This paper is concerned with the asymptotic behavior of solutions of the damped non-autonomous stochastic wave equations driven by multiplicative white noise. We prove the existence of pullback random attractors in $H^1(\mathbb{R}^n) \times L^2(\mathbb{R}^n)$ when the intensity of noise is sufficiently small. We demonstrate that these random attractors are periodic in time if so are the deterministic non-autonomous external terms. We also establish the upper semicontinuity of random attractors when the intensity of noise approaches zero. In addition, we prove the measurability of random attractors even if the underlying probability space is not complete.

The existence of a random attractor in <!-- MATH $H^1(\mathbb{R}^3) \times L^2(\mathbb{R}^3)$ --> is proved for the damped semilinear stochastic wave equation defined on the entire space <!-- MATH $\mathbb{R}^3$ --> . The nonlinearity is allowed to have a cubic growth rate which is referred to as the critical exponent. The uniform pullback estimates on the tails of solutions for large space variables are established. The pullback asymptotic compactness of the random dynamical system is proved by using these tail estimates and the energy equation method.

This paper is concerned with the asymptotic behavior of the solutions of the fractional nonclassical diffusion equations driven by nonlinear colored noise defined on the entire space . We first establish the existence of energy equations for the solutions in with , and then prove the existence and uniqueness of pullback random attractors in when the nonlinear drift and diffusion terms have polynomial growth of arbitrary order. In addition, for linear additive noise, we show the upper semi-continuity of these attractors as the correlation time of the colored noise approaches zero. The idea of energy equations due to Ball is employed to establish the pullback asymptotic compactness of the solutions in in order to overcome the weak dissipativeness of the equation as well as the non-compactness of Sobolev embeddings on unbounded domains. The result of this paper is new even in the space when the fractional Laplace operator reduces to the standard Laplace operator.