Compactness and attracting of random attractors for non-autonomous stochastic lattice dynamical systems in weighted space ${\ell }_{\rho }^p$

Abstract

In this article, some sufficient conditions on the regularity of random attractors are first provided for general random dynamical systems in the weighted space ${\ell }_{\rho }^p(p>2)$ of infinite sequences. They are then used to study the asymptotic dynamics of a class of non-autonomous stochastic lattice differential equations with spatially valued additive noises. The existences of tempered random attractors for this in both spaces ${\ell }_{\rho }^2$ and ${\ell }_{\rho }^p$ are proved respectively, which implies that the obtained ${\ell }_{\rho }^2$-random attractor is compact and attracting in the topology of ${\ell }_{\rho }^p$ space. To solve this, a common embedding space of ${\ell }_{\rho }^2$ and ${\ell }_{\rho }^p$ is constructed and some new estimates are also developed here.

Introduction

The theory of attractors is a powerful tool to depict the asymptotic dynamics of an infinite-dimensional system. By definition, a compact attractor is an invariant and attracting subset, which is often finite-dimensional despite the phase space may be infinite-dimensional [29]. In order to capture its nice topological properties, it is meanful to study the regularity of attractors. There are a large volume of literature regarding this topic, for example, [39] in the deterministic case (without white noises) and [25], [33] in the random case, and so on. Most of the crucial things of the regularity problems are the compactness and attracting properties of attractors in some non-initial spaces or terminate spaces [25], where we call the initial values located space as an initial space. However, to the best of our knowledge, there are few articles to discuss the regularity problem on lattice dynamical system generated by differential equations on infinite lattices, even in the deterministic case.

In this article, we investigate the asymptotic regularity of solutions for the following non-autonomous lattice differential equation perturbed by additive white noise:

$$\begin{array}{ccc}& & \frac{d{u}_i}{dt}+\lambda u_i-\nu (u_{i-1}-2u_i+u_{i+1})=f_i(t,u_i)+g_i\left(t\right)+a_i\frac{d{\text{w}}_i}{dt},i\in \mathbb{Z},t>\tau ,\tau \in \mathbb{R},\hfill \end{array}$$

with initial data

$$\begin{array}{c}\hfill u_i\left(\tau \right)=u_{i,\tau },i\in \mathbb{Z},\end{array}$$

where $u={\left\{u_i\right\}}_{i\in \mathbb{Z}}\in {\ell }_{\rho }^2,$$\mathbb{Z}$ is an integer set, ν and λ are positive constants, $f(t,u)={\left\{f_i(t,u_i)\right\}}_{i\in \mathbb{Z}}$ is a nonlinearity satisfying some dissipative conditions, $g\left(t\right)={\left\{g_i\left(t\right)\right\}}_{i\in \mathbb{Z}}$ is a given time-dependent sequence, $a={\left\{a_i\right\}}_{i\in \mathbb{Z}}$ is a constant sequence, $\text{w}\left(t\right)=\{{\text{w}}_i\left(t\right):i\in \mathbb{Z}\}$ are two-sided Brownian motions defined on a standard probability space. This system should be understood in the Stratonovich integral sense.

Lattice differential equations (LDEs) are a family of ordinary differential equations or difference equations indexed by points in a lattice, usually, a integer set. They arise from spatial discretization of continuum models, but more importantly, LDEs model the real problems where the spatial structure has a discrete character, such as chemical reaction, pattern formation, propagation of nerve pulse and so on, see e.g. [10], [18]. Due to the potential applications of lattice dynamical systems, LDEs have received wide attention over the past two decades. For example, the chaotic properties of solutions were investigated in [10], and the traveling wave solutions were obtained in [2], [18]. In the deterministic case, the existence of exponential attractors and global attractors are investigated in [1], [6], [31], [34] and the references therein. The study of random attractor is first introduced in [4] for LDEs with white noises as well as autonomous term, followed by a lot of developments in [7], [8], [20], [21], [22], [23]. When the stochastic LDEs has a time-dependent forcing, the existence of random attractor is established in [5], where the bounds of attractors from above and below are considered.

The theory of attractors for the deterministic partial differential equations has been developed intensively since late seventies of the last century, see Temam [29] for the autonomous equations and Carvalho et al. [9] for the non-autonomous ones, meanwhile, the notion of random attractor for the random dynamical systems (RDSs) was also initiated in [26], which captures the essential dynamics of realistic systems with possible extremely wide fluctuations. The theory of random attractors was further developed in [12], [13], [19], [27] to deal with the random systems with time-independent forcing. For the non-autonomous random dynamical systems, the existence of random attractors was established in [17], [32]. The theory on regularity of random attractors is also richly developed. For example, the existences of random attractors in the p-times integrable function spaces were investigated in [33] on bounded domains and in [24], [25], [36], [37], [38] on unbounded domains. As for the upper semi-continuity and fractal dimension of random attractors we may refer to [15], [24], [25], [35] and the references cited there. Recently, [28] studied the upper semi-continuity of pullback attractors for the lattice diffusion equation with delays.

In this article, we are interested in the regularity of random attractors for stochastic lattice dynamical systems (SLDSs) in infinite dimensional sequences space with a general weight including the usual weight functions. For this purpose, a general criterion on the regularity of random attractors in the general Banach space are first presented, which implies that the notion of K-pullback asymptotical compactness, which is weaker than the traditional $\mathfrak{D}$-pullback asymptotical compactness or $\mathfrak{D}$-omega-limit compactness [25], is sufficient to depict the compactness and attracting of random attractors in the non-initial spaces, see Theorem 2.7. Furthermore, a general standard on the regularity of random attractors in the weighted space ${\ell }_{\rho }^p(p>2)$ is also provided for the general SLDSs defined on the infinite integer set $\mathbb{Z},$ see Theorem 3.3. As an application, we study the regularity (mainly compactness and attraction) of random attractors for the non-autonomous stochastic system (1.1) and (1.2). We first prove that system (1.1) and (1.2) generates a random cocycle on ${\ell }_{\rho }^2$ space. Then, the asymptotical nullness of corresponding random cocycle in ${\ell }_{\rho }^p(p>2)$ is established by an induction technique, and by means of this and along with some absorption argument we show that the obtained ${\ell }_{\rho }^2$-random attractor is compact and attracting in the topology of ${\ell }_{\rho }^p,$ which indicates a strong asymptotic property of this system. All of these are based on a fundamental fact that the interaction space ${\ell }_{\rho }^2\cap {\ell }_{\rho }^p$ is a standard Banach space.

This paper is organized as follows. In the next section, we introduce some concepts of RDSs and derive an abstract result about the regularity of random attractors in the general Banach space. In Section 3, some sufficient conditions on the regularity of random attractors for the general RDSs are proved in the weighted space ${\ell }_{\rho }^p$ (p > 2). In Section 4, we give the assumptions on nonlinearity and non-autonomous term, and obtain a random cocycle for system (1.1) and (1.2). In Section 5, we obtain the random attractors for system (1.1) and (1.2) in weighted space ${\ell }_{\rho }^2,$ and show the regularity of the obtained attractors in ${\ell }_{\rho }^p$ (p > 2).