Abstract and Applied Analysis

Volume 2012 (2012), Article ID 469382, 12 pages

doi:10.1155/2012/469382

Research Article

Strong Global Attractors for 3D Wave Equations with Weakly Damping

Fengjuan Meng1,2

¹Department of Mathematics, Nanjing University, Nanjing 210093, China

²Department of Mathematics, Taizhou College, Nanjing Normal University, Taizhou 225300, China

Received 30 March 2012; Accepted 13 April 2012

Academic Editor: Shaoyong Lai

Copyright © 2012 Fengjuan Meng. 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.

We consider the existence of the global attractor for the 3D weakly damped wave equation. We prove that is compact in and attracts all bounded subsets of with respect to the norm of . Furthermore, this attractor coincides with the global attractor in the weak energy space .

1. Introduction

Let be a bounded domain with smooth boundary . We consider the following weakly damped wave equation:

with the boundary condition

and initial conditions:

where , is the nonlinear term, and is a given external forcing term.

Nonlinear wave equation of the type (1.1) arises as an evolutionary mathematical model in many branched of physics, for example, (i) modeling a continuous Josephson junction with ; (ii) modeling a relativistic quantum mechanics with . A relevant problem is to investigate the asymptotic dynamical behavior of these mathematical models. The understanding of the asymptotic behavior of dynamical systems is one of the most important problems of modern mathematical physics. One way to treat this problem is to analyse the existence of its global attractor.

The existence of global attractors for the classical wave equations in and the regularities of the global attractors has been studied extensively in many monographs and lectures, for example, see [1–7] and references therein.

However, to our knowledge, the research about the stronger attraction of global attractors for the damped wave equations with respect to the norm of is fewer, only has been found in [8–10]. In the above three papers, the global attractors in strong topological space were established, the attraction with respect to the norm of was proved by the asymptotic compactness of the operator semigroup.

Recently, we consider (1.1) in dimensional space where the nonlinear term without polynomial growth is in [11].

In this paper, our aim is to prove the existence of a global attractor for (1.1) in strong topological space where the nonlinear term with some polynomial growth. For simplicity, we consider the space dimension is 3, as we know, when the space dimension is lagerer than 3, the case is similar as in 3D, when the space dimension is 1 or 2, the case is more easier. The attraction with respect to the norm of will be proved by a method different from [8–10]. Furthermore, this attractor coincides with the global attractor in the weak energy space .

The basic assumptions about the external forcing term and the nonlinear term are as follows. Let be independent of time, and let the nonlinear term satisfy the following assumptions:

moreover, there exists a constant such that

Throughout this paper, we use the following notations. Let be a bounded subset of with sufficiently smooth boundary, . , , and with the corresponding norms , and , respectively. The norms in are denoted by , the scalar products of are denoted by

respectively. We have and are the dual spaces of and , respectively, and each space is dense in the following one and the injections are continuous. Then, we introduce the product Hilbert spaces , endowed with the standard product norms:

Denote by any positive constant which may be different from line to line and even in the same line, we also denote the different positive constants by , for special differentiation.

The rest of the paper is organized as follows. In the next section, for the convenience of the reader, we recall some basic concepts about the global attractors and recapitulate some abstract results. In Section 3, we present our main results.

2. Preliminaries

In this section, we first recall some basic concepts and theorems, which are important for getting our main results. We refer to [2, 5, 6, 12, 13] and the references therein for more details. Then, we outline some known results about (1.1)–(1.3).

Definition 2.1. The mappings , where , is said to be a semigroup on , if satisfies

(1) for all ;(2) for all and ;(3)the mapping is continuous.

Definition 2.2. Let be a semigroup on a metric space . A subset of is called a global attractor for the semigroup, if is compact and enjoys the following properties:

(1) is invariant, that is, , ;(2) attracts all bounded sets of . That is, for any bounded subset of ,

where is the semidistance of two sets B and A:

Definition 2.3. A semigroup in a Banach space is said to satisfy the condition if for any and for any bounded set of , there exist and a finite dimensional subspace of such that is bounded and

where is a bounded projector.

Definition 2.4. Let be a semigroup on a metric space . A set is called an absorbing set for the semigroup , if and only if for every bounded set , there exists a such that for all .

Theorem 2.5. Let be a Banach space and let be a semigroup in . Then, there is a global attractor for in if the following conditions hold true:

(1) satisfies the condition , and(2)there is a bounded absorbing set .

In [12], the authors have discussed the relations between Condition and -limit compact and proved that, in uniformly convex Banach space, Condition is equivalent to -limit compact, if the semigroup has a bounded absorbing set.

