Nonlinear Analysis 74 (2011) 3876–3883

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Nonlinear Analysis

journal homepage: www.elsevier.com/locate/na

Long-time behavior of reaction–diffusion equations with dynamicalboundary condition

Lu Yang a, Meihua Yang b,∗

a School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, PR Chinab School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, PR China

a r t i c l e i n f o

Article history:Received 15 April 2010Accepted 14 February 2011

MSC:37L0535B4035B41

Keywords:Reaction–diffusion equationDynamical boundary conditionAsymptotic regularityAttractors

a b s t r a c t

In this paper, we study the long-time behavior of the reaction–diffusion equation withdynamical boundary condition, where the nonlinear terms f and g satisfy the polynomialgrowth condition of arbitrary order. Some asymptotic regularity of the solution has beenproved. As an application of the asymptotic regularity results, we can not only obtainthe existence of a global attractor A in (H1(Ω) ∩ Lp(Ω)) × Lq(Γ ) immediately, but alsocan show further that A attracts every L2(Ω) × L2(Γ )-bounded subset with (H1(Ω) ∩

Lp+δ(Ω))× Lq+κ (Γ )-norm for any δ, κ ∈ [0,∞).© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

In this paper, we consider the asymptotic behavior of the solution of the following reaction–diffusion equation withdynamical boundary condition:

ut −1u + f (u) = h(x), inΩ,

ut +∂u∂ν

+ g(u) = 0, on Γ ,u(x, 0) = u0(x),

(1.1)

whereΩ is a bounded domain of RN(N ⩾ 3) with a smooth boundary Γ , h(x) ∈ L2(Ω). The functions f and g ∈ C1(R,R),satisfy the following conditions:

C1|s|p − k1 ≤ f (s)s ≤ C2|s|p + k2, p > 2, (1.2)

C3|s|q − k3 ≤ g(s)s ≤ C4|s|q + k4, q > 2, (1.3)

and

f ′(s) ≥ −l, g ′(s) ≥ −m, (1.4)

here l,m > 0, Ci > 0, ki > 0, i = 1, 2, 3, 4.

This work was supported by the NSFC Grant 10901063, the Mathematical Tianyuan Foundation of China Grant 10926089 and the Fund of Physics &Mathematics of Lanzhou University Grant LZULL200903.∗ Corresponding author.

E-mail addresses: yanglu@lzu.edu.cn (L. Yang), yangmeih@gmail.com, yangmeih@mail.hust.edu.cn (M. Yang).

0362-546X/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2011.02.022

L. Yang, M. Yang / Nonlinear Analysis 74 (2011) 3876–3883 3877

The reaction–diffusion equation with dynamical boundary conditions arises in hydrodynamics and the heat transfertheory, and this problem has strong background in mathematical physics. Dynamic boundary conditions are very natural inmanymathematicalmodels, such as heat transfer in a solid in contactwith amoving fluid, thermoelastic distortion, diffusionphenomena, heat transfer in two medium, problems in fluid dynamics, etc. (See [1–8] and the references therein).

In recent years the asymptotic behavior of the reaction–diffusion equation has been studied extensively in manymonographs and lectures (see, e.g., [9–15]). For the general Dirichlet boundary condition, the reaction–diffusion equationhas made fast progress, the existence and uniqueness of solution has been proven in [13,15] by the Faedo–Galerkin method,and the long-time behavior has been studied in terms of the global attractor; see, for instance, [9,12,13,15–19]. In [20,21],for the parabolic equation with nonlinear boundary conditions, the authors investigated some nonlinear balance conditionson internal and boundary nonlinear terms, and then the long-time behavior of solutions has been discussed.

As for our problem (1.1), recently, under assumptions (1.2)–(1.4), Fan and Zhong [22] proved the existence of theglobal attractor in (H1(Ω) ∩ Lp(Ω)) × Lp(Γ ). The key ingredient in their proof is to obtain the asymptotic compactnessabout the nonlinearity via a priori estimate methods (introduced in [19]). In [23], Gal and Warma studied a quasi-linearparabolic equation with nonlinear dynamic boundary conditions, the well-posedness and the asymptotic behavior of thesolutions were dealt with successfully. In [24], in the case of non-autonomous systems (i.e., the external force h(x, t) is timedependent), the existence and structure of a uniform attractor has been considered. Chueshov and Schmalfuss [2] stated theexistence of a random attractor for stochastic parabolic equation with dynamical boundary conditions.

