The Exponential Stability For A Class Of Stochastic Partial Differential Equations

In this paper,we present the exponential stability for a class of nonlinear stochastic partial differential equations with local monotone coefficients and driven by multiplica-tive noise under the generalized variational framework.Firstly,we study the existence and uniqueness of the stationary solution X_∞to the corresponding deterministic equa-tion,and prove the exponential stability of the stationary solution.Then we assume that the stationary solution X_∞ is also the stationary solution to this class of stochastic partial differential equations when the noise term satisfies certain conditions.On this basis,we further study two stabilities of this equation which satisfies local monotonic condition and growth condition.In addition,we obtain that the equation which dissat-isfies the mean square exponential stability still has pathwise exponential stability by using a multiplicative It? noise of sufficient intensity.This paper is divided into the following four chapters:In Chapter 1,we introduce the background and research status of stochastic partial differential equations and exponential stability problem,and briefly describe the main contents of this paper.In Chapter 2,the basic knowledge required for this study is given,including some basic concepts and classical variational framework.In Chapter 3,the mean square exponential stability and pathwise exponential sta-bility for a class of stochastic partial differential equations are proved.In Chapter 4,the main results of this paper are applied to the stochastic 2D hydro-dynamical type systems.