Limiting Behavior For Stochastic Heat Equations With Memory On Thin Domains

This paper deals with the dynamical behavior of a stochastic heat equation driven by additive noise defined on thin domains.We prove the existence and uniqueness of random attractors for the equation in an(n+1)-dimensional narrow domain,and then establish the upper semicontinuity of these attractors when a family of(n+1)-dimensional thin domains collapses into an rn-dimensional domain.Due to the fact that the memory term includes the whole past history of the phenomenon,we are not able to prove compactness of the generated RDS,but its asymptotic compactness can be proved by the splitting method.This paper is organized as follows:In Chapter 1,we introduce some research backgrounds on reaction-diffusion equation,then we state the main work of this paper.In Chapter 2,we introduce some concepts and lemmas on random dynamical system and random attractor.In Chapter 3,we study stochastic heat equations with memory on thin domains.We first prove the existence of random attractor in an(n+1)-dimensional thin domain,then we obtain the existence of random attractor in an n-dimensional domain,finally we discuss the upper-semicontinuity of random attractors on thin domain.