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ISRN Applied Mathematics
Volume 2012 (2012), Article ID 956291, 16 pages
http://dx.doi.org/10.5402/2012/956291
Research Article

Global Attractor for Doubly Nonlinear Parabolic Equation

School of Mathematics, Lanzhou City University, Lanzhou 730070, China

Received 26 March 2012; Accepted 10 May 2012

Academic Editors: M. Hermann, G. Kyriacou, and S. Utyuzhnikov

Copyright © 2012 Yongjun Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Our aim in this paper is to study the long-time behavior for a class of doubly nonlinear parabolic equations. First we show that the problem has a unique solution. Then we prove that the semigroup corresponding to the problem is norm-to-weak continuous in 𝐿𝑞 and 𝐻10. Finally we establish the existence of global attractor of the problem in 𝐿𝑞 and 𝐻10.

1. Introduction

We study the long-time behavior (in terms of attractors) of the solution of the following problem: 𝜕𝛼(𝑢)𝜕𝑡Δ𝑢+𝑓(𝑢)=𝑔(𝑥),𝑥Ω,𝑢(𝑥,𝑡)|𝜕Ω𝑢=0,(𝑥,0)=𝑢0(𝑥).(1.1) Here Ω is a bounded smooth domain in 𝑁(𝑁1) and 𝑔(𝑥) is a given function in 𝐿2(Ω).

Such equations appear, for example, in the study of gas filtration (so-called porous medium equation [1]). It has been extensively studied when 𝛼 is linear [15], and the existence of attractors has been proved in [611] (for 𝛼=𝐼).

Our aim in this paper is to extend the result of [14] to the more general equation (1.1). We make the following assumptions: 𝛼(𝑠)𝒞1(),𝛼(0)=0,𝛼(𝑠)𝛼0,𝛼0𝛼>0,𝑠,(1.2)1|𝑠|𝑟+2𝛼3𝛼(𝑠)𝑠𝛼2|𝑠|𝑟+2+𝛼3,𝛼1,𝛼2>0,𝛼3𝛼0,𝑟0,(1.3)(𝑠)𝛼4|𝑠|𝑟+𝛼5,𝛼4>0,𝛼50,𝑠,(1.4)𝑓(𝑠)𝒞(),𝛾1|𝑠|𝑞𝛾3𝑓(𝑠)𝑠𝛾2|𝑠|𝑞+𝛾3,𝑠,𝛾1,𝛾2>0,𝛾30,𝑞𝑟+2.(1.5) There exists a constant 𝐶00, such that 𝐶0𝛼(𝑠)+𝑓(𝑠)isincreasing.(1.6)

By hypotheses (1.2)–(1.5), 𝛼 and 𝑓 are nonlinear functions with polynomial growth of arbitrary order. Here 𝛼 is more general than in [14](where 𝛼 is linear growth), which is an essential difficulty in proving the existence of global attractor. To problem (1.1), the key points are to obtain the continuous and compactness of semigroup. By using Legendre transform and the asymptotic a priori estimate method introduced in [10, 11], we show the existence of global attractor.

This paper is organized as follows. In Section 2, we recall some basic concepts about the global attractor. In Section 3, we show the uniqueness of solution and norm-to-weak continuous semigroup for (1.1). In Section 4, we verify the asymptotic compactness of the semigroup 𝑆(𝑡) in 𝐿𝑞(Ω) and prove the existence of the (𝐿𝑟+2(Ω),𝐿𝑞(Ω))-global attractor under the hypotheses (1.2)–(1.6). Finally, in Section 5, we prove the existence of the (𝐿𝑟+2(Ω),𝐻10(Ω))-global attractor for 𝑁2.

Throughout this paper we use the following notation: 𝐻=𝐿2(Ω), and the norms in 𝐻10(Ω) and 𝐿𝑝(Ω)(1𝑝) are denoted by 𝑢2=Ω|𝑢|2𝑑𝑥 and |𝑢|𝑝𝑝=Ω|𝑢|𝑝𝑑𝑥, respectively; Ω(𝑢𝑀)={𝑥Ω𝑢(𝑥)𝑀} and Ω(𝑢𝑀)={𝑥Ω𝑢(𝑥)𝑀}; 𝑚(Ω) or |Ω| denotes Lebesgue measure of Ω; sometimes for special differentiation, we denote the different positive constants by 𝑐1, 𝑐2,.

2. Preliminaries

In this section, we recall some basic concepts about the global attractors.

Definition 2.1 (see [68]). Let {𝑆(𝑡)}𝑡0 be a semigroup on a Banach space 𝑋. A subset 𝒜𝑋 is called a global attractor for the semigroup if 𝒜 is compact in 𝑋 and enjoys the following properties:(1)𝒜 is an invariant set, that is, 𝑆(𝑡)𝒜=𝒜 for any 𝑡0;(2)𝒜 attracts all bounded sets of 𝑋, that is, for any bounded subset 𝐵 of 𝑋dist(𝑆(𝑡)𝐵,𝒜)0,as𝑡,(2.1)where dist(𝐴,𝐵) is the Hausdorff semidistance of the two sets 𝐴 and 𝐵: dist(𝐴,𝐵)=sup𝑥𝐴inf𝑦𝐵𝑥𝑦𝑋.(2.2)
And a subset 𝐵0 of 𝑋 is called a bounded absorbing set of the semigroup {𝑆(𝑡)}𝑡0 in 𝑋, if for any bounded 𝐵 of 𝑋, there exists some 𝑇𝐵0, such that 𝑆(𝑡)𝐵𝐵0 for any 𝑡𝑡0.

