Global Attractor For Stochastic Strongly Damped Semilinear Wave Equations On Unbounded Domain

In this paper,we study the asymptotic behavior of the stochastic semilinear strongly damped wave equation with nonlinear weak damping term on the unbounded domain R3:The nonlinear term g satisfies the critical growth condition,the external force term f ?L2(R3),Wj(j= 1,2,...,m)are independent two-sided real-valued Winer process.Such equations model various processes with frequency depending attenuation,such as acoustics,structural vibration,seismic wave propagation,viscous dampers in seismic isolation of buildings,anomalous diffusions occurring in porous media,just to mention a few.Long-time behavior for analogous equations on bounded domain have been investigated by many authors in recent years.The existence of global attractor and exponential attractor as well as the boundedness of the fractal dimension and Housdorff dimension of attractor have been proved on bounded domain in many papers.For the unbounded domain,due to the lack of compactness,there arc many difficulties in the study of the existence of attractors for wave equations with strong damping term and random term.In the weakly damped case,Feireisl has studied the existence of global attractors of wave equations with the decomposition method of solutions.Feireisl's technique relies on the fact that the propagation speed of initial disturbances is finite.In our situation,Feireisl's approach does not work.Indeed,the strongly damped wave equations partially have the parabolic characters.Hence the equations have more regularization,but also an infinite propagation speed of initial disturbances.M.Conti,V.Pata and M.Squassina have introduced a method to solve the existence of the global attractor of the strongly damped wave equation with nonlinear weak damping term on the unbounded region by using the decomposition method of appropriate cut-off functions.In this paper,we have encountered similar problems in proving the existence of the global attractor,and we use the decomposition method to prove the existence of the attractor of the nonlinear wave equation,which has both random term and strong attenuation term.The aim of this paper is to extend this result to the case of stochastic strongly damped wave equations with nonlinear damping on unbounded domain.We first prove the existence of weak solutions and bounded absorption sets,and then prove the asymptotic compactness by using the decomposition method of appropriate cut-off functions.