Stationary approximations of stochastic wave equations on unbounded domains with critical exponents

Abstract

This paper is concerned with the stationary approximations of the stochastic wave equations defined on Rn with critical exponents. For a class of nonlinear diffusion terms, we prove the existence and uniqueness of tempered pullback attractors of the random wave equations driven by a stationary process as an approximation to the white noise. For a linear multiplicative noise, we prove the upper semicontinuity of these attractors as the step size of the approximations approaches zero. The asymptotic compactness of the solutions on Rn is established by the idea of energy equations and the uniform estimates on the tails of the solutions.