Definition 2.2 (see [11]). Let 𝑋 be a Banach space and let {𝑆(𝑡)}𝑡0 be a family of operator in 𝑋. We say that {𝑆(𝑡)}𝑡0 is norm-to-weak continuous semigroup in 𝑋, if {𝑆(𝑡)}𝑡0 satisfies(1)𝑆(0)=𝐼 (the identity);(2)𝑆(𝑡)𝑆(𝑠)=𝑆(𝑡+𝑠),𝑡,𝑠0;(3)𝑆(𝑡𝑛)𝑥𝑛𝑆(𝑥)𝑥 if 𝑡𝑛𝑡 and 𝑥𝑛𝑥 in 𝑋.

Definition 2.3 (see [11]). A set 𝒜𝑋, which is invariant, closed in 𝑋, compact in 𝑍 and attracts the bounded subsets of 𝑋 in the topology of 𝑍, is called an (𝑋,𝑍)-global attractor.

Definition 2.4 (see [11]). Let {𝑆(𝑡)}𝑡0 be a semigroup on Banach space 𝑋. {𝑆(𝑡)}𝑡0 is called (𝑋,𝑍)-asymptotically compact, if for any bounded (in𝑋) sequence{𝑥𝑛}𝑛=1𝑋 and 𝑡𝑛0, 𝑡𝑛 as 𝑛, {𝑆(𝑡𝑛)𝑥𝑛}𝑛=1 has a convergence subsequence with respect to the topology of 𝑍.

Definition 2.5 (see [11]). Let {𝑆(𝑡)}𝑡0 be a semigroup on Banach space 𝑋. A bounded subset 𝐵0 of 𝑍 is called an (𝑋,𝑍)-bounded absorbing set, if for any bounded set 𝐵 of 𝑋, there exists some 𝑡𝐵0, such that 𝑆(𝑡)𝐵𝐵0 for any 𝑡𝑡𝐵.

Theorem 2.6 (see [11]). Let 𝑋 be a Banach space and let {𝑆(𝑡)}𝑡0 be a norm-to-weak continuous semigroup on 𝑋. Then {𝑆(𝑡)}𝑡0 has a global attractor in 𝑋 provided that the following conditions hold:(1){𝑆(𝑡)}𝑡0 has a bounded absorbing set 𝐵0 in 𝑋;(2){𝑆(𝑡)}𝑡0 is asymptotically compact in 𝑋.

Theorem 2.7 (see [11]). Let {𝑆(𝑡)}𝑡0 be a norm-to-weak continuous semigroup on 𝑋. One assumes that 𝑆(𝑡)𝑋𝐿𝑝(Ω). Then {𝑆(𝑡)}𝑡0 has an (𝑋,𝐿𝑝(Ω))-global attractor provided that the following conditions hold:(1){𝑆(𝑡)}𝑡0 has an (𝑋,𝐿𝑝(Ω))-bounded absorbing set 𝐵0𝐿𝑝(Ω);(2)there is a 𝑞(1𝑞𝑝) such that {𝑆(𝑡)}𝑡0 is (𝑋,𝐿𝑞(Ω))-asymptotically compact;(3)for any 𝜀>0 and for any bounded 𝐵𝑋, there exist constants 𝑀=𝑀(𝜀,𝐵) and 𝑇=𝑇(𝜀,𝐵), such that Ω(|𝑆(𝑡)𝑢0|𝑀)||𝑆(𝑡)𝑢0||𝑝𝑑𝑥<𝜀forany𝑢0𝐵and𝑡𝑇.(2.3)

Lemma 2.8 (see [11]). Let {𝑆(𝑡)}𝑡0 be a semigroup on 𝐿𝑝(Ω)(𝑝1) and suppose that {𝑆(𝑡)}𝑡0 has a bounded absorbing set in 𝐿𝑝(Ω). Then for any 𝜀>0 and any bounded subset 𝐵𝐿𝑝(Ω), there exist positive constants 𝑇=𝑇(𝐵) and 𝑀=𝑀(𝜀) such that 𝑚Ω||𝑆(𝑡)𝑢0||𝑀𝐶𝜀forany𝑢0𝐵and𝑡𝑇,(2.4) where the positive constant 𝐶 is independent of 𝐵, 𝑇, and 𝜀.

3. Uniqueness of Solution and Norm-to-Weak Continuous Semigroup

The existence of weak solution for (1.1) can be obtained by the standard Faedo-Galerkin approximation method. Here we only state the result.

Lemma 3.1. Assume that 𝑔(𝑥)𝐿2(Ω), 𝛼 and 𝑓 satisfying (1.3)–(1.5), 𝑢0(𝑥)𝐿𝑟+2(Ω). Then for any initial data 𝑢0(𝑥)𝐿𝑟+2(Ω), there exists solution 𝑢(𝑥,𝑡) for (1.1) which satisfies 𝑢(𝑥,𝑡)𝐶0,𝑇;𝐿2(Ω)𝐿20,𝑇;𝐻10(Ω)𝐿𝑞(0,𝑇;𝐿𝑞(Ω)).(3.1)

We now show that the solution is uniqueness and continuous dependent on initial conditions.

