Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2008, Article ID 354652, 8 pages
http://dx.doi.org/10.1155/2008/354652
Research Article

Exponential Attractor for a First-Order Dissipative Lattice Dynamical System

Department of Mathematics, Southwest Jiaotong University, Chengdu 610031, China

Received 22 November 2007; Accepted 2 March 2008

Academic Editor: Andrei Agrachev

Copyright © 2008 Xiaoming Fan. 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 construct an exponential attractor for a first-order dissipative lattice dynamical system arising from spatial discretization of reaction-diffusion equations in . And we obtain fractal dimension of the exponential attractor.

1. Introduction

Lattice systems arise in many applications, for example, in chemical reaction theory, image processing, pattern recognition, material science, biology, electrical engineering, laser systems, and so forth. A lattice dynamical system (LDS) is an infinite system of ordinary differential equations (lattice ODEs) or of difference equations. In some cases, they arise from spatial discretizations of partial differential equations (PDEs), but they possess their own form.

Let be a fixed positive integer. Denotewhere is the set of integers. Define a linear operator acting on in the following way: for any , ,

In this paper, we will consider the following first-order lattice dynamical system:where , , , denote the first-order derivative, and , . Then, problem (1.3) can be regarded as a discrete analogue of the following reaction diffusion equation in :One example is the Chafee-Infante equation.

Bates [1] and his collaborators made some results on a global attractor for lattice dynamical system (LDS). Zhou [2] applied them to a first-order dissipative lattice dynamical systems analogue to problem (1.3), proved the existence of the global attractor for the LDS, and considered the finite-dimensional approximation of the attractor. Wang [3] and Zhao and Zhou [4] studied asymptotic behavior of nonautonomous lattice systems. In standard definition of exponential attractor, a compact and positively invariant set is needed for the semigroup , and the system possesses a global attractor . More specifically, the semigroup is not compact for all positive . So, it is difficult to find a compact and positively invariant which is not the attractor . The first-order dissipative lattice dynamical systems analogue to problem (1.3) is such an example. Babin and Nicolaenko [5] consider reaction-diffusion systems in unbounded domains, prove the existence of exponential attractors for such systems, and estimate their fractal dimension. In [5], the compactness assumption plays a relatively minor role in the whole construction. In [6], Eden et al. provide constructions of exponential attractor for a Lipschitz -contraction on a closed bounded that satisfies the discrete squeezing property, where is not assumed to be compact.

The main novelty of this work is that we make an improvement in the constructions of exponential attractors is indicated in [6] that if a map is asymptotically compact on a closed bounded that satisfies the discrete squeezing property, then possesses an exponential attractor. is not assumed to be -contraction in the result. We apply the result to study an exponential attractor for a first-order dissipative lattice dynamical system. We not only construct an exponential attractor for the lattice dynamical system and consider its finite-dimensional approximation, but also obtain an upper bound of its fractal dimension.

2. A Key Theorem

Let be a separable Hilbert space with the norm , be nonempty closed bounded set, and be a Lipschitz continuous map with Lipschitz constant . In this paper, we will always denote dist the Hausdorff semi-distance of sets as follows:

Definition 2.1. is asymptotically compact on if for any , there is a convergent subsequence of in .

Remark 2.2. If is an -contraction on , then is asymptotically compact on .

Definition 2.3. is said to satisfy the discrete squeezing property on if there exists an orthogonal projection of rank such that for every and in ,

Definition 2.4. A compact set is called as an exponential attractor for if (i), where is the global attractor;(ii), that is positively invariant under ;(iii) has finite fractal dimension; and(iv) there exist universal constants , such that for every , for every natural number , .

Let be the orthogonal projection chosen as in the definition of the squeezing property. Denotefor the inclusion relation. From the definition of , we know is one-to-one on . Clearly, is a bounded closed set of a finite dimensional vector space, and therefore, it is compact. So, as the preimage under the continuous map must also be compact.

Let be a subset of the set , which is formed by a finite union of exceptional sets of the form , which is described above, hence all are compact.

Lemma 2.5. If is asymptotically compact on , then is relatively compact.

Proof. Let be a sequence in . Then, two cases will appear as follows. Case 1. There exists a natural number such that all are in ;Case 2. There exists a subsequence (still denoted by ) satisfying for every , there exists such that . In Case 1, since is compact and is continuous, there exists a convergent subsequence of that converges in . In Case 2, since is asymptotically compact on , it is immediate that we can extract from a subsequence that converges in . So, is relatively compact.

