H ds) ε.

18 531 Li By Lemma 3.5 and (3.43), we have (3.54) and 2β 2β e 2λ(t s) f(s), P 1 U(s, t k)u(t k, n )(u n, υ n ) ds e 2λ(t s) f(s), P 1 U(s, t k)ω k ds, n, (3.55) 2α 2α e 2λ(t s) g(s), P 2U(s, t k)u(t k, n )(u n, υ n ) ds e 2λ(t s) g(s), P 2 U(s, t k)ω k ds, n. Considering (3.50)-(3.55) and letting m, we get from (3.48) that (3.56) lim sup(β P1U(t, n n )(u, υ n n ) 2 +α P 2U(t, n )(u n, υ n ) 2 ) e 2λk δ 1(R(t k)) 2 + 2α(λ δ) 2βν 2β + 2β + 2α + ε lim sup( n e 2λ(t s) P 1U(s, t k)ω k 2 ds e 2λ(t s) P 2U(s, t k)ω k 2 ds e 2λ(t s) R n h(p 1 U(s, t k)ω k )P 1 U(s, t k)ω k ds e 2λ(t s) f(s), P 1U(s, t k)ω k ) ds e 2λ(t s) g(s), P 2 U(s, t k)ω k ) ds Combining (3.44) and (3.47), we can obtain (3.57) e 2λ(t s) P 1 U(s, n )(u n, υ n ) 2(r+1) H 1 ds) ε. (β 1 2 u0, α 1 2 υ0 ) 2 H H= (β 1 2 P1 U(t, t k)ω k, α 1 2 P2 U(t, t k)ω k ) 2 = e 2λk (β u k 2 +α υ k 2 ) + 2α(λ δ) 2βν e 2λ(t s) P 1U(s, t k)ω k 2 ds e 2λ(t s) P 2 U(s, t k)ω k 2 ds

19 2β + 2β + 2α Pullback D-attractors 532 e 2λ(t s) h(p 1U(s, t k)ω k ), P 1U(s, t k)ω k ) ds e 2λ(t s) f(s), P 1 U(s, t k)ω k ) ds e 2λ(t s) g(s), P 2 U(s, t k)ω k ) ds. By (3.56) and (3.57), we can deduce that (3.58) lim n sup(β P1U(t, n )(u n, υ n ) 2 +α P 2U(t, n )(u n, υ n ) 2 ) e 2λk δ 1 (R(t k)) 2 + (β u 0 2 +α υ 0 2 ) e 2λk (β u k 2 +α υ k 2 ) +ε lim sup( n e 2λ(t s) P 1 U(s, n )(u n, υ n ) 2(r+1) H 1 ds) ε. Finally, we only need to prove that for any t R, lim sup( e 2λ(t s) P 1 U(s, n )(u n n, υ n ) 2(r+1) H ds) < +. From (3.21) we get that P 1 U(s, n )(u n, υ n ) 2(r+1) H 1 Ce σ(r+1)s (e σ n (β u n 2 +α υ n 2 )) r+1 + Ce σ(r+1)s ( e σs ( f(s) 2 + g(s) 2 )ds) r+1. Then we can easily obtain that for 0 < σ < min{λ, 2λ r+1 }, n e 2λ(t s) P 1U(s, n )(u n, υ n ) 2(r+1) H 1 C[(e σ n (β u n 2 + α υ n 2 )) r+1 + ( e 2λ(t s) e σ(r+1)s ds Ce σ(r+1)t 2λ σ(r + 1) [(eσ n (β u n 2 +α υ n 2 )) r+1 + ( + g(s) 2 )ds) r+1 ]. ds e σs ( f(s) 2 + g(s) 2 )ds) r+1 ] e σs ( f(s) 2

