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.

The random dynamics in $ H^s(\mathbb{R}^n) $ with $ s\in (0,1) $ is investigated for the fractional nonclassical diffusion equations driven by colored noise. Both existence and uniqueness of pullback random attractors are established for the equations with a wide class of nonlinear diffusion terms. In the case of additive noise, the upper semi-continuity of these attractors is proved as the correlation time of the colored noise approaches zero. The methods of uniform tail-estimate and spectral decomposition are employed to obtain the pullback asymptotic compactness of the solutions in order to overcome the non-compactness of the Sobolev embedding on an unbounded domain.

This paper is concerned with the asymptotic behaviour of solutions to a class of non-autonomous stochastic nonlinear wave equations with dispersive and viscosity dissipative terms driven by operator-type noise defined on the entire space $\mathbb {R}^n$ . The existence, uniqueness, time-semi-uniform compactness and asymptotically autonomous robustness of pullback random attractors are proved in $H^1(\mathbb {R}^n)\times H^1(\mathbb {R}^n)$ when the growth rate of the nonlinearity has a subcritical range, the density of the noise is suitably controllable, and the time-dependent force converges to a time-independent function in some sense. The main difficulty to establish the time-semi-uniform pullback asymptotic compactness of the solutions in $H^1(\mathbb {R}^n)\times H^1(\mathbb {R}^n)$ is caused by the lack of compact Sobolev embeddings on $\mathbb {R}^n$ , as well as the weak dissipativeness of the equations is surmounted at light of the idea of uniform tail-estimates and a spectral decomposition approach. The measurability of random attractors is proved by using an argument which considers two attracting universes developed by Wang and Li (Phys. D 382: 46–57, 2018).