Discrete Dynamics in Nature and Society

Volume 2014 (2014), Article ID 237027, 10 pages

http://dx.doi.org/10.1155/2014/237027

Research Article

Pullback Exponential Attractor for Second Order Nonautonomous Lattice System

Shengfan Zhou,¹ Hong Chen,¹ and Zhaojuan Wang²

¹Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

²School of Mathematical Science, Huaiyin Normal University, Huaiyin 223300, China

Received 7 January 2014; Accepted 27 February 2014; Published 3 April 2014

Academic Editor: Zengji Du

Copyright © 2014 Shengfan Zhou et al. 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 first present some sufficient conditions for the existence of a pullback exponential attractor for continuous process on the product space of the weighted spaces of infinite sequences. Then we prove the existence and continuity of a pullback exponential attractor for second order lattice system with time-dependent coupled coefficients in the weighted space of infinite sequences. Moreover, we obtain the upper bound of fractal dimension and attracting rate for the attractor.

1. Introduction

Lattice dynamical systems (LDSs), which include coupled systems of infinite ordinary differential equations and coupled map lattices, have drawn more and more attention because these systems appear in various fields [1, 2]. In recent years, global attractors, uniform attractors, pullback attractors (or kernel sections), and random attractor for autonomous, nonautonomous, and stochastic LDSs have been studied; see [3–12]. However, these attractors sometimes attract orbits at a relatively slow speed, so that it might take an unexpected long time to be reached. Besides, it is usually difficult to estimate the attracting rate in terms of physical parameters of the model. Appropriate alternatives are exponential attractors and pullback exponential attractors which contain the global attractors and pullback attractors and attract all bounded sets exponentially [13–20]. For LDSs, [13, 16, 21] studied the existence of exponential attractors for first order, second order, and partly dissipative autonomous LDSs, respectively. Zhou and Han [20] presented some sufficient conditions for the existence of pullback exponential attractors for LDSs in and provided applications to first order and partly dissipative nonautonomous LDSs. As we know, there are no results on pullback exponential attractors for second order nonautonomous LDSs which have been studied extensively [4–7, 9–13, 21–23]. In this paper, we study the following second order nonautonomous LDSs:

where for , , ; ; , are locally integrable in , and , are positive constants. First we present some sufficient conditions for the existence of a pullback exponential attractor for a continuous process on the product space of infinite sequences, which are direct results of [20]. Then we prove the existence of a pullback exponential attractor for system (1) in weighted space ; moreover, we obtain the upper bound of fractal dimension and attracting rate for the attractor.

2. Preliminaries

In this section, we present some sufficient conditions for the existence of a pullback exponential attractor for a continuous process on the product space of weighted space of infinite sequences.

Let be positive-valued function such that , where are positive constants, , and let

which is Banach space with norm for , . Let be a product space of spaces , . Write

Then is -dimensional subspaces of . Let and define a bounded projection as follows: for , ,

Consider a two-parameter continuous process on : , , satisfying the following: , for all ; , ; is continuous for .

Definition 1. A family of subsets of is called a pullback exponential attractor for the continuous process , if

(i)each is a compact set of and its fractal dimension is uniformly bounded in ; that is, ;(ii)it is positively invariant; that is, for all ;(iii)there exist an exponent and two positive-valued functions , such that for any bounded set

where “” is the Hausdorff semidistance between two subsets of .

As a direct consequence of Theorem  2 of [20], we have the following theorem.

Theorem 2. Let be a continuous process in . Assume that there exists a uniformly bounded absorbing set for : for any bounded set , there exists a constant such that for all , . For any , set . If

there exist and such that, for every , any , and ,

there exist a positive constant and a -dimensional bounded projection such that, for every and ,

Then possesses a pullback exponential attractor satisfying

and, for any bounded set ,

where is the minimal number of closed balls of with radius covering the closed unit ball of centered at , and with , , and .

3. Pullback Exponential Attractor for Second Order Nonautonomous Lattice System

In this section, we study the existence of a pullback exponential attractor for the continuous processes associated with the second order nonautonomous lattice system (1). Note that system (1) with initial conditions can be written as a vector form

where , , , , .

Throughout this section, let be a positive weight function from to satisfying

(P0), , , for some positive constants and .

Consider the weighted Hilbert space of infinite sequences

endowed with inner product and norm for , . For any , , define an inner product on by , then the norm include by is equivalent to the norm include by . Let

We make the following assumptions on , , , and , for , .

