Table of Contents
ISRN Applied Mathematics
Volume 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

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 [36]. 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.

References

  1. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences, Springer, New York, NY, USA, 2nd edition, 1997. View at MathSciNet
  2. Y. D. Shang, “Initial-boundary value problem for the equation utt-Δu-Δut-Δutt=f(u),” Acta Mathematicae Applicatae Sinica, vol. 23, no. 3, pp. 385–393, 2000 (). View at Google Scholar · View at MathSciNet
  3. I. L. Bogolubky, “Some examples of inelastic solution interaction,” Computer Physics Communications, vol. 13, 1977. View at Google Scholar
  4. W. G. Zhu, “Nonlinear waves in elastic rods,” Acta Solid Mechanica Sinica, vol. 1, no. 2, pp. 247–253, 1980. View at Google Scholar
  5. C. E. Seyler and D. L. Fanstermacher, “Asymmetric regularized long wave equation,” Physics of Fluids, vol. 27, no. 1, pp. 58–66, 1984. View at Google Scholar
  6. P. A. Clarkson, R. J. LeVeque, and R. Saxton, “Solitary-wave interactions in elastic rods,” Studies in Applied Mathematics, vol. 75, no. 2, pp. 95–121, 1986. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. Y. Xie and C. Zhong, “The existence of global attractors for a class nonlinear evolution equation,” Journal of Mathematical Analysis and Applications, vol. 336, no. 1, pp. 54–69, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. Y. Xie and C. Zhong, “Asymptotic behavior of a class of nonlinear evolution equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 11, pp. 5095–5105, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. A. O. Çelebi, V. K. Kalantarov, and M. Polat, “Attractors for the generalized Benjamin-Bona-Mahony equation,” Journal of Differential Equations, vol. 157, no. 2, pp. 439–451, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet