Random Attractors And Estimate Of The Dimension For Stochastic Sine-Gordon Equations With Additive White Noises

This paper studies the existence of random attractor of a Sine-Gordon equation with additive white noise and nonlinear damping(on velocity) and estimate its dimension. The paper has three parts as the following.The first chapter is a summary of the paper. The history of random dynamical systems and the background of the paper are presented. The results and some preliminaries of the paper are in the chapter.The second chapter investigates the long-time behavior of the solution for of a Sine-Gordon equation with additive white noise and nonlinear damping(on velocity) under smooth boundary conditions. By several transformations, the stochastic differential equation is transformed into a random differential equation with no noise, in which the sample is regarded only as a common parameter. By the deterministic theory, the random equation implies the existence a RDS determined by the stochastic equation. A positively definite matrix operator is introduced to obtain the existence of absorbing set and the asymptotical compactness of the RDS, which lead to the existence of random attractor.The aim of the third chapter is to estimate the Hausdorff dimension of the random attractor. Firstly, it is checked that the RDS is Lipschitz continuous and qusi-differentiable on the attractor. Secondly, an upper bound of the Hausdorff dimension of the random attractor is present.