Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 869621, 6 pages

http://dx.doi.org/10.1155/2013/869621

Research Article

Exponential Attractor for Lattice System of Nonlinear Boussinesq Equation

Min Zhao and Shengfan Zhou

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

Received 14 July 2013; Accepted 13 August 2013

Academic Editor: Zhan Zhou

Copyright © 2013 Min Zhao and Shengfan Zhou. 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 lattice dynamical system of a nonlinear Boussinesq equation. We first verify the Lipschitz continuity of the continuous semigroup associated with the system. Then, we provide an estimation of the tail of the difference between two solutions of the system. Finally, we obtain the existence of an exponential attractor of the system.

1. Introduction

Lattice dynamical systems (LDSs) have a wide range of applications in many areas such as electrical engineering, chemical reaction theory, laser systems, material science, and biology [1, 2]. In recent years, many works about the asymptotic behavior of LDSs have been done, which include the global attractor, see [3–11] and the references therein. However, the global attractor sometimes attracts orbits at a relatively slow speed and it might take an unexpected long time to be reached. For this reason, the exponential attractor having finite fractal dimension and attracting all bounded sets exponentially was introduced, and it has been studied for a large class of LDSs, see [12–15] and the references therein. Han presented in [13] some sufficient conditions for the existence of exponential attractor for LDSs in the weighted space of infinite sequences and applied the result to obtain the existence of exponential attractors for some LDSs. Zhou and Han in [15] presented some sufficient conditions for the existence of uniform exponential attractor for LDSs, which is easier to verify the existence of exponential attractor for some LDSs. Abdallah in [3] considered the following initial problem of lattice system of nonlinear Boussinesq equation:

where , , , and   are positive constants, is a real constant; for , ; and , , , and   are linear operators (see Section 3 for details). Equation (1) can be regarded as a spatial discretization of the following nonlinear damped Boussinesq equation on :

which appears in many fields of physics and mechanics, for example, long waves in shallow water, nonlinear elastic beam systems, thermomechanical phase transitions, and some Hamiltonian mechanics. Abdallah has in [3] investigated the existence and finite-dimensional approximation of the global attractor for (1) under the following conditions:

In this paper, motivated by the ideas of [13, 15], we will further prove the existence of an exponential attractor for the system (1) under the condition (4).

The paper is organized as follows. In Section 2, we present some preliminaries. Section 3 is devoted to the existence of an exponential attractor for (1).

2. Preliminaries

In this section, we present the definition of an exponential attractor and some sufficient conditions for the existence of an exponential attractor for a semigroup in a separable Hilbert space from [13, 15].

Let be a separable Hilbert space, let be a bounded subset of , and let   be a semigroup acting on which satisfy: , , for all , , and for , where is the identity operator on .

Definition 1. A set is called an exponential attractor for the semigroup on , if

(i) is compact; (ii), where is the global attractor; (iii), ;(iv) has a finite fractal dimension; (v)there exist two positive constants and such that dist for all , .

Let be a -dimensional subspace of . We define the bounded -dimensional orthogonal projection from into and .

As a direct consequence of [13, Theorem 2.5] and [15, Theorem 2.1], we have the following theorem.

Theorem 2. Let be a continuous semigroup on and let be a closed bounded subset of such that , for . If there exist , a constant and a -dimensional subspace of such that for any ,,

Then,

(i) has an exponential attractor on with , where is a constant; (ii) is an exponential attractor for on that , and there exist two positive constants and such that for all , .

3. Exponential Attractor for System (1)

Let and equip it with the inner product and norm as

Then, is a separable Hilbert space. The linear operators , , , and   are defined from into as follows: for any ,

then, , .

The system (1) with initial data (2) is equivalent to the following vector form:

where , , ,  , .

Letting

then, the system (8) can be written as the following initial value problem:

where

We define

Then, the bilinear form is an inner product on and the induced norm is equivalent to . Let and let , then, is a separable Hilbert space with the following norm:

In this section, we will study the existence of an exponential attractor of (10) in the space .

Lemma 3 (see [3]). Assume (4) holds. Then, there exist small and , such that

Moreover,

(1) for any initial data , there exists a unique solution of (10), such that , and the solution map

 generates a continuous semigroup on .(2) The semigroup possesses a closed bounded absorbing ball , where , , ,   . Therefore, there exists a constant such that , for .(3) For any , there exist and such that the solution of (10) with satisfies

(4) The semigroup of (10) possesses a global attractor .

