Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear colored noise on RN

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 R-N. We first establish the existence of energy equations for the solutions in H-s(R-N) with s is an element of (0, 1], and then prove the existence and uniqueness of pullback random attractors in H-s(R-N) 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 H-s(R-N) 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 H-1(R-N) when the fractional Laplace operator reduces to the standard Laplace operator.