In this paper, inspired by the method (idea) in [18], we are interested in analyzing the dynamical behavior of thereaction–diffusion equation with dynamical boundary condition, where the growth orders of internal and boundarynonlinearities f and g are arbitrary.

The main goal of the present paper is to study the asymptotic regularity of solutions of systems (1.1), which excel theregularity allowed by the corresponding elliptic equation.

When h(x) ∈ L2(Ω), consider the following elliptic equation−1φ + f (φ) = h(x), inΩ,∂φ

∂ν+ g(φ) = 0, on Γ .

(1.5)

Due to assumptions (1.2)–(1.4), from the classical results about elliptic equation, we know (e.g., see [25]) that (1.5) at leasthas one solution φ(x)with

φ(x) ∈ H1(Ω) ∩ Lp(Ω) and γ (φ) ∈ Lq(Ω). (1.6)

Observe that: herewe only said thatφ(x) satisfies (1.6), and so inmost situations, to the best of our knowledge, the globalattractor A of (1.1) were only obtained (considered) in the space (H1(Ω) ∩ Lp(Ω))× Lq(Γ ) (see, e.g., [2,22,24]).

In this paper, we will show that: if we shift the solution u(x, t) of (1.1) by a proper (fixed) point φ(x), then u(x, t)− φ(x)will be uniformly (w.r.t. time t and initial data) bounded in some higher (than (H1(Ω) ∩ Lp(Ω))× Lq(Γ )) regular space ast is sufficiently large; that is, u(x, t)− φ(x) is bounded in (H1(Ω) ∩ Lp+δ(Ω))× Lq+κ(Γ ) for any δ, κ ∈ [0,∞) as t is largeenough. As a direct application of the asymptotic regularity results, we can obtain the existence of a global attractor A in(H1(Ω) ∩ Lp(Ω))× Lq(Γ ) immediately. Furthermore, we can prove that A attracts every L2(Ω)× L2(Γ )-bounded subsetwith (H1(Ω) ∩ Lp+δ(Ω))× Lq+κ(Γ )-norm.

Comparedwith the case of the Dirichlet boundary condition, since here wewill consider the asymptotic behavior both inthe internal and on the boundary, it seems difficult to apply directly the method of [18] to verify the asymptotic regularityin (H1(Ω) ∩ Lp+δ(Ω))× Lq+κ(Γ ) for any δ, κ ∈ [0,∞). So, some new estimates (techniques) seem to be needed.

The main result of this paper is the following:

Theorem 1.1. Let Ω be a bounded domain in RN with smooth boundary Γ , h(x) ∈ L2(Ω), f and g satisfy (1.2)–(1.4), andsuppose that S(t)t≥0 is the semigroup generated by the solutions of Eq. (1.1)with initial data (u0, ψ0) ∈ L2(Ω)× L2(Γ ). Then,for any δ, κ ∈ [0,∞), there exists a bounded subset Bδ,κ satisfying the following properties:

Bδ,κ =(w, γ (w)) ∈ (H1(Ω) ∩ Lp+δ(Ω))× Lq+κ(Γ ) : ‖w‖H1(Ω)∩Lp+δ(Ω) + ‖γ (w)‖Lq+κ (Γ ) ⩽ Λp,q,N,δ,κ < ∞

,

and for any bounded subset B ⊂ L2(Ω)× L2(Γ ), there exists a T = T (‖B‖L2(Ω)×L2(Γ ), δ, κ) such that

S(t)B ⊂ (φ, γ (φ))+ Bδ,κ for all t ⩾ T , (1.7)

where the constant Λp,q,N,δ,κ depends only on p, q,N, δ, κ; φ(x) is a fixed solution of (1.5).

