ISRN Applied MathematicsVolume 2012 (2012), Article ID 956291, 16 pagesdoi:10.5402/2012/956291

Research Article

Global Attractor for Doubly Nonlinear Parabolic Equation

Yongjun Li, Suyun Wang, and Yanhong Zhang

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 and . Finally we establish the existence of global attractor of the problem in and .

1. Introduction

We study the long-time behavior (in terms of attractors) of the solution of the following problem: Here is a bounded smooth domain in () and is a given function in .

Such equations appear, for example, in the study of gas filtration (so-called porous medium equation [1]). It has been extensively studied when is linear [1–5], and the existence of attractors has been proved in [6–11] (for ).

Our aim in this paper is to extend the result of [1–4] to the more general equation (1.1). We make the following assumptions: There exists a constant , such that

By hypotheses (1.2)–(1.5), and are nonlinear functions with polynomial growth of arbitrary order. Here is more general than in [1–4](where 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 in and prove the existence of the -global attractor under the hypotheses (1.2)–(1.6). Finally, in Section 5, we prove the existence of the -global attractor for .

Throughout this paper we use the following notation: , and the norms in and are denoted by and , respectively; and ; or denotes Lebesgue measure of ; sometimes for special differentiation, we denote the different positive constants by , ,.

2. Preliminaries

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

Definition 2.1 (see [6–8]). Let be a semigroup on a Banach space . A subset is called a global attractor for the semigroup if is compact in and enjoys the following properties:(1) is an invariant set, that is, for any ;(2) attracts all bounded sets of , that is, for any bounded subset of where is the Hausdorff semidistance of the two sets and : And a subset of is called a bounded absorbing set of the semigroup in , if for any bounded of , there exists some , such that for any .

Definition 2.2 (see [11]). Let be a Banach space and let be a family of operator in . We say that is norm-to-weak continuous semigroup in , if satisfies(1) (the identity);(2);(3) if and in .

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

Definition 2.4 (see [11]). Let be a semigroup on Banach space . is called -asymptotically compact, if for any bounded sequence and , as , has a convergence subsequence with respect to the topology of .

Definition 2.5 (see [11]). Let be a semigroup on Banach space . A bounded subset of is called an -bounded absorbing set, if for any bounded set of , there exists some , such that for any .

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

Theorem 2.7 (see [11]). Let be a norm-to-weak continuous semigroup on . One assumes that . Then has an -global attractor provided that the following conditions hold: (1) has an -bounded absorbing set ;(2)there is a such that is -asymptotically compact;(3)for any and for any bounded , there exist constants and , such that

Lemma 2.8 (see [11]). Let be a semigroup on and suppose that has a bounded absorbing set in . Then for any and any bounded subset , there exist positive constants and such that where the positive constant is independent of , , and .

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 , and satisfying (1.3)–(1.5), . Then for any initial data , there exists solution for (1.1) which satisfies

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

Theorem 3.2. Assume that , and satisfying (1.3)–(1.6). Then there exists a unique solution of (1.1).

Proof. Suppose that be two solutions of (1.1) with initial conditions , ; then that is We define the sign function by Multiplying (3.3) by and integrating in , we obtain Using  (1.6), we get Since , by dominated convergence theorem, we have So By Gronwall inequality, we get From (1.2), we have which gives continuous dependence on initial conditions and uniqueness of solution in .By Theorem 3.2, we can define the operator semigroup in as the following: which is continuous in .Since is a continuous increasing function with , we define for , Then the Legendre transform is defined by Note that

Theorem 3.3. Assume that the conditions (1.2)–(1.6) are satisfied, . Then the semigroup is norm-to-weak continuous in and .

Proof. Let in , be the solutions of (1.1) corresponding to initial date , . In (1.1), replace by . Multiplying (1.1) by and integrating in , we get Applying Young inequality, we have So Integrating from 0 to , we obtain in , so there existence , such that . Is bounded in and . Therefore, there exists weak convergent subsequence of in and . Let obviously, be a solution of (1.1) satisfying the initial value condition . By the unique of solution for (1.1), we have , that is, in and . By Definition 2.2, Theorem 3.3 holds.

