Random Attractor Of Stochastic Heat Equation With Dynamical Boundary Conditions

Abstract This work is concerned with the long time dynamical behavior of a stochastic heat equation with a dynamical boundary condition. We first analyze the following problems: nonlinear heat equation, dynamical boundary, the white noise in the system, and the white noise in the boundary, and then derive some of their characteristics and relationship. Then according to these data, we will study the spectrum of the operator and the effect of white noise for the stochastic system in some appropriates paces. After that, we will apply partial differential equation, functional analysis, and stochastic analysis to establish the priori esti-mates of solutions for the stochastic system in two classes of spaces, which finally yields the existence of the random attractor of the system.The first chapter introduces the theory of dynamical system and the relative knowledge and background, especially for the stochastic dynamical system. Then we narrow down to the Brownian motion, the basic theory of stochastic analysis and so on. Also we give out the function spaces and some inequalities used here.The second chapter states the stochastic nonlinear heat equation with dynamical bound-ary condition. And we introduce the background and the known studies, next we give some preliminaries including estimating the nonlinearities f(u) and g(u). Using an appropriate O-U process, we transfer the original system (Stochastic partial differentia equations) into a random system (Random partial differential equations), then establish the existence and uniqueness by Galerkin method which implies the existence of the random dynamical system. The third chapter shows that the absorbing sets are established in Η and in ν such that the random attractor exists in ν.The final chapter expresses the next research plan and our future work.