Regularity And Upper Semi-continuity Of Random Attractors For Stochastic Reaction-Diffusion Equation With Multiplicative Noise On P-times Integrable Space

This paper mainly studies the reaction diffusion equations with multiplicative per-turbation and the properties of its stochastic dynamic system and random attractor. Based on the equation of the unique solution one can generate a random dynamic system and its (L2, LP)-random attractor.By uniformly asymptotic estimation, we prove that when disturbance quantity belong to the finite range, the random dy-namical system is the upper continuous in any non-negative point. We investigate the following equation: where x ∈ Rn, t≥ 0, u= u(x, t) with the initial condition:u(x,0)= u0(x). λ is a positive constant,ε≥0, g ∈ L2(Rn)∩Lp(Rn).W(t) is a two-sided real-valued Wiener process on a probability space (Ω,F,P) and f is a nonlinear function satis-fying the following conditions: where α1, α2 and β are positive constants, ψ1 ∈ L1(Rn) ∩ L∞(Rn), ψ2 ∈ L2(Rn) ∩ Lq(Rn) with 1/p+1/q= 1, and ψ3 ∈ L2(Rn). This article is divided into four chapters altogether:In the first chapter, we mainly describe the concept of random attractor and stochastic dynamical systems, and its important significance for studies on stochastic partial differential equations, We then introduced the present domestic and foreign research status of stochastic partial differential equations, especially emphasizes the significance of this paper which has been done, At the last,we write briefly about the research contents and methods of this article in this paper.In the second chapter, we introduce a few basic definitions and some proved ab-stract results for random attractor and random dynamical system which are required for this article.In the third chapter, through altering stochastic reaction diffusion equations into a determined partial differential equation, By the existence and uniqueness of solution of the function we can generate a random dynamical system and (L2,Lp) random attractor. Finally we can conclude that the system satisfies the upper semi-continuity under some conditions.In the fourth chapter, according to the determination conditions of upper semi-continuity of stochastic dynamic system condition, we first prove random dynamical system is absorbing in L2, Lp (lemma 4.1, lemma 4.2). To prove the asymptotic compactness of the system on Lp over any finite interval (lemma 4.6), we must prove three auxiliary lemma (lemma 4.3, lemma 4.4, lemma 4.5), finally, we prove that the random dynamic system in L2 is convergent at any non-negative point. Thus we pass all proofs.