This theorem shows some asymptotic regularity of the solutions of (1.1). Consequently, combining with the (L2(Ω) ×

L2(Γ ), L2(Ω) × L2(Γ ))-asymptotic compactness (obtained in [22]) and interpolation inequality, we know that S(t)t⩾0is asymptotically compact in (H1(Ω) ∩ Lp(Ω)) × Lq(Γ ), and then we can deduce the main result of [22,24] immediately.Moreover, the global attractor A has the decomposition A = φ(x)+ B with B bounded in (H1(Ω) ∩ Lp+δ(Ω))× Lq+κ(Γ )for any δ, κ ∈ [0,∞); see Corollary 3.1.

3878 L. Yang, M. Yang / Nonlinear Analysis 74 (2011) 3876–3883

For convenience, hereafter γ denotes the trace operator u → u|Γ and ‖ · ‖ is the norm in L2(Ω). C, Ci denotes a generalpositive constant, i = 1, . . . ,which will be different in different estimates.

This paper is organized as follows: in Section 2, we give some preparations for our consideration; in Section 3, we proveour main result, Theorem 1.1.

2. Preliminaries

In this section, we give some auxiliary results which will be used later.We start with the following general existence and uniqueness result. The proof is based on the normal Faedo–Galerkin

methods, readers can refer to [23,26] for more details.

Theorem 2.1 ([23]). Let Ω be a bounded domain in RN with smooth boundary Γ , h(x) ∈ L2(Ω), f and g satisfy (1.2)–(1.4).Then for any initial data (u0, ψ0) ∈ L2(Ω)× L2(Γ ) and any T > 0, there exists a unique solution (u, ψ) for Eq. (1.1), in the sensethat

u ∈ C([0, T ]; L2(Ω)) ∩ L2(0, T ;H1(Ω)) ∩ Lp(0, T ; Lp(Ω)),ψ ∈ C([0, T ]; L2(Γ )) ∩ Lq(0, T ; Lq(Γ )),γ (u(t)) = ψ(t), a.e. t ∈ (0, T )

and ∫Ω

u(t)vdx +

∫Γ

ψ(t)γ (v)dx +

∫Ω

∇u(t)∇v(t)dx +

∫Ω

f (u(t))vdx +

∫Γ

g(ψ(t))γ (v)dx

=

∫Ω

u0vdx +

∫Γ

ψ0γ (v)dx +

∫Ω

hvdx,

for all t ⩾ 0 and any v ∈ H1(Ω) ∩ Lp(Ω) such that γ (v) ∈ Lq(Γ ).Furthermore, for any t ⩾ 0 the mapping (u0, ψ0) → (u(t), ψ(t)) is continuous on L2(Ω)× L2(Γ ).

By Theorem 2.1, we can define the operator semigroup S(t)t⩾0 in L2(Ω)× L2(Γ ) as follows:

S(t)(u0, ψ0) : L2(Ω)× L2(Γ )× R+→ L2(Ω)× L2(Γ ), (2.1)

which is continuous in L2(Ω)× L2(Γ ).In order to verify that the semigroup S(t)t⩾0 is asymptotically compact, the authors in [17] have given an equivalent

condition which is called condition C, that is

Definition 2.2 ([17]). A semigroup S(t)t⩾0 is called satisfying condition C if and only if for any bounded set B of X and forany ε > 0, there exist a positive constant tB > 0 and a finite-dimensional subspace X1 of X , such that(i) PS(t)x | x ∈ B, t ⩾ tB is bounded; and(ii) ‖(I − P)S(t)x‖ < ε, for any t ⩾ tB and x ∈ B,where P : X → X1 is the canonical projector.

Theorem 2.3 ([19]). Let X be a Banach space and S(t)t⩾0 be a norm-to-weak continuous semigroup on X. Then S(t)t⩾0 hasa global attractor in X provided that the following conditions hold true:(i) S(t)t⩾0 has a bounded absorbing set in X; and(ii) S(t)t⩾0 satisfies condition C.

Next, exactly as in [24, Theorem 5.3], we have the following dissipative results:

Lemma 2.4 ([22,24]). Under the assumption of Theorem 2.1, the semigroup S(t)t⩾0 has a positively invariant (L2(Ω) ×

L2(Γ ), (H1(Ω) ∩ Lp(Ω)) × Lq(Γ ))-bounded absorbing set, that is, there is a positive constant M, such that for any boundedsubset B ⊂ L2(Ω)× L2(Γ ), there exists a positive constant T which depends only on the L2(Ω)× L2(Γ )-norm of B such that∫

Ω

|∇u(t)|2dx +

∫Ω

|u(t)|pdx +

∫Γ

|ψ(t)|qdx ⩽ M for all t ⩾ T and (u0, ψ0) ∈ B,

where ψ(t) = γ (u(t)), a.e., t ∈ (0, T ).