(H1)There exist two positive constants and such that

(H2)Let satisfy the following:

(H2a) in ;(H2b)there exists a continuous positive valued function such that

where is a positive constant;(H2c)there exist and such that

(H3)For all , let satisfy the following:

(H3a) is differentiable in and continuous in ;(H3b)for any , , ;(H3c)there exist functions and such that

(H4), , where

and , , denotes the space of all continuous bounded functions from into .

Letting

then the system (10) is equivalent to the following evolution equation:

where

Definition 3. The function is called a mild solution of the following lattice differential equations:

if and

Theorem 4. Assume that (P0) and (H1)–(H4) hold. Then for any fixed and any initial data , problem (19) admits a unique mild solution with and being continuous in , and the mapping

generates a continuous process on .

Proof. Let

and then is continuous in and locally integrable in from into . For any bounded set with , let

Then for , , , by (P0), (H1)–(H4), we have

By the approximating method [24], it follows that problem (19) possesses a unique local mild solution . Similar to the proof of Theorem 5 below concerning the existence of a uniformly bounded absorbing set, we can prove that .

In the following part of this section, we assume that (P0) and (H1)–(H4) and

hold.

Theorem 5. Assume that (P0) and (H1)–(H4) hold. There exists a uniform bounded closed absorbing ball of centered at with radius (independent of , ) such that, for any bounded subset , there exists yielding for all .

Proof. Let be a mild solution of (19), . Since the set of continuous functions is dense in , by (H2a), there exist sequences of continuous functions , , such that

Consider the following lattice differential equations:

where

Combining the continuity of in with the proof of Theorem 4, (36) has a unique strong solution , satisfying

Taking the inner product of (29) with , we obtain

By some computation, we have that

Then we have that, for ,

Applying Gronwall’s inequality to (35) on (), we obtain

It then follows that, for ,

for some independent of and , and then, for any , ,

implies the equicontinuity of . By the Arzela-Ascoli Theorem, there exists a convergent subsequence of , such that

and is continuous in ; moreover, by (36), for . By the Lebesgue Convergence Theorem, we have

Thus, by replacing with in (31) and letting , we have

By the uniqueness of the mild solutions of (19), we have for . By replacing with and letting in (36), we have that for, ,

where

By (H2b), there exists such that for, and ,

Let

and then

Thus, for any bounded subset of and ,

where

Again, by (H2b),

It then follows that

is a uniformly bounded closed absorbing set for .

Lemma 6. Assume that (P0) and (H1)–(H4) hold. For any , there exist and such that, for , the mild solution , of system (19) with satisfies

Proof. Let be a smooth increasing function which satisfies

Let be a solution of (29) with . Let ( is in (H2c)) be a suitable large integer, and set

Taking the inner product of (29) with , we have

And we have the following estimates:

where

Therefore,

Applying Gronwall’s inequality to (57) on , we have that, for ,

By (44)-(45), for , we have

as . This means that, for any , there exists such that when ,

By , , and (H2b), there exists such that, for ,

From (47),

By (44), we have

And, for any , there exists such that, for ,

Let and . It follows immediately that

In the following, we prove the existence of a pullback exponential attractor for by applying Theorem 2. For any , set

where is the uniform absorbing set defined in Theorem 5.

Theorem 7. Assume that (P0) and (H1)–(H4) hold.

(a)There exists a continuous positive value function in such that, for every ,

(b)There exist positive constants , , and a -dimensional orthogonal projection , such that, for every and ,

(c)For each fixed , .

Proof. For any , initial data , . Let , , and ; then, , , .

(a) For any , let , be two solutions of (29) with initial data , and set , and then satisfies

Taking the inner product of (69) with , we have

Since , , by (36) and (66), for , , , thus , , and for . By (33), we have

and, by (H2)-(H3),

Thus,

Applying Gronwall’s inequality to (73) on , , we have

that is,

where

By replacing with in (75) and letting , we have

(b) For , let and , where is as in (52) and . Taking the inner product of (69) with in , we have that, for ,

For the terms in the right hand of (78), we have

where

By the continuity of and (see (H3c)), there exists such that

By Lemma 6 and for , there exist , such that, for ,

implying that, for ,

and thus

For and ,

Therefore, we obtain that, for all , ,

Applying Gronwall’s inequality to (86) from to , we have

Thus, for , we have

where

where

Thus, for , , we have

Taking

then

Thus, by (91), we have

(c) It follows from the following estimate:

and the continuity of , .