Theorem 3.2. Assume that 𝑔(𝑥)𝐿2(Ω),𝑢0(𝑥)𝐿𝑟+2(Ω), 𝛼 and 𝑓 satisfying (1.3)–(1.6). Then there exists a unique solution of (1.1).

Proof. Suppose that 𝑢(𝑡),𝑣(𝑡) be two solutions of (1.1) with initial conditions 𝑢0(𝑥), 𝑣0(𝑥); then 𝜕(𝛼(𝑢)𝛼(𝑣))𝜕𝑡Δ(𝑢𝑣)+𝑓(𝑢)𝑓(𝑣)=0,(3.2) that is 𝜕(𝛼(𝑢)𝛼(𝑣))𝐶𝜕𝑡Δ(𝑢𝑣)+0𝐶𝛼(𝑢)+𝑓(𝑢)0𝛼(𝑣)+𝑓(𝑣)=𝐶0(𝛼(𝑢)𝛼(𝑣)).(3.3) We define the sign function by sign(𝜏)=1if𝜏>0,0if𝜏=0,1if𝜏<0.(3.4) Multiplying (3.3) by sign(𝑢𝑣) and integrating in Ω, we obtain 𝑑𝑑𝑡Ω||||𝛼(𝑢)𝛼(𝑣)𝑑𝑥Ω+Δ(𝑢𝑣)sign(𝑢𝑣)Ω𝐶0𝐶𝛼(𝑢)+𝑓(𝑢)0𝛼(𝑣)+𝑓(𝑣)sign(𝛼(𝑢)𝛼(𝑣))𝑑𝑥=𝐶0Ω||||𝛼(𝑢)𝛼(𝑣)𝑑𝑥.(3.5) Using  (1.6), we get Ω𝐶0𝐶𝛼(𝑢)+𝑓(𝑢)0𝛼(𝑣)+𝑓(𝑣)sign(𝛼(𝑢)𝛼(𝑣))𝑑𝑥0.(3.6) Since sign(𝑢𝑣)=lim𝜀0+(𝑢𝑣)/(𝜀+|𝑢𝑣|), by dominated convergence theorem, we have ΩΔ(𝑢𝑣)sign(𝑢𝑣)𝑑𝑥=lim𝜀0+ΩΔ(𝑢𝑣)𝑢𝑣𝜀+|𝑢𝑣|𝑑𝑥=lim𝜀0+Ω(𝑢𝑣)𝑢𝑣𝜀+|𝑢𝑣|𝑑𝑥0.(3.7) So 𝑑𝑑𝑡Ω||||𝛼(𝑢)𝛼(𝑣)𝑑𝑥𝐶0Ω||||𝛼(𝑢)𝛼(𝑣)𝑑𝑥.(3.8) By Gronwall inequality, we get Ω||||𝛼(𝑢(𝑡))𝛼(𝑣(𝑡))𝑑𝑥Ω||𝛼𝑢0𝑣𝛼0||𝑑𝑥.(3.9) From (1.2), we have Ω||||1𝑢(𝑡)𝑣(𝑡)𝑑𝑥𝛼0Ω||𝛼𝑢0𝑣𝛼0||𝑑𝑥,(3.10) which gives continuous dependence on initial conditions and uniqueness of solution in 𝐿1(Ω).
By Theorem 3.2, we can define the operator semigroup {𝑆(𝑡)}𝑡0 in 𝐿1(Ω) as the following: 𝑆(𝑡)𝑢0+×𝐿𝑟+2(Ω)𝐿1(Ω),(3.11) which is continuous in 𝐿1(Ω).
Since 𝛼 is a continuous increasing function with 𝛼(0)=0, we define for 𝑡, 𝜓(𝑡)=𝑡0𝛼(𝜏)𝑑𝜏.(3.12) Then the Legendre transform 𝜓 is defined by 𝜓(𝜏)=sup𝑠{𝜏𝑠𝜓(𝑠)}.(3.13) Note that 𝜓(𝜏)0,𝜓(𝛼(𝜏))+𝜓(𝜏)=𝜏𝛼(𝜏),𝜓(𝛼(𝜏))𝜏𝛼(𝜏).(3.14)

Theorem 3.3. Assume that the conditions (1.2)–(1.6) are satisfied, 𝑔(𝑥)𝐿2(Ω). Then the semigroup {𝑆(𝑡)}𝑡0 is norm-to-weak continuous in 𝐿𝑞(Ω) and 𝐻10(Ω).

