Random Attractors Of Wave Equations With Multiplicative Noise

In this paper, we consider the wave equations with Dirichlet bounded under multi-plicative noise: where Q denote an open bounded set of R with smooth boundary г1ui(x,t)=ui,i=1,2, x∈Ω,t∈R are real valued functions. sin(u1+u2), sin (u1-u2)∈Cb(R2, R).Cb(X,Y) is the set of continuous bounded functions from X into YMultiplicative noise is described by O-U process. Two sided Wiener process on the probability space (Ω,F,P), where Ω={ω∈C(R,R):ω(0)=0},F is the Borel Sigma-algebra induced by the compact-open topology of Q, P is a Wiener measure. We associate with the initial conditions: ui(x,t)=0,г×R,i=1,2, ui(x,Ï„)=ui0∈H01(Ω), ui’(x,Ï„)=ui’(Ï„)∈L2(Ω).The existence and uniqueness of solution of the equations with the boundary initial conditions is proved. The existence of the random attractors of the random dynamical system (RDS) generated by the unique solution is also proved. At first, there is no stochastic differential appears for introducing O-U process in the equations. The unique solution ψ=ψ(t,w,ψ(Ï„)) of equations by using the Galerkin approximate method is obtained, which generates a RDS S(t,w). The existence of random attractors of the RDS is established by using the Sobolev compact embedding theorem. This paper is divided into four chapters.In Chapter One:The background on the theory of RDS and global attractors,the study on global attractor of the wave equations with multiplicative noise are introduced. And some preliminary definitions,results and inequalities are presented.In Chapter Two:The stochastic differential will not exist. The equations can be solved pathwise by writing in an abstract operator form and using the Galerkin approxi-mate method.Moreover,the unique solution of equations will generate a RDS.In Chapter Three:The existence of random attractors of the RDS is established by using the Sobolev compact embedding theorem.Firstly,a random absorbing set B(0,γ(ω))for RDS S(t,w)is got.Theorem3.1.2There exist random variables β(t,w)=γ1(t,w)+γ2(t,w) and T>0. Let X,Y,ψ be a solution of the equations(2.3.2),(2.3.3),(2.3.1)with the initial values X(Ï„)=(u10,u1’(Ï„)+εu10)Ï„,Y(Ï„)=(u20,u2’(Ï„)+εu20)T,ψ(Ï„)=(u0,u’(Ï„)+εu0)T,for any bounded set B’(?) E’ and any t>T,there is‖ψ(0,t,w;ψ(Ï„))‖E’,≤γ(t,w). Besides,there are the mappings tâ†'γ3(t,w) and tâ†'γ4(t,w)tempered in R,for any fixed t and small enough Ï„,there are‖x(t,w;x(Ï„))‖E≤γ3(t,w),‖y(t,w;y(Ï„))‖E≤γ4(t,w).Theorem3.2.2There are random variables γ’(t,w)=γ6(t,w)+γ7(t,w)>0,such that the solution of equation(2.3.1)satisfies:So we obtain the random attractor of the RDS by using the Sobolev compact em-bedding theorem.In Chapter Four:More study about this equations are discussed.