Now, we will give some a priori estimates about ut .

Lemma 2.5. Under the assumption of Theorem 2.1, for any bounded subset B ⊂ L2(Ω)× L2(Γ ), there exists a positive constantT1 which depends only on the L2(Ω)× L2(Γ )-norm of B such that∫

Ω

|ut(s)|2dx +

∫Γ

|ut(s)|2dx ⩽ ρ for all s ⩾ T1 and (u0, ψ0) ∈ B, (2.2)

where ρ is a positive constant which depends on M.

L. Yang, M. Yang / Nonlinear Analysis 74 (2011) 3876–3883 3879

Proof. The idea comes from [19] and the details are similar to [22].By differentiating (1.1) in time and denoting v = ut , we have

vt −1v + f ′(u)v = 0, x ∈ Ω, (2.3)

vt +∂v

∂ν+ g ′(u)v = 0, x ∈ Γ . (2.4)

Multiplying (2.3) by v, we obtain that

12

ddt

∫Ω

|v|2dx +12

ddt

∫Γ

|v|2dx +

∫Ω

|∇v|2dx +

∫Ω

f ′(u)v2dx +

∫Γ

g ′(u)v2dx = 0. (2.5)

From (1.4), this yields

12

ddt

∫Ω

|v|2dx +12

ddt

∫Γ

|v|2dx +

∫Ω

|∇v|2dx ≤ l∫Ω

|v|2dx + m∫Γ

|v|2dx

≤ C0

∫Ω

|v|2dx +

∫Γ

|v|2dx

. (2.6)

On the other hand, multiplying (1.1) by ut , let F(s) = s0 f (τ )dτ ,G(s) =

s0 g(τ )dτ , we deduce that

∫Ω

|ut |2dx +

∫Γ

|ut |2dx +

12

ddt

∫Ω

|∇u|2dx +ddt

∫Ω

F(u)dx +

∫Γ

G(u)dx

=

∫Ω

h(x)utdx. (2.7)

Integrating the inequality above from t to t + 1, by using the similar arguments as in [19, Theorem 5.9] and noticingLemma 2.4, we have∫ t+1

t

∫Ω

|v|2dx +

∫Γ

|v|2dx

≤ CM , (2.8)

where CM depends on M . Combining (2.6) and (2.8), and using the uniform Gronwall lemma, we can get (2.2)immediately.

Hereafter, from Lemma 2.4, we denote one of the positively invariant absorbing set by B0 with

B0 ⊂ (u, ψ) ∈ (H1(Ω) ∩ Lp(Ω))× Lq(Γ ) : ‖u‖2H1(Ω)

+ ‖u‖pLp(Ω) + ‖ψ‖

qLq(Γ ) ⩽ M,

note that here positively invariant means S(t)B0 ⊂ B0 for any t ⩾ 0.

3. Asymptotic regularity

In this section, we begin to consider the asymptotic regularity for system (1.1).Here φ(x) always denotes a fixed solution of Eq. (1.5). Then, for the solution u(x, t) of (1.1), we decompose u(x, t) as

follows:

u(x, t) = φ(x)+ w(x, t), (3.1)

wherew(x, t) solves the following equation:wt −1w + f (u)− f (φ) = 0 inΩ,

wt +∂w

∂ν+ g(u)− g(φ) = 0, on Γ ,

w(x, 0) = u0(x)− φ(x).

(3.2)

It is easy to see that (3.2) is also globally well-posed. Moreover, thanks to Lemma 2.4, without loss of generality, hereafterwe assume that (u0, ψ0) ∈ B0 and so (w(x, 0), γ (w(x, 0))) ∈ (H1(Ω) ∩ Lp(Ω))× Lq(Γ ).

At the same time, from the positive invariance of B0 and (1.6) we have that

‖w(x, t)‖H1(Ω)∩Lp(Ω) + ‖w(x, t)‖Lq(Γ ) ⩽ M1 for all t ⩾ 0 (3.3)

with some positive constantM1.

