Random dynamics of fractional nonclassical diffusion equations driven by colored noise

Abstract

The random dynamics in \begin{document}$ H^s(\mathbb{R}^n) $\end{document} with \begin{document}$ s\in (0,1) $\end{document} 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.