Limiting Behavior For A Class Of Delay Stochastic Evolutionary Equations With Multiplicative Noise

In this thesis,we consider the existence,smoothness and convergence of ran-dom center manifolds and unstable manifolds for a class of stochastic evolutionary equations with delay driven by multiplicative white noise.First,the Ornstein-Uhlenbeck process,It(?)formula and random transformation T are used to transform the delay stochastic evolutionary equations with multiplicative noise into random partial differential equations with delay,and then the random dynamical system generated by the solutions of the equations is obtained.Next,in order to show the existence of random center manifolds and unstable manifolds for the equations with delay,we need to transform the nonlinear terms with delay,introduce new spectral gap conditions and deal with the coefficients generated by the effect of delay.Finally,Lyapunov-Perron method is used to investigate the smoothness,Lipschitz convergence andC~ksmooth convergence of center manifolds and unstable manifolds for the equations with delay.This thesis is organized as follows:In Chapter 1,the research background of stochastic evolutionary equations with delay is introduced,and the main work of this thesis is briefly stated.In Chapter 2,the concepts and theorems of random dynamical system,random invariant manifolds and each operator in the equation are given.In Chapter 3,the existence,smoothness and convergence of random center manifolds for a class of stochastic evolutionary equations with delay driven by multiplicative white noise are proved.In Chapter 4,the existence,smoothness and convergence of random unstable manifolds for a class of stochastic evolutionary equations with delay driven by multiplicative white noise are considered.In Chapter 5,the example illustrating the results obtained above is given.In Chapter 6,the main results are summarized and some problems for future research are proposed.