Existence of attractors for stochastic diffusion equations with fractional damping and time-varying delay

Abstract

This paper deals with the well-posedness and existence of attractors of a class of stochastic diffusion equations with fractional damping and time-varying delay on unbounded domains. We first prove the well-posedness and the existence of a continuous non-autonomous cocycle for the equations and the uniform estimates of solutions and the derivative of the solution operators with respect to the time-varying delay. We then show pullback asymptotic compactness of solutions and the existence of random attractors by utilizing the Arzelà–Ascoli theorem and the uniform estimates for the derivative of the solution operator in the fractional Sobolev space Hα(Rn), with 0 < α < 1.