Dynamics for subcritical fractional nonclassical diffusion equations with nonlinear Wong-Zakai noise and delays

Abstract

The multi-scale stability of random attractors in the space of continuous functions from the delay interval into the fractional Sobolev space for a class of nonclassical diffusion equations on $ \mathbb{R}^n $ with time-delay and nonlinear Wong-Zakai noise is studied. First, we prove the existence of a unique pullback random attractor, which is a family of compact sets depending on two parameters: the step size of noise and the current time. Secondly, we study the semicontinuity of the pullback random attractor as the current time goes to positive infinity. Thirdly, we consider the semicontinuity of the pullback random attractor as the delay time approach zero. Finally, we prove the semicontinuity of the pullback random attractor as the step size of noise tends to positive infinity. In order to overcome the lack of compact Sobolev embeddings on $ \mathbb{R}^n $ and the weak dissipation of equations, we employ the spectral method and idea of uniform tail-estimates to prove that the solution operator is pullback asymptotic compactness over a time-uniformly tempered attracting universe.