Theorem 2.6. Let be a separable Hilbert space and let be a nonempty closed bounded subset of . Assume that (i) is a Lipschitz continuous map with Lipschitz constant on ;(ii) is asymptotically compact on ;(iii) satisfies the discrete squeezing property on (with rank ), then has an exponential attractor on :where is a global attractor for on , is as the above-mentioned. Moreover, the fractal dimension of satisfies

Proof. Note that all the limits point of belong to . Together with Lemma 2.5, the proof follows exactly in the same way as the proof of Theorem 2.1 in [7].

Remark 2.7. In Theorem 2.6, there are two advantages than all the previous results on the existence of exponential attractor for :(i) is not assumed to be compact;(ii) if possesses a global attractor, then is at least asymptotically compact. So, we only check that if satisfies the Lipschitz property and the discrete squeezing property to obtain the existence of an exponential attractor for .

3. Exponential Attractor

For , we will always denote in the following discussion. For any , , define the operators , , and , from to itself as follows: , ,Then, we have

For any , , we define inner product and norm of as follows:then is a Hilbert space. It is obvious that any , ,

We always make the following assumptions on :(H1) and .(H2)There exists an increasing continuous function with such thatwhere

Similar to [2, Theorem 1], we have.

Theorem 3.1. For any initial data , there exists a unique local solution of problem (1.3) with such that for any .

In fact, it will be showed in Lemma 3.2 below that the local solution of problem (1.3) exists globally, that is, . It implies that the mapgenerates a continuous semigroup from to itself.

Lemma 3.2. Let be a closed bounded ball of , centered at with radius where For any bounded set of , there exists such that

Proof. The proof is easily obtained.

Corollary 3.3. For any , .

We obtain the following lemma after some simple computation.

Lemma 3.4. Let be a solution of problem (1.3) with initial data . Then, for any ,

From Lemmas 3.2 and 3.4, we have the following.

Theorem 3.5. The semigroup is asymptotically compact in and possesses a nonempty compact global attractor . Furthermore, .
Let and . Since , , by Corollary 3.3, , , for . Let . Then, satisfies

After some simple computation, we obtain the following.

Lemma 3.6 (Lipschitz Property). For any , and any ,

Let be a positive integer. SetFor convenience, we always denotewith the same inner product and norm as those of .

Let be the inverse function of in (H2). Set

Suppose is an orthogonal projection of rank on such that .

Lemma 3.7 (Discrete Squeezing Property). For any , , if then

Proof. Denote , and . Taking the inner product in (3.10) with , we havewhere . By the mean value theorem,where , . By (H2) and Lemma 3.4, for ,which impliesBy (3.18), (3.21), and the Gronwall inequality, we havefor all . So, for any , , ifthen

From Theorems 2.6 and 3.5, Lemmas 3.6 and 3.7 in this article, and [7, Theorem 3.1], we obtain.

Theorem 3.8. The semigroup determined by problem (1.3) with (H1)-(H2) possesses anexponential attractor on : whose fractal dimension satisfieswhere is as (3.14), , is defined as in Section 2 and is as (3.15), .

Remark 3.9. Indeed, when , by Lemma 3.6, we easily know that has an exponential attractor of dimension zero on , which is an equilibrium point of problem (1.3) (the global attractor for ).

References

  1. P. W. Bates, K. Lu, and B. Wang, “Attractors for lattice dynamical systems,” International Journal of Bifurcation and Chaos, vol. 11, no. 1, pp. 143–153, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. S. Zhou, “Attractors for first order dissipative lattice dynamical systems,” Physica D, vol. 178, no. 1-2, pp. 51–61, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. B. Wang, “Asymptotic behavior of non-autonomous lattice systems,” Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 121–136, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. C. Zhao and S. Zhou, “Compact kernel sections for nonautonomous Klein-Gordon-Schrödinger equations on infinite lattices,” Journal of Mathematical Analysis and Applications, vol. 332, no. 1, pp. 32–56, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. A. Babin and B. Nicolaenko, “Exponential attractors of reaction-diffusion systems in an unbounded domain,” Journal of Dynamics and Differential Equations, vol. 7, no. 4, pp. 567–590, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. A. Eden, C. Foias, and V. Kalantarov, “A remark on two constructions of exponential attractors for α-contractions,” Journal of Dynamics and Differential Equations, vol. 10, no. 1, pp. 37–45, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. A. Eden, C. Foias, B. Nicolaenko, and R. Temam, “Inertial sets for dissipative evolution equations,” 1991, IMA preprint series. View at Google Scholar