3880 L. Yang, M. Yang / Nonlinear Analysis 74 (2011) 3876–3883

3.1. Proof of Theorem 1.1

Proof of Theorem 1.1. We will use the Moser–Alikakos iteration technique to prove the following induction estimates(e.g., see also [13,16,18]) about the solution of (3.2).

For clarity, we separate our proof into two steps. The following estimates will be deduced by a formal argument. This canbe justified by means of the Galerkin approximation procedure which is devised in the proof of [23, Theorem 2.6].

Step 1: We first claim thatFor each k = 0, 1, 2, . . . , there exist two positive constants Tk and Mk, which depend only on k, p, q,N and

‖B0‖(H1(Ω)∩Lp(Ω))×Lq(Γ ), such that for any (u0, ψ0) ∈ B0 and t ⩾ Tk, we have∫Ω

|w(t)|2

N−1N−2

kdx +

∫Γ

|w(t)|2

N−1N−2

kdx ⩽ Mk, (Ak)

and

∫ t+1

t

∫Ω

|w(s)|2

N−1N−2

k+1

dx

N−2N−1

ds +

∫ t+1

t

∫Γ

|w(s)|2

N−1N−2

k+1

dx

N−2N−1

ds ⩽ Mk (Bk)

wherew(t) is the solution of Eq. (3.2).(i) Initialization of the induction (k = 0).

From [22, Corollary 4.1], we can deduce (A0) immediately. To prove (B0), we multiply (3.2) byw and integrating overΩ ,we get that

12

ddt

∫Ω

|w|2dx +

12

ddt

∫Γ

|w|2dx +

∫Ω

|∇w|2dx +

∫Ω

(f (u)− f (φ))wdx +

∫Γ

(g(u)− g(φ))wdx = 0. (3.4)

From (1.4) we have∫Ω

(f (u)− f (φ))wdx ⩾ −l∫Ω

|w|2dx, (3.5)∫

Γ

(g(u)− g(φ))wdx ⩾ −m∫Γ

|w|2dx. (3.6)

Inserting (3.5)–(3.6) into (3.4), we obtain

12

ddt

∫Ω

|w|2dx +

12

ddt

∫Γ

|w|2dx +

∫Ω

|∇w|2dx ⩽ l

∫Ω

|w|2dx + m

∫Γ

|w|2dx

⩽ C0

∫Ω

|w|2dx +

∫Γ

|w|2dx

. (3.7)

Then, for any t ⩾ 0, integrating the above inequality over [t, t + 1] and using (3.3), we deduce that∫ t+1

t

∫Ω

|∇w(x, s)|2dxds ⩽ CM,M1 for all t ⩾ 0. (3.8)

Using the Sobolev embedding (e.g., see [27, Theorem 5.22])

H1(Ω) → L2N−2N−2 (Ω), H1(Ω) → L

2N−2N−2 (Γ ),

from (3.8) we have, for all t ⩾ 0,

∫ t+1

t

∫Ω

|w(x, s)|2N−2N−2 dx

N−2N−1

ds ⩽ C1

∫ t+1

t

∫Ω

|∇w(x, s)|2dxds ⩽ CM,M1,N , (3.9)

∫ t+1

t

∫Γ

|w(x, s)|2N−2N−2 dx

N−2N−1

ds ⩽ C2

∫ t+1

t

∫Ω

|∇w(x, s)|2dxds ⩽ CM,M1,N , (3.10)

where C1, C2 are constants of embedding H1(Ω) → L2N−2N−2 (Ω) and H1(Ω) → L

2N−2N−2 (Γ ), note that here C1, C2 depend only

on N . This implies that (B0) holds.(ii) The induction argument.

L. Yang, M. Yang / Nonlinear Analysis 74 (2011) 3876–3883 3881

We now assume the (Ak) and (Bk) hold for k ⩾ 1, and we need to prove that (Ak+1) and (Bk+1) hold.

