Random Attrator For A Class Of Abstract Stochastic Evolution Equations

We consider the longtime dynamical behavior of the following abstract stochastic evolution equations with mulcative noise on a separable Hilbert space H Where A is an nonlinear operator which satisfies the standard of monotonicity and coercivity conditions, s, n are positive constants,tN is a standard two-sided real valued Wiener process.Firstly, we present some concepts and the main results of the theory of random dynamical systems. Secondly we prove the existence of the solution for the above nonlinear stochastic evolution equations by using the method of variational approach to the equations.Thirdly, we use the uniform estimates method to prove the existence of random absorbing set of the above equation. Finally, we use the compact embedding theorem and the limit of the approximate norm to prove the pullback asymptotic compactness of the random dynamical system. From these results, we obtain the existemce of the ramdom attractor for the above abstract stochastic evolution equations. The results can be applied to various types of nonlinear SPDE, such as random nonlinear reaction diffusion equations with mulcative noise,stochastic Laplasse equations and stochastic porous media equations.