Random uniform exponential attractor for stochastic non-autonomous reaction-diffusion equation with multiplicative noise in ℝ3

Zongfei Han,Shengfan Zhou

doi:10.1515/math-2019-0099

Abstract

Abstract We first introduce the concept of the random uniform exponential attractor for a jointly continuous non-autonomous random dynamical system (NRDS) and give a theorem on the existence of the random uniform exponential attractor for a jointly continuous NRDS. Then we study the existence of the random uniform exponential attractor for reaction-diffusion equation with quasi-periodic external force and multiplicative noise in ℝ3.

Highlights

The concept of the exponential attractor was introduced by A
We rst introduce the concept of the random uniform exponential attractor for a jointly continuous non-autonomous random dynamical system (NRDS) and give a theorem on the existence of the random uniform exponential attractor for a jointly continuous NRDS
We study the existence of the random uniform exponential attractor for reaction-di usion equation with quasi-periodic external force and multiplicative noise in R

Summary

Introduction

The concept of the exponential attractor was introduced by A. Eden et al, which is a compact positively invariant set with nite fractal dimension and attracts trajectories exponentially fast, see [1]. It can describe the asymptotic behavior of trajectories of autonomous dynamical system or the solutions to dissipative autonomous evolution equations. An alternative extension to the case of non-autonomous dynamical system of the concept of the exponential attractor was based on the work [7] of Chepyzhov and Vishik (see [8, Chapter 4]), in which they introduced an approach to study a family of non-autonomous evolution equations of the form du dt

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Existence of random uniform exponential attractors

Taking the inner product with ξ

Note that where