Multiplying (3.2) by |w|2N−1N−2

k+1−2w and integrating overΩ , we then obtain that

12

N − 2N − 1

k+1 ddt

∫Ω

|w|2N−1N−2

k+1

dx +12

N − 2N − 1

k+1 ddt

∫Γ

|w|2N−1N−2

k+1

dx

+

2N − 1N − 2

k+1

− 1

N − 2N − 1

2(k+1) ∫Ω

∇w N−1N−2

k+12 dx+

∫Ω

f (u)− f (φ)

|w|

2N−1N−2

k+1−2wdx +

∫Γ

(g(u)− g(φ))|w|2N−1N−2

k+1−2wdx = 0. (3.11)

Similar to (3.5)–(3.6), we have∫Ω

f (u)− f (φ)

|w|

2N−1N−2

k+1−2wdx ⩾ −l

∫Ω

|w|2N−1N−2

k+1

dx (3.12)

and ∫Γ

(g(u)− g(φ))|w|2N−1N−2

k+1−2wdx ⩾ −m

∫Γ

|w|2N−1N−2

k+1

dx, (3.13)

so that

12

N − 2N − 1

k+1 ddt

∫Ω

|w|2N−1N−2

k+1

dx +12

N − 2N − 1

k+1 ddt

∫Γ

|w|2N−1N−2

k+1

dx

+

2N − 1N − 2

k+1

− 1

N − 2N − 1

2(k+1) ∫Ω

∇w N−1N−2

k+12 dx⩽ C0

∫Ω

|w|2N−1N−2

k+1

dx +

∫Γ

|w|2N−1N−2

k+1

dx

. (3.14)

Then, combining with (Bk), applying the uniform Gronwall lemma (e.g., see [15]) to (3.14) we can get (Ak+1) immediately.For (Bk+1), we integrate the above inequality over [t, t + 1] and using (Ak+1), we have∫ t+1

t

∫Ω

∇w N−1N−2

k+12 dxds ⩽ Mk+1 for all t ⩾ 0, (3.15)

where Mk+1 depends on k, p, q,N,M,M1. By the embedding H1(Ω) → L2N−2N−2 (Ω), H1(Ω) → L

2N−2N−2 (Γ ) again, we have∫

Ω

|w|

N−1N−2

k+1·2N−2N−2 dx

N−2N−1

⩽ C1

∫Ω

∇w N−1N−2

k+12 dx, (3.16)

∫Γ

|w|

N−1N−2

k+1·2N−2N−2 dx

N−2N−1

⩽ C2

∫Ω

∇w N−1N−2

k+12 dx, (3.17)

combining (3.15)–(3.17), we deduce (Bk+1) immediately.Step 2: Based on Step 1, since N ⩾ 3, we have N−1

N−2 > 1 and thenN − 1N − 2

k

→ ∞ as k → ∞.

Hence, for any δ ∈ [0,∞), we can take k1 ⩾ log N−1N−2

p+δ2 , such that

p + δ ⩽ 2N − 1N − 2

k1

,

similarly, for any κ ∈ [0,∞), we choose k2 ⩾ log N−1N−2

q+κ2 , such that

q + κ ⩽ 2N − 1N − 2

k2

.

3882 L. Yang, M. Yang / Nonlinear Analysis 74 (2011) 3876–3883

Let k = maxk1, k2, then, combining with (Ak) and (3.3), we can define Bδ,κ as follows:

Bδ,κ =(z, γ (z)) ∈ (H1(Ω) ∩ Lp+δ(Ω))× Lq+κ(Γ ) : ‖z − φ(x)‖H1(Ω)

+ ‖z‖Lp+δ(Ω) + ‖γ (z)‖Lq+κ (Γ ) ⩽ ΛM1,Mk,p,q,N,δ,κ < ∞, (3.18)

where the constantsM1 and Mk come from (3.3) and (Ak), respectively; and recall that φ(x) is a fixed solution of (1.5).(1.7) follows from (1.5) and (Ak) immediately.

Hence, from Theorem 1.1, using interpolation inequality, we can obtain immediately the following results:

Corollary 3.1. Under the assumptions of Theorem 1.1, the semigroup S(t)t⩾0 has a compact global attractor A in (H1(Ω) ∩

Lp(Ω))× Lq(Γ ). Moreover, A attracts every L2(Ω)× L2(Γ )-bounded subset with (H1(Ω)∩ Lp+δ(Ω))× Lq+κ(Γ )-norm for anyδ, κ ∈ [0,∞); and A allows the decomposition A = φ(x) + B with B is bounded in (H1(Ω) ∩ Lp+δ(Ω)) × Lq+κ(Γ ) for anyδ, κ ∈ [0,∞), and φ(x) is a fixed solution of (1.5).