Proof. Let 𝑢𝑚0(𝑥)𝑢0(𝑥) in 𝐿𝑟+2(Ω), 𝑢𝑚(𝑡),𝑢(𝑡) be the solutions of (1.1) corresponding to initial date 𝑢𝑚0(𝑥), 𝑢0(𝑥). In (1.1), replace 𝑢(𝑡) by 𝑢𝑚(𝑡). Multiplying (1.1) by 𝑢𝑚(𝑡) and integrating in Ω, we get 𝑑𝑑𝑡Ω𝜓𝛼𝑢𝑚(||𝑡)𝑑𝑥+𝑢𝑚||22+𝑓𝑢𝑚,𝑢𝑚=𝑔,𝑢𝑚.(3.15) Applying Young inequality, we have ||||Ω𝑔(𝑥)𝑢𝑚||||𝛾𝑑𝑥12||𝑢𝑚||𝑞𝑞+𝑐Ω||||𝑔(𝑥)𝑞1𝑑𝑥.𝑞+1𝑞.=1(3.16) So 𝑑𝑑𝑡Ω𝜓𝛼𝑢𝑚(||𝑡)𝑑𝑥+𝑢𝑚||22+𝛾12||𝑢𝑚||𝑞𝑞𝛾3||Ω||+𝑐Ω||𝑔||𝑞𝑑𝑥.(3.17) Integrating from 0 to 𝑇, we obtain Ω𝜓𝛼𝑢𝑚(𝑇)𝑑𝑥+𝑇0||𝑢𝑚||22𝛾𝑑𝑡+12𝑇0||𝑢𝑚||𝑞𝑞𝑑𝑡Ω𝜓𝛼𝑢𝑚0𝑑𝑥+𝑐𝑇Ω𝑢𝑚0𝛼𝑢𝑚0𝑑𝑥+𝛼3||Ω||+𝑐𝑇𝛼2||𝑢𝑚0||𝑟+2𝑟+2+𝑐𝑇+𝛼3||Ω||.(3.18)𝑢𝑚0𝑢0 in 𝐿𝑟+2(Ω), so there existence 𝑀>0, such that |𝑢𝑚0|𝑟+2𝑟+2𝑀. {𝑢𝑚(𝑡)} Is bounded in 𝐿2(0,𝑇;𝐻10(Ω)) and 𝐿𝑞(0,𝑇;𝐿𝑞(Ω)). Therefore, there exists weak convergent subsequence {𝑢𝑚𝑘(𝑡)} of {𝑢𝑚(𝑡)} in 𝐿2(0,𝑇;𝐻10(Ω)) and 𝐿𝑞(0,𝑇;𝐿𝑞(Ω). Let 𝑢𝑚(𝑡)𝑣(𝑡) obviously, 𝑣(𝑡) be a solution of (1.1) satisfying the initial value condition 𝑣(0)=𝑢0(𝑥). By the unique of solution for (1.1), we have 𝑢(𝑡)=𝑣(𝑡), that is, 𝑢𝑚𝑘𝑢(𝑡) in 𝐿2(0,𝑇;𝐻10(Ω)) and 𝐿𝑞(0,𝑇;𝐿𝑞(Ω)). By Definition 2.2, Theorem 3.3 holds.

Remark 3.4. The semigroup {𝑆(𝑡)}𝑡0 is norm-to-weak continuous in 𝐿2(Ω).

4. (𝐿𝑟+2(Ω),𝐿𝑞(Ω))-Global Attractor

By Theorem 3.3, we can define operator semigroup {𝑆(𝑡)}𝑡0 as the following: 𝑆(𝑡)𝑢0+×𝐿𝑟+2(Ω)𝐿𝑞(Ω),𝑆(𝑡)𝑢0+×𝐿𝑟+2(Ω)𝐻10(Ω).(4.1)

Theorem 4.1. Assume that the conditions (1.2)–(1.6) are satisfied, 𝑔(𝑥)𝐿2(Ω). Then the semigroup {𝑆(𝑡)}𝑡0 exists bounded absorbing sets in 𝐿𝑞(Ω) and 𝐻10(Ω); that is, for arbitrary bounded set 𝐵𝐿𝑟+2(Ω), there exist 𝑀>0, 𝑇(𝐵)>0, 𝑡𝑇, 𝑢0(𝑥)𝐵. We have ||||𝑢(𝑡)𝑞𝑞+||||𝑢22𝑀.(4.2)

