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This paper is concerned with the stationary approximations of the stochastic wave equations defined on R n 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 R n is established by the idea of energy equations and the uniform estimates on the tails of the solutions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic behavior of the solutions of the nonautonomous fractional stochastic reaction-diffusion equations on R n with continuous (but not necessarily differentiable) nonlinear drift terms, which leads to the nonuniqueness of solutions and hence a multivalued random dynamical system with the solution operators of the equation. We first show the existence and uniqueness of random attractors for such a dynamical system and then establish the upper semicontinuity of these attractors as the intensity of noise approaches zero. The measurability of the random attractors is proved by the method of weak upper semicontinuity of multivalued functions, and the pullback asymptotic compactness of the system is derived by the idea of uniform estimates on the tails of the solutions. [ABSTRACT FROM AUTHOR]

This paper deals with the asymptotic behavior of solutions of parabolic equations on unbounded domains which contain stochastic terms as well as non-autonomous deterministic terms. We define a continuous random dynamical system for the equations in weighted Sobolev spaces that allow functions to have certain polynomial growth rate at infinity and hence include all bounded solutions. We prove pullback asymptotic compactness of solutions as well as the existence and uniqueness of tempered random attractors in the weighted spaces. The structures of the tempered attractors are fully characterized by tempered complete solutions. In the case where the non-autonomous deterministic terms are periodic in time, we show that the tempered random attractors are also periodic in time. To overcome the difficulty of non-compactness of Sobolev embeddings on unbounded domains, the idea of uniform pathwise estimates on the tails of solutions is employed to show asymptotic compactness. [ABSTRACT FROM AUTHOR]

Deals with the asymptotic behavior of solutions for the Klein-Gordon-Schrodinger equation. Proof for the existence of compact global attractors for the model in the space for each integer; Establishment of the continuity property of the solutions on initial data; Asymptotic compactness of solutions for each integer; Difficulty in obtaining the compactness for weakly dissipative equation.