Dynamics Of Fractional Stochastic Ginzburg-Landau Equation On Unbounded Domains Driven By Nonlinear Noise

This paper studies the dynamical behavior of fractional stochastic Ginzburg-Landau equations driven by nonlinear noise on unbounded domains.In this paper,we gradually prove the existence and uniqueness of the solution of the equation and the uniform estimation of the solution driven by regular additive noise,general additive noise,globally Lipschitz continuous noise and locally Lipschitz continuous noise,and then construct random mean dynamical system and prove the existence of weak pull-back random attractor.There are mainly the following difficulties in the process of proving the existence and uniqueness of the solution of the equation,the derivation of uniform estimation and the existence and uniqueness of random attractor.Firstly,the standard Sobolev embedding on unbounded domain is non-compactness.Secondly,random noise is nonlinear;Thirdly,the nonlinear noise term is locally Lipschitz continuous.For these difficulties,this paper adopts the tail uniform estimation method of the solution to prove the compactness of the random distribution of the solution,so as to overcome the non-compactness of the Sobolev embedding on unbounded domain.For the second difficulty,this paper adopts the concept of random mean dynamical system and weak pullback random attractor instead of the concept of path random dynamical system and mean attractor,which is generally applied to the stochastic equations driven by additive and multiplicative noise,instead of the nonlinear noise equations discussed in this paper.For the last difficulty,this paper uses a sequence of functions of the globally Lipschitz continuous sequence to approximate the locally Lipschitz continuous nonlinear noise term.The composition of the article is as follows:The first chapter is an introductory part,which mainly introduces stochastic dynamical system,the research background and significance of fractional stochastic Ginzburg-Landau equation as well as summarizes the main work of this paper.Chapter 2 is the preparatory knowledge part.This chapter gives the concept of weak pull-back attractor,inequalities and theorems.Chapter 3 deals with the existence and uniqueness of solution of the stochastic equation driven by regular additive noise,general additive noise,globally Lipschitz continuous noise and locally Lipschitz continuous noise,respectively,and derive the uniform estimation of the solution respectively.Finally,we prove the existence and uniqueness of the solution of the stochastic fractional Ginzburg-Landau equation driven by nonlinear noise.Chapter 4 constructs the random mean dynamical system of the equation and prove the existence and uniqueness of the weak pull-back attractor of the equation driven by nonlinear noise.Chapter 5 is the summary and prospect of this paper.