Proof. Let 𝑢(𝑡) be the solution of (1.1) with initial date 𝑢0(𝑥); taking scalar product with 𝑢(𝑡) in (1.1), we deduce that 𝑑𝑑𝑡Ω𝜓(||||𝛼(𝑢))𝑑𝑥+𝑢22+(𝑓(𝑢),𝑢)=(𝑔,𝑢).(4.3) By (1.5), we obtain 𝑑𝑑𝑡Ω𝜓(||||𝛼(𝑢))𝑑𝑥+𝑢22+𝛾1|𝑢|𝑞𝑞𝛾3||Ω||(𝑔,𝑢).(4.4) Using Young inequality, we get ||(||𝛾𝑔,𝑢)12Ω|𝑢|𝑞𝑑𝑥+𝑐1Ω||𝑔||𝑞𝑑𝑥.(4.5) By (4.4), we have 𝑑𝑑𝑡Ω𝜓(||||𝛼(𝑢))𝑑𝑥+𝑢22+𝛾12|𝑢|𝑞𝑞𝛾3||Ω||+𝑐1Ω||𝑔||𝑞𝑑𝑥.(4.6) Since 𝑞𝑟+2, by (1.3), there exist 𝑐2,𝑐3>0 such that 𝛾14Ω|𝑢|𝑞𝑑𝑥𝑐3Ω𝜓(𝛼(𝑢))𝑑𝑥𝑐2.(4.7) Hence, 𝑑𝑑𝑡Ω𝜓(||||𝛼(𝑢))𝑑𝑥+𝑢22+𝛾14|𝑢|𝑞𝑞+𝑐3Ω𝜓(𝛼(𝑢))𝑑𝑥𝑐4.(4.8) We get 𝑑𝑑𝑡Ω𝜓(𝛼(𝑢))𝑑𝑥+𝑐3Ω𝜓(𝛼(𝑢))𝑑𝑥𝑐4.(4.9) By the Gronwall lemma, we have Ω𝜓(𝛼(𝑢(𝑡)))𝑒𝑐3𝑡Ω𝜓𝛼𝑢0𝑐𝑑𝑥+4𝑐3.(4.10) Therefore, Ω𝜓𝛼𝑢0𝑑𝑥Ω𝑢0𝛼𝑢0𝑑𝑥.(4.11) We obtain Ω𝜓(𝛼(𝑢(𝑡)))𝛼2||𝑢0||𝑟+2𝑟+2𝑒𝑐3𝑡+𝑐4𝑐3+𝛼3||Ω||𝑐5.(4.12) It follows from (4.8) that there exists 𝑇1>0,𝑡𝑇1; we get 𝑡𝑡+1||||𝑢22𝑑𝑠+𝑡𝑡+1|𝑢|𝑞𝑞𝑑𝑠𝑐6.(4.13) Multiplying (1.1) by 𝑢𝑡 and integrating over Ω, we get Ω𝛼(𝑢)𝑢2𝑡𝑑𝑑𝑥+𝑑𝑡Ω||||𝑢2+𝐹(𝑢)𝑑𝑥=𝑔,𝑢𝑡12Ω𝛼(𝑢)𝑢2𝑡1𝑑𝑥+2𝛼0||||𝑔(𝑥)22,(4.14) where 𝐹(𝑢)=𝑢0𝑓(𝑠)𝑑𝑠. Now by (1.2) we get 𝑑𝑑𝑡Ω||||𝑢21+𝐹(𝑢)𝑑𝑥2𝛼0||||𝑔(𝑥)22.(4.15) It follows from (1.5) that there exist 𝛾1,𝛾2>0,𝛾30 such that 𝛾1|𝑠|𝑞𝛾3𝐹(𝑠)𝛾2|𝑠|𝑞+𝛾3.(4.16) Using the uniform Gronwall Lemma that there exists 𝑇2>0,𝑡𝑇2, we have Ω||||𝑢2+𝐹(𝑢)𝑑𝑥𝑐7.(4.17) By (4.16), 𝑡𝑇=max{𝑇1,𝑇2}, we obtain ||||𝑢22+|𝑢|𝑞𝑞𝑐8.(4.18) Therefor, the semigroup {𝑆(𝑡)} exists bounded absorbing set in 𝐻10(Ω) and 𝐿𝑞(Ω); it follows from Theorem 2.6 that we have the following.

Theorem 4.2. Assume that the conditions (1.2)–(1.6) are satisfied, 𝑔(𝑥)𝐿2(Ω). Then the semigroup {𝑆(𝑡)}𝑡0 has a (𝐿𝑟+2(Ω),𝐿2(Ω))-global attractor, which is nonempty, compact, invariant in 𝐿2(Ω) and attracts every bounded subset of 𝐿𝑟+2(Ω) with respect to 𝐿2(Ω) norm.

In the following, we will give the asymptotic a priori estimate of {𝑆(𝑡)}𝑡0 with respect to 𝐿𝑞(Ω)-norm, which plays a crucial role in the proof of the (𝐿𝑟+2(Ω),𝐿𝑞(Ω))-global attractor.

Lemma 4.3. Assume that the conditions (1.2)–(1.6) are satisfied, 𝑔(𝑥)𝐿2(Ω), 𝑞max{2𝑟,𝑟+2}, 𝐵𝐿𝑟+2(Ω). For any 𝜀>0, there exist positive constants 𝑀(=𝑀(𝜀)) and 𝑇(=𝑇(𝜀)) such that Ω(|𝑢(𝑡)|𝑀)||||𝑢(𝑡)𝑞𝑑𝑥<𝑐𝜀,forany𝑡𝑇and𝑢0(𝑥)𝐵.(4.19)