Proof. At first, we verify that S(t)t⩾0 with initial data (u0, ψ0) ∈ L2(Ω) × L2(Γ ) satisfies condition C in H1(Ω). Letω1, ω2, . . . be the orthogonal basis in L2(Ω)× L2(Γ ) obtained in [22,24].

We denote the orthonormal projector PN by

PN : L2(Ω)× L2(Γ ) → spanω1, ω2, . . . , ωN,

u =

∞−i=1

αiωi → uN =

N−i=1

αiωi.

Then we denote u = u1 + u2, where u1 = PNu and u2 = (I − PN)u.Multiplying (1.1) by u2, we can get

12

ddt

∫Ω

|u2|2dx +

12

ddt

∫Γ

|u2|2dx +

∫Ω

|∇u2|2dx +

∫Ω

f (u)u2dx +

∫Γ

g(u)u2dx

=

∫Ω

h(x)u2dx. (3.19)

Noticing (1.2)–(1.3), we deduce

∫Ω

|∇u2|2dx ≤

∫Ω

|ut2|2

12∫

Ω

|u2|2

12

+

∫Γ

|ut2|2

12∫

Γ

|u2|2

12

+ C(1 + ‖u‖p−1Lp(Ω))‖u2‖Lp(Ω) + C(1 + ‖u‖q−1

Lq(Γ ))‖u2‖Lq(Γ ) + ‖u‖L2(Ω)‖h‖L2(Ω), (3.20)

where C depends on ‖u2‖L2(Ω), ‖u2‖L2(Γ ), ‖h‖L2(Ω).Based on Lemmas 2.4 and 2.5, similar completely to that in [22, Theorem 4.6], we know that S(t)t⩾0 with initial

data (u0, ψ0) ∈ L2(Ω) × L2(Γ ) satisfies condition C in H1(Ω) × L2(Γ ), and then S(t)t⩾0 is asymptotically compact inH1(Ω)× L2(Γ ).

Therefore, from Theorem 1.1, combining with the (L2(Ω)× L2(Γ ), L2(Ω)× L2(Γ ))-asymptotic compactness (obtainedin [22]) and interpolation inequality, one obtains immediately the (L2(Ω)×L2(Γ ), Lp(Ω)×Lq(Γ ))-asymptotic compactness.Then, using (3.20), the (L2(Ω) × L2(Γ ), (H1(Ω) ∩ Lp(Ω)) × Lq(Γ ))-asymptotic compactness holds. Hence, according toTheorem 2.3, the existence of a compact global attractor A in (H1(Ω)∩ Lp(Ω))× Lq(Γ ) is verified. Note that although A isonly bounded in (H1(Ω) ∩ Lp(Ω))× Lq(Γ ), the attraction is w.r.t. (H1(Ω) ∩ Lp+δ(Ω))× Lq+κ(Γ )-norm.

Remark 3.2. In this paper, we only said that φ(x) satisfies (1.6), however, under our assumptions (1.2)–(1.4), if the solutionof (1.5) has a better smoothness than (1.6), that is, φ(x) ∈ H2(Ω) ∩ L2p−2(Ω), then, combining with the estimate aboutwt(t) = ut(t) (e.g., given in [22, Lemma 4.5]), we can deduce further that the solution w(t) of (3.2) will be bounded withstronger topology; that is, w(t) is bounded in (W 2,α(Ω) ∩ Lp+δ(Ω)) × Lq+κ(Γ ) for some proper constant α > 2 and anyδ, κ ∈ [0,∞). Combining with Theorem 1.1, these results form a basis for further considering the long-time behavior ofEq. (1.1).

Acknowledgement

The authors would like to express their sincere thanks to the referee for his/her valuable comments and suggestions,especially for Theorem 1.1.

L. Yang, M. Yang / Nonlinear Analysis 74 (2011) 3876–3883 3883

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