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Mathematical Problems in Engineering
Volume 2012, Article ID 658052, 7 pages
http://dx.doi.org/10.1155/2012/658052
Research Article

Existence of Global Attractors for the Nonlinear Plate Equation with Memory Term

School of Mathematics, Taiyuan University of Technology, Yingze 79 West Street, Shanxi, Taiyuan 030024, China

Received 21 July 2012; Accepted 6 September 2012

Academic Editor: Sebastian Anita

Copyright © 2012 Lifang Niu and Jianwen Zhang. 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

A two-dimensional nonlinear plate equation is revisited, which arises from the model of the viscoelastic thin rectangular plate with four edges supported. We establish that the system is exponentially decayed if the memory kernel satisfies the condition of the exponential decay. Furthermore, we show the existence of the global attractor by verifying the condition (C).

1. Introduction

In this paper, we investigate the nonlinear plate equation with memory type: verifying the initial conditions: and the boundary conditions: where ,   are nonnegative constants, is a bounded domain with boundary and for every .

The asymptotical behavior of solutions for the nonlinear plate equations had been studied by many authors [110]; of those, Santos and Junior [5] studied a kind of plate equation with memory type. Yang and Zhong [11] studies the plate equation: where is a bounded domain and proves the existence of a global attractor in the space . After Yang and Zhong [11], Yue and Zhong [12] obtained the existence of a global attractor about some equations similar to (1.4). Xiao [13] discusses the long-time behavior of the plate equation: on the unbounded domain and show that there exists a compact global attractor for the above equation satisfying certain initial-boundary data. Wang and Zhang [14] prove that the two-dimensional nonlinear equation has a global attractor in space .

2. Preliminaries

We denote by endowed with the scalar product and the norm on and , respectively, where ,  . Define , where . For the operator , we assume that are isomorphism, and there exists such that , for all . We also define the power of for which operates on the spaces , and we write . This is a Hilbert space with the inner product and norm defined and is an isomorphism from onto for all . In particular, , where   are the dual space, respectively, and each space is dense in the following one and the injections are continuous. Using the Poincáre inequality we have where denotes the first eigenvalue of .

Let and assume that the memory kernel is required to satisfy the following assumptions:  , for all ;   ;   for all .

In view of , let be the Hilbert space of -valued functions on , endowed with the following inner product and the norm: and . Finally, we introduce the following Hilbert spaces:

We define and (1.1) is transformed into the system where the second equation is obtained by differentiating (2.5). The corresponding initial-boundary value conditions are then given by where .

According to the classical Faedo-Galerkin method it is easy to obtain the existence and uniqueness of solutions and the continuous dependence to the initial value, so we omit it and only give the following theorem.

Theorem 2.1 (see [15, 16]). Let hold and . Then given any time interval , problems (2.6)-(2.7) have a unique solution in with initial data , and the mapping is continuous in .

Thus, it admits to define a semigroup and they map into themselves.

In addition, the following abstract results will be used in our consideration.

Theorem 2.2 (see [17]). A semigroup in a Banach space is said to satisfy condition which arised by [17] if for any and for any bounded set of , there exists and a finite dimensional subspace of , such that is bounded and where is a bounded projector.

Lemma 2.3 (see [17]). Let be a semigroup in a Hilbert space . Then has a global attractor if and only if (1) satisfies the condition ; (2)there exists a bounded absorbing subset of .

3. Global Attractor in

Theorem 3.1. Assume hold. Then the ball of , centered at 0 with , is a bounded absorbing set in for the semigroup .

Proof. We fixed and take , where .
First, taking the inner product of the first equation of (2.6) with , after computation we conclude where , and we can easily obtain Combining with the second equation of (2.6) we have According to , we conclude Integrating with (3.4), from (3.3), entails Write , from (2.7), (3.2), and (3.5), we obtain Take small enough, such that . Write ,  , thus in line with (3.6), we have By the Gronwall Lemma, we conclude Due to (3.8), write , we have Write , we end up with

Theorem 3.2. Suppose and conditions are hold. Then the solution semigroup associated with system (2.6) and (2.7) has a global attractor in , and it attracts all bounded subsets of , in the norm of .

Proof. Applying Lemma 2.3, we only to prove that the condition holds in .
Let be an orthonormal basis of which consists of eigenvectors of . It is also an orthonormal basis of , respectively. The corresponding eigenvalues are denoted by with , for all . We write . For any , we decompose that , where , and is the orthogonal projector. Since , for any , there exists some , such that Taking the scalar product of the first equation of (2.6) in with , combining with the second equation and using the same way with Theorem 3.1, we find where . According to Theorem 3.1, some and exist, for any , such that . By the Gronwall Lemma, we conclude Take large enough, such that , so we conclude Thus satisfies the condition .

Acknowledgments

Project supported by the Nature Science Foundation of Shanxi province (Grant no. 200611005) and Natural Science Foundation of China (Grant no. 10772131).

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