Proof. By (1.3) and (1.5), we find that there exists 𝑀1>0,|𝑢|𝑀1 such that 𝛾𝑓(𝑢)𝑢12|𝑢|𝑞,𝛼12|𝑢|𝑟+1||||𝛼(𝑢)2𝛼2|𝑢|𝑟+1.(4.20)
Letting 𝑀2=max{1,𝛼1/(2)|𝑀1|𝑟+1}, when |𝛼(𝑢)|𝑀2, then |𝑢|𝑀1. Multiplying (1.1) with |(𝛼(𝑢)𝑀2)+|(𝑞/(𝑟+1))2(𝛼(𝑢)𝑀2)+, we get 𝑟+1𝑞𝑑𝑑𝑡Ω||𝛼(𝑢)𝑀2+||𝑞/(𝑟+1)𝑑𝑥+Ω||𝑢𝛼(𝑢)𝑀2+||𝑞/(𝑟+1)2𝛼(𝑢)𝑀2++𝑑𝑥Ω||𝑓(𝑢)𝛼(𝑢)𝑀2+||𝑞/(𝑟+1)2𝛼(𝑢)𝑀2+=𝑑𝑥Ω||𝑔(𝑥)𝛼(𝑢)𝑀2+||𝑞/(𝑟+1)2𝛼(𝑢)𝑀2+𝑑𝑥,(4.21) where (𝛼(𝑢)𝑀2)+ denote the positive part of (𝛼(𝑢)𝑀2), that is: 𝛼(𝑢)𝑀2+=𝛼(𝑢)𝑀2,𝛼(𝑢)𝑀2,0,𝛼(𝑢)<𝑀2.(4.22) Thus we have Ω||𝑢𝛼(𝑢)𝑀2+||𝑞/(𝑟+1)2𝛼(𝑢)𝑀2+=𝑑𝑥Ω(𝛼(𝑢)𝑀)||𝑢𝛼(𝑢)𝑀2||𝑞/(𝑟+1)1=𝑞𝑑𝑥𝑟+11Ω(𝛼(𝑢)𝑀)𝛼||(𝑢)𝛼(𝑢)𝑀2||𝑞/(𝑟+1)2||||𝑢2𝑑𝑥0,Ω||𝑓(𝑢)𝛼(𝑢)𝑀2+||𝑞/(𝑟+1)2𝛼(𝑢)𝑀2+𝛾𝑑𝑥12Ω|𝑢|𝑞1||𝛼(𝑢)𝑀2+||𝑞/(𝑟+1)1𝑑𝑥𝑐Ω||||𝛼(𝑢)(𝑞1)/(𝑟+1)||𝛼(𝑢)𝑀2+||𝑞/(𝑟+1)1𝑑𝑥𝑐Ω||𝛼(𝑢)𝑀2+||(2𝑞𝑟2)/(𝑟+1)||||𝑑𝑥,Ω||𝑔(𝑥)𝛼(𝑢)𝑀2+||𝑞/(𝑟+1)2𝛼(𝑢)𝑀2+||||𝑑𝑥Ω||||||𝑔(𝑥)𝛼(𝑢)𝑀2+||𝑞/(𝑟+1)1𝑐𝑑𝑥2Ω||𝛼(𝑢)𝑀2+||(2𝑞𝑟2)/(𝑟+1)𝑑𝑥+𝑐Ω(𝛼(𝑢)𝑀2)||||𝑔(𝑥)(2𝑞𝑟2)/(𝑞1)𝑑𝑥.(4.23) We obtain 𝑑𝑑𝑡Ω||𝛼(𝑢)𝑀2+||𝑞/(𝑟+1)𝑑𝑥+𝑐Ω||𝛼(𝑢)𝑀2+||(2𝑞𝑟2)/(𝑟+1)𝑑𝑥𝑐Ω(𝛼(𝑢)𝑀2)||||𝑔(𝑥)(2𝑞𝑟2)/(𝑞1)𝑑𝑥.(4.24) Since 𝑞𝑟+2, we have 2𝑞𝑟2𝑞𝑟+1𝑟+1,1<2𝑞𝑟2𝑞12.(4.25) So 𝑑𝑑𝑡Ω||𝛼(𝑢)𝑀2+||𝑞/(𝑟+1)𝑑𝑥+𝑐Ω||𝛼(𝑢)𝑀2+||𝑞/(𝑟+1)𝑑𝑥𝑐Ω(𝛼(𝑢)𝑀2)||||𝑔(𝑥)(2𝑞𝑟2)/(𝑞1)𝑑𝑥.(4.26) By Gronwall inequality, there exists 𝑇(𝐵)>0,𝑡𝑇(𝐵) such that Ω||𝛼(𝑢(𝑡))𝑀2+||𝑞/(𝑟+1)𝑑𝑥𝑒𝑐(𝑡𝑇)Ω||𝛼(𝑢(𝑇))𝑀2+||𝑞/(𝑟+1)𝑑𝑥+𝑐Ω(𝛼(𝑢)𝑀2)||||𝑔(𝑥)(2𝑞𝑟2)/(𝑞1)𝑑𝑥.(4.27)
For any 𝜀>0 there exists 𝛿>0; for any 𝑒Ω, 𝑚(𝑒)<𝛿, we have 𝑒||||𝑔(𝑥)(2𝑞𝑟2)/(𝑞1)𝑑𝑥<𝜀.(4.28)
Leting 𝑇(𝑡)=𝛼(𝑢(𝑡)), obviously {𝑇(𝑡)}𝑡0 is a semigroup in 𝐿1(Ω). By (1.5) and (4.18), {𝑇(𝑡)}𝑡0 has a bounded absorbing set in 𝐿1(Ω). Combining Lemma 2.8, for any 𝜀>0, there exist 𝑀𝑀2,𝑇1𝑇(𝐵), for any 𝑡>𝑇1, we have 𝑚(Ω(𝛼(𝑢)𝑀))<min{𝛿,𝜀}.(4.29) Hence Ω||𝛼(𝑢(𝑡))𝑀2+||𝑞/(𝑟+1)𝑑𝑥<𝑐𝜀,(4.30) that is Ω(𝛼(𝑢)𝑀)||||𝛼(𝑢(𝑡))𝑀𝑞/(𝑟+1)𝑑𝑥<𝑐𝜀.(4.31)
By repeating the same step above and multiplying (1.1) with |(𝛼(𝑢)+𝑀2)|𝑞/(𝑟+1)2(𝛼(𝑢)+𝑀2), we get Ω(𝛼(𝑢)𝑀)||||𝛼(𝑢(𝑡))𝑀𝑞/(𝑟+1)𝑑𝑥<𝑐𝜀,(4.32) where 𝛼(𝑢)+𝑀2=𝛼(𝑢)+𝑀2,𝛼(𝑢)𝑀2,0,𝛼(𝑢)>𝑀2.(4.33)
Combining (4.31) and (4.32), we have Ω(|𝛼(𝑢)|2𝑀)||||𝛼(𝑢(𝑡))𝑞/(𝑟+1)=𝑑𝑥Ω(|𝛼(𝑢)|2𝑀)||||𝛼(𝑢(𝑡))𝑀+𝑀𝑞/(𝑟+1)𝑑𝑥𝑐Ω(|𝛼(𝑢)|2𝑀)||||𝛼(𝑢(𝑡))𝑀𝑞/(𝑟+1)𝑑𝑥+Ω(|𝛼(𝑢)|2𝑀)𝑀𝑞/(𝑟+1)𝑑𝑥𝑐Ω(|𝛼(𝑢)|𝑀)||||𝛼(𝑢(𝑡))𝑀𝑞/(𝑟+1)𝑑𝑥+Ω(|𝛼(𝑢)|𝑀)||||𝛼(𝑢(𝑡))𝑀𝑞/(𝑟+1)𝑑𝑥𝑐𝜀.(4.34)
Thanks to (4.20), we have Ω(|𝑢(𝑡)|𝑀)||||𝑢(𝑡)𝑞𝑑𝑥<𝑐𝜀.(4.35)
From Theorem 4.2, we know that the semigroup {𝑆(𝑡)}𝑡0 is asymptotically compact in 𝐿2(Ω); using Theorem 4.1, Lemma 4.3, and Theorem 2.7, we get the following.

