ISRN Applied Mathematics

Volume 2013 (2013), Article ID 372507, 3 pages

http://dx.doi.org/10.1155/2013/372507

Research Article

A Remark on the Global Attractors of the Nonlinear Evolution Equations

Chang Ya-ya and Ma Qiao-zhen

School of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China

Received 18 September 2013; Accepted 13 November 2013

Academic Editors: H. Homeier and J. Kou

Copyright © 2013 Chang Ya-ya and Ma Qiao-zhen. 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 study the existence of global attractor of the nonlinear elastic rod oscillation equation when the forcing term belongs only to ; furthermore, we prove that the fractal dimension of global attractor is finite.

1. Introduction

Let be an open bounded set of with smooth boundary . We consider the following equation:

where and . The nonlinear term , , and satisfies the following:

where is the first eigenvalue of in and is a positive constant.

In line with the Galerkin methods introduced in [1], we know that (1) has a unique solution , , for . The proof has no essential difference between and , so we omit it; see [2].

Equation (1), which appears as a class of nonlinear evolution equations, like the strain solitary wave equation and dispersive-dissipative wave equation, is used to represent the propagation problems of a lengthwise wave in nonlinear elastic rods and lon-sonic of space transformation by weak nonlinear effect; see [3–6]. For (1), when , in [2], the author has discussed the existence of global strong solutions in ; in [7, 8], the authors have obtained the existence of global attractors in the weak topological space and the strong topology space, respectively. Recently, existence of the uniform compact attractors has been proved about the nonautonomous case of (1); that is, . In this paper, we prove existence of global attractor and its fractal dimension for (1) under the condition that only satisfies the lower regularity.

2. The Main Results

Without loss of generality, we denote , , and , is, respectively, the dual space of , . Write . Let and ; we define ; is Hilbert space family, and its inner product and norm are

The following results will be used later.

Lemma 1 (see [8]). Assume that satisfies (2) and (3), ; then, the solution semigroup has a bounded absorbing set in ; that is, for any bounded subset , there exists such that

Lemma 2. Let be a bounded domain with smooth boundary, and one assumes that satisfies (2) and (3), ; then, the semigroup possesses a global attractor on .

Proof. Since is dense, for any , there exists such that

The remained proof of Lemma 2 is similar to that of [7], so we omit it.

Lemma 3 (see [9]). Let be a bounded subset in Hilbert space , the mapping , such that , and satisfy

where is orthogonal mapping and is spanned subspace by the former Nth eigenvector of ; then, the fractal dimension of satisfies

where is the Gaussian constant.

Our main result is as follows.

Theorem 4. Let be a bounded domain with smooth boundary; one assumes that satisfies (2) and (3), ; then, the fractal dimension of the global attractor of the semigroup is finite.

Proof. According to Lemma 2, provided

then,

Let satisfy the following equation:

Taking the scalar product of (11) in with , we obtain that

As the global attractor is bounded in , so there exists such that

Therefore, using (3) and (13), it follows that

By (12) and (14), we obtain

where is constant independent of ; by Gronwall's inequality, we get

For some , define ; hence, we prove that the first inequality in Lemma 3 holds true.

Taking the inner product of (11) in with , we commute the operator with the projection to get

Similar to the estimates of (13)-(14), we obtain

Then,

where is the th eigenvalue of problem (11), so we have

We let

Choosing large enough that and setting , integrating with (16) and (20), we get

Gronwall's inequality implies that

So, we have

Take a proper and large enough such that

Therefore, for , satisfies the condition of Lemma 3; then, the fractal dimension of the global attractor satisfies

This implies that the global attractor for semigroup generated by the problem (1) has a finite fractal dimension.

Acknowledgments

This work was partly supported by the NSFC (11101334) and the NSF of Gansu Province (1107RJZA223), and by the Fundamental Research Funds for the Gansu Universities.

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