Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise

Shengfan Zhou,Zhaojuan Wang

doi:10.3934/dcds.2018210

Abstract

In this paper, we first establish some sufficient conditions for the existence and construction of a random exponential attractor for a continuous cocycle on a separable Banach space. Then we mainly consider the random attractor and random exponential attractor for stochastic non-autonomous damped wave equation driven by linear multiplicative white noise with small coefficient when the nonlinearity is cubic. First step, we prove the existence of a random attractor for the cocycle associated with the considered system by carefully decomposing the solutions of system in two different modes and estimating the bounds of solutions. Second step, we consider an upper semicontinuity of random attractors as the coefficient of random term tends zero. Third step, we show the regularity of random attractor in a higher regular space through a recurrence method. Fourth step, we prove the existence of a random exponential attractor for the considered system, which implies the finiteness of fractal dimension of random attractor. Finally we remark that the stochastic non-autonomous damped cubic wave equation driven by additive white noise also has a random exponential attractor.

Highlights

It is known that the existence of attractor and the estimate of its dimension are two main topics in studying the asymptotic behavior of infinitedimensional dynamical systems [2, 3, 9, 26, 32, 39, 41, 43, 50]
We will show that the random attractors of (1) tends to the pullback attractor of deterministic nonautonomous system (1)a=0 as a → 0 in the sense of Hausdorff semi-distance between two subsets of phase space. (III) Study the regularity of random attractor by constructing a compact measurable tempered attracting set through a recurrence method and prove the boundedness of random attractor in higher regular space [H2(U ) ∩ H01(U )] × H01(U ) for the cocycle Φ. (IV) Prove the existence of a random exponential attractor for a continuous cocycle Φ in H01(U ) × L2(U ) when f satisfies ( A1)-(A2) by applying our new criterion, which implies the finiteness of fractal dimension of random attractor for (1)
The continuous property (v) of {E(τ, ω)}τ∈R,ω∈Ω follows from Theorem 3.1 in [20] and condition (H4)

Summary

Introduction

It is known that the existence of attractor and the estimate of its dimension are two main topics in studying the asymptotic behavior of infinitedimensional dynamical systems [2, 3, 9, 26, 32, 39, 41, 43, 50]. We establish some sufficient conditions for the existence and construction of a random exponential attractor for a continuous cocycle on a separable Banach space. (H1) there exists a family of tempered closed random subsets {χ(τ, ω)}τ∈R,ω∈Ω of X such that for any τ ∈ R and ω ∈ Ω, (h11) the diameter ||χ(τ, ω)||X of χ(τ, ω) is bounded by a tempered random variable Rω (independent of τ ), i.e., supτ∈R supu,v∈χ(τ,ω) ||u − v||X ≤ Rω < ∞, where Rθtω is continuous in t for all t ∈ R; (h12)χ(τ, ω) is positively invariant with respect to {θt}t∈R in the sense that Φ(t, τ − t, θ−tω)χ(τ − t, θ−tω) ⊆ χ(τ, ω) for all t ≥ 0;.

Write n n

Write Er