Theorem 4.4. Assume that the conditions (1.2)–(1.6) are satisfied, 𝑔(𝑥)𝐿2(Ω). Then the semigroup {𝑆(𝑡)}𝑡0 has a (𝐿𝑟+2(Ω),𝐿𝑞(Ω))-global attractor, which is nonempty, compact, invariant in 𝐿𝑞(Ω) and attracts every bounded subset of 𝐿𝑟+2(Ω) with respect to 𝐿𝑞(Ω) norm.

5. (𝐿𝑟+2(Ω),𝐻10(Ω))-Global Attractor

In this section, we want to prove the (𝐿𝑟+2(Ω),𝐻10(Ω))-global attractor. However, for general N-dimension space, our methods can not work, so we make the following assumption: 𝑁2,𝛼(𝑠)𝐶2(),𝑓(𝑠)𝐶1(Ω),𝑠.(5.1)

Lemma 5.1. Assume that the conditions (1.2), (1.3), (1.5), (1.6), and (5.1) are satisfied, 𝑔(𝑥)𝐿2(Ω). For any bounded set 𝐵𝐿𝑟+2(Ω), there exists 𝑡(𝐵)>0; for any 𝑡𝑡, we have Ω𝛼(||𝑢𝑢)𝑡||2𝑑𝑥𝑐.(5.2)

Proof. Multiplying (1.1) by 𝑢𝑡 and integrating over Ω, we have Ω||𝑢𝛼(𝑢)𝑡||2𝑑𝑑𝑥+𝑑𝑡Ω||||𝑢2+𝐹(𝑢)𝑑𝑥=𝑔(𝑥),𝑢𝑡Ω𝛼(𝑢)2||𝑢𝑡||2||||𝑑𝑥+𝑔(𝑥)222𝛼0,(5.3) and then Ω||𝑢𝛼(𝑢)𝑡||2𝑑+𝑐𝑑𝑡Ω||||𝑢2+𝐹(𝑢)𝑑𝑥𝑐.(5.4) By (4.16) and (4.18), there exists 𝑇(𝐵)>0 for any 𝑡𝑇(𝐵); we get Ω||||𝑢2+𝐹(𝑢)𝑑𝑥𝑐.(5.5) Integrating (5.4) from 𝑡 to 𝑡+1, we obtain 𝑡𝑡+1Ω𝛼||𝑢(𝑢)𝑡||2𝑑𝑥𝑑𝑠𝑐,𝑡𝑇(𝐵).(5.6) By differentiating (1.1) in time 𝑡, we have 𝛼(𝑢)𝑢2𝑡+𝛼(𝑢)𝑢𝑡𝑡Δ𝑢𝑡+𝑓(𝑢)𝑢𝑡=0.(5.7) Letting 𝑟(𝑡)=𝛼(𝑢)𝑢2𝑡, we get 𝑑𝑑𝑡Ω𝑟(𝑡)𝑑𝑥=Ω𝛼(𝑢)𝑢3𝑡𝑑𝑥+2Ω𝛼(𝑢)𝑢𝑡𝑢𝑡𝑡𝑑𝑥,(5.8) that is 12𝑑𝑑𝑡Ω1𝑟(𝑡)𝑑𝑥=2Ω𝛼(𝑢)𝑢3𝑡𝑑𝑥+Ω𝛼(𝑢)𝑢𝑡𝑢𝑡𝑡𝑑𝑥.(5.9) Multiplying (5.7) by 𝑢𝑡 and integrating over Ω, we get Ω𝛼(𝑢)𝑢3𝑡𝑑𝑥+Ω𝛼(𝑢)𝑢𝑡𝑡𝑢𝑡𝑑𝑥=Ω||𝑢𝑡||2𝑑𝑥Ω𝑓(𝑢)𝑢2𝑡𝑑𝑥.(5.10) Hence from (5.9), we obtain 12𝑑𝑑𝑡Ω1𝑟(𝑡)𝑑𝑥=2Ω𝛼(𝑢)𝑢3𝑡𝑑𝑥Ω||𝑢𝑡||2𝑑𝑥Ω𝑓(𝑢)𝑢2𝑡𝑑𝑥.(5.11)
Since 𝑁2, from Theorem 4.1 the semigroup {𝑆(𝑡)}𝑡0 has a bounded absorbing set in 𝐻10(Ω). Applying Sobolev embedding theorem, the semigroup {𝑆(𝑡)}𝑡0 is bounded in 𝐿(Ω); hence, 12𝑑𝑑𝑡Ω𝑟(𝑡)𝑑𝑥+Ω||𝑢𝑡||2𝑑𝑥𝑀Ω||𝑢𝑡||3𝑑𝑥+cΩ||𝑢𝑡||2𝑑𝑥.(5.12)
Since 𝑁2, the imbedding 𝐻1/3(Ω)𝐿3(Ω) is compact; hence ||𝑢𝑡||33||𝑢𝑐𝑡||3𝐻1/3(Ω).(5.13) Using interpolation inequality for Sobolev space, we get ||𝑢𝑡||3𝐻1/3(Ω)||𝑐𝑢𝑡||2||𝑢𝑡||2212||𝑢𝑡||22||𝑢+𝑐𝑡||42.(5.14) Therefore, from (5.12), we have 12𝑑𝑑𝑡Ω𝑟(𝑡)𝑑𝑥+Ω||𝑢𝑡||2||𝑢𝑑𝑥𝑐𝑡||22||𝑢+𝑐𝑡||42||𝑢𝑐𝑡||42+𝑐=𝑐Ω||𝑢𝑡||2𝑑𝑥2+𝑐𝑐Ω𝛼(||𝑢𝑢)𝑡||2𝑑𝑥2+𝑐𝑐Ω𝑟(𝑡)𝑑𝑥2+𝑐.(5.15) Thanks to uniform Gronwall inequality, there exists 𝑇(𝐵)>0 for any 𝑡𝑇(𝐵); we have Ω𝑟(𝑡)𝑑𝑥𝑐.(5.16)
Now we prove the semigroup {𝑆(𝑡)}𝑡0 is asymptotically compact in 𝐻10(Ω).

