Random attractors for second-order stochastic lattice dynamical systems

https://doi.org/10.1016/j.na.2009.06.094Get rights and content

Abstract

In this paper, the asymptotic behavior of second-order stochastic lattice dynamical systems is considered. We firstly show the existence of an absorbing set. Then an estimate on tails of the solutions is derived when the time is large enough, which ensures the asymptotic compactness of the random dynamical system. Finally, the existence of the random attractor is provided.

Introduction

Lattice dynamical systems (LDS) arise from a variety of applications in science and engineering where the spatial structure has a discrete character. Among such examples are propagation of nerve pulses in myelinated axons, electrical engineering, pattern recognition, image processing, chemical reaction theory, etc. On the other hand, lattice systems also arise from spatial discretizations of partial differential equations. In this respect, we refer the reader to Hale [1] for more details. In the past few decades, much attention has been paid to the research on LDS. For example, the chaotic properties of solutions were examined in Refs. [2], [3], [4]. As for the traveling wave solutions, we refer the readers to Refs. [5], [6], [7] and the references therein.

Recently, the existence of attractors for LDS has been considered. In Ref. [8], Bates et al. established the first result on the existence of the global attractors for LDS. Since then, much work has been done for either the first-order or the second-order LDS, see Refs. [9], [10], [11], [12], [13], [14] and the references therein.

As pointed out by Bates et al. [15], stochastic lattice dynamical systems (SLDS) arise naturally in a wide variety of applications where the spatial structure has a discrete character and uncertainties or random influences, called noises, are taken into account. These random effects are not only introduced to compensate for the defects in some deterministic models, but are also rather intrinsic phenomena. Random attractors for random dynamical systems (RDS) were first introduced by Crauel and Flandoli [16], [17] and Schmalfuss [18], with notable developments given in [19], [20], [21] and in the references there among many others. Quite recently, there have been some initial studies on the existence of global random attractor for SLDS. In Ref. [15], the existence of global random attractors was investigated for a kind of first-order LDS driven by stochastic processes on lattice Z. The results of Ref. [15] have been extended by Lv et al. [22] to a generalized system with stochastic disturbances on the lattice Zk, and to stochastic discrete Ginzburg–Landau equations [23]. The random attractors for stochastic FitzHugh–Nagumo equations in an infinite lattice with additive white noise have been considered by Huang [24].

To the best of our knowledge, there are no results on the existence of random attractors for second-order SLDS. On the basis of this, this article is devoted to the discussion of this problem. We will further develop the idea of “tail estimates” of [25], [15] to prove the asymptotic compactness for the random dynamical system. The main methods used in the proofs of the theorems are motivated by the papers [25], [15], [14], [9], [26].

This paper is organized as follows. In Section 2, we introduce basic concepts concerning random dynamical systems and global random attractors. In Section 3, we show the existence and uniqueness of the solution to the second-order SLDS. Then it generates an infinite dimensional random dynamical system. The existence of the global random attractor is given in Section 4.

Section snippets

Preliminaries

In this section, we recall some basic concepts related to random attractors for stochastic dynamical systems. The reader is referred to [16], [15], [24] for more details.

Let (X,X) be a separable Hilbert space with Borel σ-algebra B(X), and let (Ω,F,P) be a probability space.

Definition 2.1

(Ω,F,P,(θt)tR) is called a metric dynamical system if θ:R×ΩΩ is (B(R)×F,F)-measurable, θ0 is the identity on Ω, θs+t=θtθs for all s,tR and θtP=P for all tR.

Definition 2.2

A set AΩ is called invariant with respect to (θt)tR if for

Second-order SLDS

Denote by Z the set of integers. Let kN be a fixed positive integer. We denote p(p1) defined by p={u|u=(ui)iZk,i=(i1,i2,,ik)Zk,uiR and iZk|ui|p<+}, with the norm p given by up=(iZk|ui|p)1/p, for any u=(ui)iZk and v=(vi)iZkp.

In particular, 2 is a Hilbert space with the inner product (,) and norm given by (u,v)=iZkuivi,u=(iZk|ui|2)1/2, for any u=(ui)iZk and v=(vi)iZk2.

Define a linear operator A acting on 2 in the following way: for any u=(ui)iZk2, i=(i1,i

Random attractor

For our purpose, we need to convert the stochastic Eq. (3.1) with a random term into a deterministic one with a random parameter. To this end, we consider the stationary solutions of the following equation: dy+δydt=dW(t), where δ=min{ν,c1ε2p+2},andν=α1λ4λ+α22(α2+4λ+α22).

The solution to (4.1) is given by y(θtω)=δ0eδsθtω(s)ds,tR. Note that there exists a θt-invariant set ΩΩ of full P measure such that

(1) the mapping sy(θsω) is continuous for each ωΩ,

(2) the random variable y(θtω) is

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The work is supported by National Natural Science Foundation of China under Grant 10671133.

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