The Asymptotic Behavior For Stochastic Ginzburg-Landau Equation On Bounded Domains And Lattices

Random attractor is the essential concept in describing the asymptotic be-havior of random dynamical system. The present paper is devoted to the existence of the random attractors of stochastic Ginzburg-Landau equations.In Chapter 1, we take a brief view of the background of random dynamical systems and the derivation for our general Ginzburg-Landau equations. Here, we present an overview of our main work.In Chapter 2, we give the introduction of the basic concepts of random attractors which contains some inequalities that will be used in the following.In Chapter 3, it is devoted to proving the existence of the random attractors of stochas-tic Ginzburg-Landau equation with multiplicative noise defined on a bounded domain. By establishing the uniform estimate of the solution of the equation, we deduce the existence of a random bounded absorbing set.In Chapter 4, we firstly prove the existence of a pullback attractor of a stochastic Ginzburg-Landau equation with multiplicative noise on unbounded domains. Then we de-duce a random differential equation with random coefficient from the stochastic equation by Ornstein-Uhlenbeck transformation. We convert the equation into a random dynamical sys-tem and demonstrate the asymptotic compactness by using uniform estimates for far-field values of solutions.In Chapter 5, we prove the existence of a global random attractor of a stochastic discrete complex Ginzburg-Landau equation with multiplicative noise. Due to the lack of smoothness on the infinite lattice, we prove asymptotic compactness by using uniform a priori estimates for the tail of solutions and obain the existence of a compact random attractor.In Chapter 6, we summarize our results and propose some works for future consideration.