Remark 3.4. The semigroup is norm-to-weak continuous in .

4. -Global Attractor

By Theorem 3.3, we can define operator semigroup as the following:

Theorem 4.1. Assume that the conditions (1.2)–(1.6) are satisfied, . Then the semigroup exists bounded absorbing sets in and ; that is, for arbitrary bounded set , there exist , , , . We have

Proof. Let be the solution of (1.1) with initial date ; taking scalar product with in (1.1), we deduce that By (1.5), we obtain Using Young inequality, we get By (4.4), we have Since , by (1.3), there exist such that Hence, We get By the Gronwall lemma, we have Therefore, We obtain It follows from (4.8) that there exists ; we get Multiplying (1.1) by and integrating over , we get where . Now by (1.2) we get It follows from (1.5) that there exist such that Using the uniform Gronwall Lemma that there exists , we have By (4.16), , we obtain Therefor, the semigroup exists bounded absorbing set in and ; it follows from Theorem 2.6 that we have the following.

Theorem 4.2. Assume that the conditions (1.2)–(1.6) are satisfied, . Then the semigroup has a -global attractor, which is nonempty, compact, invariant in and attracts every bounded subset of with respect to norm.

In the following, we will give the asymptotic a priori estimate of with respect to -norm, which plays a crucial role in the proof of the -global attractor.

Lemma 4.3. Assume that the conditions (1.2)–(1.6) are satisfied, , , . For any , there exist positive constants and such that

Proof. By (1.3) and (1.5), we find that there exists such that Letting , when , then . Multiplying (1.1) with , we get where denote the positive part of , that is: Thus we have We obtain Since , we have So By Gronwall inequality, there exists such that For any there exists ; for any , , we have Leting , obviously is a semigroup in . By (1.5) and (4.18), has a bounded absorbing set in . Combining Lemma 2.8, for any , there exist , for any , we have Hence that is By repeating the same step above and multiplying (1.1) with , we get where Combining (4.31) and (4.32), we have Thanks to (4.20), we have From Theorem 4.2, we know that the semigroup is asymptotically compact in ; 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, . Then the semigroup has a -global attractor, which is nonempty, compact, invariant in and attracts every bounded subset of with respect to norm.

5. -Global Attractor

In this section, we want to prove the -global attractor. However, for general N-dimension space, our methods can not work, so we make the following assumption:

Lemma 5.1. Assume that the conditions (1.2), (1.3), (1.5), (1.6), and (5.1) are satisfied, . For any bounded set , there exists ; for any , we have

Proof. Multiplying (1.1) by and integrating over , we have and then By (4.16) and (4.18), there exists for any ; we get Integrating (5.4) from to , we obtain By differentiating (1.1) in time , we have Letting , we get that is Multiplying (5.7) by and integrating over , we get Hence from (5.9), we obtain Since , from Theorem 4.1 the semigroup has a bounded absorbing set in . Applying Sobolev embedding theorem, the semigroup is bounded in ; hence, Since , the imbedding is compact; hence Using interpolation inequality for Sobolev space, we get Therefore, from (5.12), we have Thanks to uniform Gronwall inequality, there exists for any ; we have Now we prove the semigroup is asymptotically compact in .

Lemma 5.2. Assume that the conditions (1.2), (1.3), (1.5), (1.6) and (5.1) are satisfied, . Then the semigroup is asymptotically compact in .

Proof. Let is a bounded absorbing set in , , as , . Now we will prove that there exists Cauchy sequence of in . By Theorem 4.2, we know that exists global attractor in , therefore there exists Cauchy sequence of in , we denote by . Hence, : In view of (1.1), we get Multiplying the above equality by and integrating over , we obtain Hence, for any , we get That is to say, the semigroup is asymptotically compact in .

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, . Then the semigroup has a -global attractor, which is nonempty, compact, invariant in and attracts every bounded subset of with respect to norm.

Acknowledgment

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

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