In the following, we first verify the Lipschitz continuity of and provide an estimation of the tail of the difference between two solutions of (10). Then, we obtain the existence of an exponential attractor of (10) by Theorem 2.

For , , , let be the solutions of (10). Set = = = , we have by (10) that

Lemma 4. Assume that (4) and (14) hold. Let

Then,

(1)

(2) There exist and , such that

where

Proof. (1) Taking the inner product of (17) with , we obtain

We can write (22) into the following form:

where

Then,

Since

thus,

From (23), (25), and (27), it follows that for ,

Applying Gronwall’s inequality to (28), we obtain

Since

From (29) to (31), it follows that for ,

(2) Choosing a smooth increasing function satisfies

where is a positive constant. For , let , , , where . Taking the inner product of (17) with , we obtain

Similar to (4.3)–(4.5) in [3], we can get

where

Then,

By (3) of Lemma 3, there exist , , such that

This implies that

Then, for , ,

Since

where . From (32), (35), (37), and (40)-(41), it follows that for , ,

where . Applying Gronwall’s inequality to (42) from to , where , we obtain that for ,

Similar to (30), we can get

Since

By (31)-(32), we obtain

From (43) to (46), it follows that for , ,

Letting

we then have

As a direct consequence of (1)-(2), (4) of Lemma 3, (1)-(2) of Lemma 4 and Theorem 2, we have our main result.

Theorem 5. Assume that (4) and (14) hold. Then, the semigroup of (10) possesses an exponential attractor on with (i) is compact; (ii) , where is the global attractor; (iii) has a finite fractal dimension , where is a constant and and are as in (48); and (iv) there exist two positive constants and such that for all .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this article.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant no. 11071165 and Zhejiang Normal University (ZC304011068).

References

H. Chate and M. Courbage, “Lattice systems,” Physica D, vol. 103, no. 1–4, pp. 1–612, 1997.

S. N. Chow, Lattice Dynamical Systems, in Dynamical System, vol. 1822 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2003.

A. Y. Abdallah, “Global attractor for the lattice dynamical system of a nonlinear Boussinesq equation,” Abstract and Applied Analysis, vol. 2005, no. 6, pp. 655–671, 2005.

A. Y. Abdallah, “Long-time behavior for second order lattice dynamical systems,” Acta Applicandae Mathematicae, vol. 106, no. 1, pp. 47–59, 2009.

P. W. Bates, K. Lu, and B. Wang, “Attractors for lattice dynamical systems,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 11, no. 1, pp. 143–153, 2001.

H. Li and S. Zhou, “Structure of the global attractor for a second order strongly damped lattice system,” Journal of Mathematical Analysis and Applications, vol. 330, no. 2, pp. 1426–1446, 2007.

J. C. Oliveira, J. M. Pereira, and G. Perla Menzala, “Attractors for second order periodic lattices with nonlinear damping,” Journal of Difference Equations and Applications, vol. 14, no. 9, pp. 899–921, 2008.

B. Wang, “Dynamics of systems on infinite lattices,” Journal of Differential Equations, vol. 221, no. 1, pp. 224–245, 2006.

C. Zhao and S. Zhou, “Upper semicontinuity of attractors for lattice systems under singular perturbations,” Nonlinear Analysis, Theory, Methods and Applications, vol. 72, no. 5, pp. 2149–2158, 2010.

S. Zhou, “Attractors for second order lattice dynamical systems,” Journal of Differential Equations, vol. 179, no. 2, pp. 605–624, 2002.

S. Zhou, “Attractors and approximations for lattice dynamical systems,” Journal of Differential Equations, vol. 200, no. 2, pp. 342–368, 2004.

X. Fan and H. Yang, “Exponential attractor and its fractal dimension for a second order lattice dynamical system,” Journal of Mathematical Analysis and Applications, vol. 367, no. 2, pp. 350–359, 2010.

X. Han, “Exponential attractors for lattice dynamical systems in weighted spaces,” Discrete and Continuous Dynamical Systems, vol. 31, no. 2, pp. 445–467, 2011.

A. Y. Abdallah, “Exponential attractors for second order lattice dynamical systems,” Communications on Pure and Applied Analysis, vol. 8, no. 3, pp. 803–813, 2009.

S. Zhou and X. Han, “Uniform exponential attractors for non-autonomous KGS and Zakharov lattice systems with quasiperiodic external forces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 78, pp. 141–155, 2013.