Next, we recall the result about the global attractor in whose proofs are omitted here, the reader is referred to [6] and the reference therein.

Theorem 2.6. Under the conditions (1.4), (1.5), (1.6), the solution semigroup of the problem (1.1)–(1.3) has a global attractor in . is included and bounded in .

3. Main Results

According to the standard Fatou-Galerkin method, it is easy to obtain the existence and uniqueness of solutions and the continuous dependence to the initial value of (1.1)–(1.3). We address the reader to [6] and the reference therein. Here, we only state the result as follows.

Lemma 3.1. Let conditions (1.4), (1.5), (1.6) hold, then for any and , there exists a unique solution of (1.1)–(1.3) such that

If, furthermore,

then satisfies

We define the mappings:

By Lemma 3.1, it is easy to see that is semigroup in the energy phase spaces and .

In order to verify the existence of the bounded absorbing set in , we need the result about the existence of the bounded absorbing set in . First, we establish the bounded absorbing set in . Its proof is essentially established in [6] and the reference therein, and we only need to make a few minor changes for our problem. Here, we only give the following lemma.

Lemma 3.2. Under the conditions (1.4), (1.5), (1.6), has a bounded absorbing set in , that is, for any and any bounded subset , there is a positive constant such that

Next, let us establish the existence of the bounded absorbing set in .

Lemma 3.3. Under the conditions (1.4), (1.5), (1.6), has a bounded absorbing set in , that is, for any and any bounded subset , there is a positive constant such that

Proof. Take the scalar product in of (1.1) with , we have

For , , by Hölder inequality, Poincaré inequality, and Cauchy inequality we have

It follows from (1.5) that, for any , there exists a constant , such that

Hence,

where is the positive constant satisfying

If belongs to a bounded set of , then is also bounded in , and for , by Lemma 3.2, we have

, are given in Lemma 3.2 Choose

it follows from (3.10) that

Combining with (3.8), (3.12), and (3.15), by the Hölder inequality and the Young inequality, we deduce from that (3.7):

Let , from the above inequality, we can obtain

Let . By the Gronwall lemma, we have

Defining by

we see that

and we conclude that

The ball of , , centered at of radius , is absorbing in for the semigroup .

We now give the property of compactness about the nonlinear operator which will be needed in the proof of the condition .

Lemma 3.4. Assume that and are defined by

for all . Then, is continuous compact.

Proof. Let be a bounded sequence in . Without loss of generality, we assume that weakly converges to in , since is reflexive. By the Sobolev embedding theorem, we know that and the embedding is compact in . Hence, we have that

Furthermore, there exists a constant such that

It is sufficient to prove that converges to in :

On the one hand, for the first term in (3.24), combining with (3.23) and the continuity of , we have

On the other hand, for the second term in (3.24), using the continuity of ,

follows immediately by dominated convergence theorem.

Also, considering (3.22), passing to the limit in (3.24), we can obtain

This completes the proof.

Lemma 3.5. Suppose the conditions (1.4), (1.5), (1.6) hold, the solution semigroup of the problem (1.1)–(1.3) satisfies the condition in .

Proof. Let be an orthonormal basis of which consists of eigenvalues of . The corresponding eigenvalues are denoted by :

with

Let   span in and let be an orthogonal projector. We write

Taking the scalar product of (1.1) in with , we find

Choose , similar to (3.8), we have

Since , is compact by Lemma 3.4, for any , there exists some such that

where is given by Lemma 3.2.

By exploiting the Hölder inequality and Cauchy inequality, we have

By (3.33) and (3.36), we have

where .

Hence, combining with (3.32), (3.35), and (3.37), we obtain from (3.31) that

Let , from the above inequality, similar to (3.17), we can obtain

Let . By the Gronwall lemma, we have

Choosing , it follows that

That is,

for all , where .

Thus we complete the proof.

We are now in a position to state our main results as follows.

Theorem 3.6. Under the conditions (1.4), (1.5), (1.6), problem (1.1)–(1.3) has a global attractor in ; it attracts all bounded subsets of with respect to the norm of .

Proof. By Lemmas 3.1, 3.3, and 3.5, the conditions of Theorem 2.5 are satisfied. The proof is complete.

Corollary 3.7. The global attractor in is coincides with in .

Proof. By Theorem 2.6, is a bounded set of , combining with Theorem 3.6, we can easily get .

Acknowledgment

The author of this paper would like to express her gratitude to Professor Chengkui Zhong for his guidance and encouragement, and to the reviewer, for valuable comments and suggestions. This work was supported in part by the Scientific Research Foundation of Graduate School of Nanjing University (2012CL21) and Taizhou Natural Science and Technology Development Project 2011.

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