As a direct consequence of Theorem 2, Theorem 4, Theorem 5, Theorem 7, and Theorem  3.1 of [15], we have the following result.

Theorem 8. Assume that (P0) and (H1)–(H4) hold. The process associated with (19) possesses a pullback exponential attractor with the following properties:

(i)for any , ;(ii);(iii)and for any bounded set , , , ;(iv);

where

and is the minimal number of closed balls of with radius covering the closed unit ball of centered at .

Remark 9. It should be pointed out that, for some special cases, (H2) and (27) can be removed. For example, if the coupled linear operator in (10) is a constant operator satisfying , where , for all (constant),, and is the adjoint of . A simple example of such an operator is and . Choose a positive weight function satisfying the properties (P0) and , for all for some small . Two examples of such weight functions are , , and , , with being a parameter. If the number satisfies

where , then conditions (H2) and (27) can be removed.

Conflict of Interests

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

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant no. 11071165, 11326114 and Zhejiang Normal University (ZC304011068).

References

H. Chate and M. Courbage, “Lattice systems,” Physica D, vol. 10, no. 3-4, pp. 1–612, 1997.

S.-N. Chow, “Lattice dynamical systems,” in Dynamical Systems, vol. 1822 of Lecture Notes in Mathematics, pp. 1–102, Springer, Berlin, Germany, 2003.

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.

X. Fan and Y. Wang, “Attractors for a second order nonautonomous lattice dynamical system with nonlinear damping,” Physics Letters A, vol. 365, no. 1-2, pp. 17–27, 2007.

X. Han, “Random attractors for second order stochastic lattice dynamical systems with multiplicative noise in weighted spaces,” Stochastics and Dynamics, vol. 12, no. 3, Article ID 1150024, 20 pages, 2012.

X. Han, “Asymptotic behaviors for second order stochastic lattice dynamical systems on in weighted spaces,” Journal of Mathematical Analysis and Applications, vol. 397, no. 1, pp. 242–254, 2013.

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 & Applications A, vol. 72, no. 5, pp. 2149–2158, 2010.

X.-Q. Zhao and S. Zhou, “Kernel sections for processes and nonautonomous lattice systems,” Discrete and Continuous Dynamical Systems B, vol. 9, no. 3-4, pp. 763–785, 2008.

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

S. Zhou and L. Wei, “A random attractor for a stochastic second order lattice system with random coupled coefficients,” Journal of Mathematical Analysis and Applications, vol. 395, no. 1, pp. 42–55, 2012.

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.

A. Eden, C. Foias, B. Nicolaenko, and R. Teman, Exponential Attractors for Dissipative Evolution Equation, Research in Applied Mathematics, vol. 37, Masson/John Wiley, Paris, France, 1994.

M. Efendiev, Y. Yamamoto, and A. Yagi, “Exponential attractors for non-autonomous dissipative system,” Journal of the Mathematical Society of Japan, vol. 63, no. 2, pp. 647–673, 2011.

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

J. A. Langa, A. Miranville, and J. Real, “Pullback exponential attractors,” Discrete and Continuous Dynamical Systems A, vol. 26, no. 4, pp. 1329–1357, 2010.

M. Efendiev, A. Miranville, and S. Zelik, “Exponential attractors for a nonlinear reaction-diffusion system in ,” Comptes Rendus de l'Académie des Sciences, vol. 330, no. 8, pp. 713–718, 2000.

M. Efendiev, A. Miranville, and S. Zelik, “Global and exponential attractors for nonlinear reaction-diffusion systems in unbounded domains,” Proceedings of the Royal Society of Edinburgh A, vol. 134, no. 2, pp. 271–315, 2004.

S. Zhou and X. Han, “Pullback exponential attractors for non-autonomous lattice systems,” Journal of Dynamics and Differential Equations, vol. 24, no. 3, pp. 601–631, 2012.

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. Wang, S. Li, and D. Xu, “Random attractors for second-order stochastic lattice dynamical systems,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 72, no. 1, pp. 483–494, 2010.

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

A. Pazy, Semigroups of Linear Operator and Applications to Partial Differential Equations, Springer, New York, NY, USA, 2007.