Upper Semicontinuity Of Random Attractors For The Stochastic Sine-Gordon Equations

In this paper,we study the upper semicontinuity of random attractors for the stochastic Sine-Gordon equations defined in the(H01((Ω)×L2(Ω))2.We will inves-tigate the following generalized Sine-Gordon equations with white noise: with the initial conditions:Where E is a small positive parameter.Ω(?)R is a bounded set with a smooth boundary (?)Ω,uit=(?)ui/(?)t,ui=ui(x,f),i=1,2 are real-valued functions defined on Ω×[τ,+∞),τ∈R. fi(x)∈H2(Ω)∩H01(Ω) is time-independent. ω(t) is a two-sided real-valued Wiener process on a probability space(θ,F,P).The outline of this paper is as follow:In the first chapter,we introduce the background and research status of ran-dom dynamical system,random attractor and the meaning of researching the upper semicontinuity of the stochastic Sime-Gordon equations.And we also show the basic random attractors theory which will be used.In the second chapter, we get the existence of a unique solution of stochastic Sine-Gordon equations by using O-U process to eliminate the random disturbance of the equations, which generates a continuous random dynamical system.In the third chapter, by the use of the uniform estimates of solutions for the stochastic Sine-Gordon equations, we prove that the existence of bounded absorbing set and uniform asymptotically compactness of the system in (H01(Ω)×L2(Ω))2. And then we get the unique random attractor of the system in (H01(Ω)×L2(Ω))2In the forth chapter, by the convergence of random dynamical system in (H01(Ω)× L2(Ω))2,the upper semicontinuity of random attractors for the stochastic Sine Gordon equations is obtained.