20 533 Li Since (u n, υ n ) D 0 ( n ), by (3.13), we have (3.59) lim sup( e 2λ(t s) P 1 U(s, n )(u n n, υ n ) 2(r+1) H ds) n σ(r+1)t 2Ce 2 ( 2λ σ(r + 1) <. By (3.58) and (3.59), we can get (3.60) e σs ( f(s) 2 + g(s) 2 )ds) r+1 2 lim sup(β P 1U(t, n )(u n n, υ n ) 2 +α P 2U(t, n )(u n, υ n ) 2 ) e 2λk δ 1(R(t k)) 2 + (β u 0 2 +α υ 0 2 ) e 2λk (β u k 2 +α υ k 2 ) Note that σ(r+1)t + ε 2Ce 2 t ( e σs ( f(s) 2 + g(s) 2 )ds) r ε. 2λ σ(r + 1) e 2λk (R(t k)) 2 = 2e σt e (2λ σ)k β ( γ λ C 0 0, as k. k e σs f(x, s) 2 ds + α δ k e σs g(x, s) 2 ds) Letting ε 0 and k in (3.60) and considering (3.1), we get (3.46). This completes the proof. Remark 3.7. In this paper, we only get the existence of pullback D-attractor for system (1.1)-(1.3). For more information on pullback D-attractor obtained in Theorem 3.6 such as dimension, regularity and inner structure, we leave them for future study. Acknowledgements This work was supported by the NNSF of China Grant References [1] A.V. Babin and M.I. Vishik, Attractors of Evolution Equaitons, North-Holland, Amsterdam, [2] A.V. Babin and M.I. Vishik, Attractors of partial differential evolution equations in an unbounded domain, Proc. Roy. Soc. Edinburgh Sect. A 116 (1990), no. 3-4, [3] J.M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst. 10 (2004), no. 1-2,

21 Pullback D-attractors 534 [4] T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal. 64 (2006), no. 3, [5] D.N. Cheban, P.E. Kloeden and B. Schmalfuss, The relationship between pullback, forward and global attractors of nonautonomous dynamical systems, Nonlinear Dyn. Syst. Theory 2 (2002) [6] V.V. Chepyzhov and M.I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc. Colloq. Publ. Vol. 49, Amer. Math. Soc. Providence, RI, [7] H. Crauel, A. Debussche and F. Flandoli, Random attractor, J. Dynam. Differential Equations 9 (1997), no. 2, [8] M.A. Efendiev and S.V. Zelik, The attractor for a nonlinear reaction-diffusion system in an unbounded domain, Comm. Pure. Appl. Math. 54 (2001), no. 6, [9] J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors in V for nonautonomous 2D-Navier-Stokes equations and their tempered behaviour, J. Differential Equations 252 (2012), no. 8, [10] J.K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, [11] J.L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites Non Linéaires, Dunod Gauthier-Villars, Paris, [12] P. Marín-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems, Nonlinear Anal. 71 (2009), no. 9, [13] I. Moise, R. Rosa and X. Wang, Attractors for noncompact nonautonomous systems via energy equations, Discrete Contin. Dyn. Syst. 10 (2004), no. 1-2, [14] J.C. Robinson, Infinite-dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge Univ. Press, Cambridge, [15] A. Rodriguez-Bernal and B. Wang, Attractors for partly dissipative reaction diffusion systems in R n, J. Math. Anal. Appl. 252 (2000), no. 2, [16] R. Rosa, The global attractor for the 2D Navier-Stokes flow on some unbounded domains, Nonlinear Anal. 32 (1998), no. 1, [17] H. Song, Pullback attractors of non-autonomous reaction-diffusion equations in H 1 0, J. Differential Equations 249 (2010), no. 10, [18] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer- Verlag, 2nd edition, New York, [19] B. Wang, Attractors for reaction-diffusion equations in unbounded domains, Phys. D 128 (1999), no. 1, [20] B. Wang, Pullback attractors for the non-autonomous FitzHugh-Nagumo system on unbounded domains, Nonlinear Anal. 70 (2009), no. 11, [21] Y. Wang, L. Wang and W. Zhao, Pullback attractors for nonautonomous reactiondiffusion equations in unbounded domains, J. Math. Anal. Appl. 336 (2007), no. 1, (Xiaojun Li) School of Science, Hohai University, Nanjing, Jiangsu , China. address: lixjun05@mailsvr.hhu.edu.cn; lixiaoj@yahoo.com