The Existence Of Random Attractor For A Class Of Stochastic Strongly Damped Wave Equation

Let U???Rn be a bounded open set with a smooth boundary aU.We consider the following strongly damped stochastic wave equation with additive noise on U x?0,+??:u_tt+u_t +f?u?-?u+a?-??^?ut= g + sum from j=1 to m( h_jdW_j,? ??0,1]?.Where ? is the Laplacian with respect to the variable x? U,u = u?x,t?is a real function of x?U and t ? 0,a>0 is the strong damping coefficient,g ? H¹?U?is a given external force,f is a nonlinear term.For the above equation,we consider the Neumann boundary condition on aU.It's well known that the differential operator A =-? with the Neumann boundary condition is not positive in general Sobolev space.As a result,when f = 0,g =0,h_i = 0,i =1,2,…,m,the above equation does not have a attractor.In this paper,we investigate the existence of a random attractor for the random dynamical system associated with the above equation in a orthogonal complement space of a 1-dimensional space.To do this,firstly,we prove that the above equation generates a random dynamical system in a proper functional space.Our method is to convert the above strongly damped wave equation to an evolution equation of the first order by replacing the variables,and then turn into equivalent deterministic dynamical systems with random parameters.There for we get a random dynamic system.Secondly,we prove that the random dynamic system has a compact attracting set.Finally,we prove the compact attracting set is tempered and then we get the existence of a unique random attractor of the random dynamical system.