Asymptotic behavior of stochastic discrete wave equations with nonlinear noise and damping

Abstract

In this article, we study the asymptotic behavior for a class of discrete wave equations with nonlinear noise and damping defined on a k-dimensional integer set. The well-posedness of the system is established when the nonlinear drift function and the nonlinear diffusion term are only locally Lipschitz continuous. The mean random dynamical system associated with the non-autonomous system is shown to possess a unique tempered weak pullback mean random attractor in L2(Ω,F,ℓ2×ℓ2). The existence of invariant measures for the autonomous system is also derived by using the Krylov–Bogolyubov method. The difficulty in proving the tightness of a family of distribution laws of the solutions is overcome by using the idea of uniform estimates on the tails of solutions.