Lemma 5.2. Assume that the conditions (1.2), (1.3), (1.5), (1.6) and (5.1) are satisfied, 𝑔(𝑥)𝐿2(Ω). Then the semigroup {𝑆(𝑡)}𝑡0 is asymptotically compact in 𝐻10(Ω).

Proof. Let 𝐵 is a bounded absorbing set in 𝐻10(Ω), 𝑥𝑛𝐵, 𝑡𝑛 as 𝑛, 𝑢𝑛(𝑡)=𝑆(𝑡𝑛)𝑥𝑛. Now we will prove that there exists Cauchy sequence of {𝑢𝑛(𝑡)} in 𝐻10(Ω). By Theorem 4.2, we know that {𝑆(𝑡)}𝑡0 exists global attractor in 𝐿2(Ω), therefore there exists Cauchy sequence of {𝑢𝑛(𝑡)} in 𝐿2(Ω), we denote by {𝑢𝑛(𝑡)}. Hence, 0<𝜀<1,𝑁,𝑛,𝑚>𝑁: ||𝑢𝑛𝑢𝑚||22<𝜀2.(5.17) In view of (1.1), we get 𝛼𝑢𝑚𝑢𝑚𝑡𝛼𝑢𝑛𝑢𝑛𝑡𝑢Δ𝑚𝑢𝑛𝑢+𝑓𝑚𝑢𝑓𝑛=0.(5.18) Multiplying the above equality by 𝑢𝑚𝑢𝑛 and integrating over Ω, we obtain ||𝑢𝑚𝑢𝑛||22Ω𝛼𝑢𝑚𝑢𝑚𝑡𝛼𝑢𝑛𝑢𝑛𝑡𝑢𝑚𝑢𝑛𝑑𝑥Ω𝑓𝑢𝑚𝑢𝑓𝑛𝑢𝑚𝑢𝑛𝑑𝑥𝑐Ω||𝑢𝑚𝑡||||𝑢𝑚𝑢𝑛||𝑑𝑥+Ω||𝑢𝑛𝑡||||𝑢𝑚𝑢𝑛||𝑑𝑥Ω𝑓||𝑢(𝜉)𝑚𝑢𝑛||2||𝑢𝑑𝑥𝑐𝑚𝑡||2||𝑢𝑚𝑢𝑛||2+||𝑢𝑛𝑡||2||𝑢𝑚𝑢𝑛||2||𝑢+𝑐𝑚𝑢𝑛||22.(5.19) Hence, for any 𝑛,𝑚>𝑁, we get ||𝑢𝑚𝑢𝑛||22<𝑐𝜀.(5.20) That is to say, the semigroup {𝑆(𝑡)}𝑡0 is asymptotically compact in 𝐻10(Ω).

Thus from Theorem 4.1 and Lemma 5.2, we have the following.

Theorem 5.3. Assume that the conditions (1.2), (1.3), (1.5), (1.6) and (5.1) are satisfied, 𝑔(𝑥)𝐿2(Ω). Then the semigroup {𝑆(𝑡)}𝑡0 has a (𝐿𝑟+2(Ω),𝐻10(Ω))-global attractor, which is nonempty, compact, invariant in 𝐻10(Ω) and attracts every bounded subset of 𝐿𝑟+2(Ω) with respect to 𝐻10(Ω) norm.

Acknowledgment

This work is supported in part by the NSFC Grant (11161026).

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