Asymptotic dynamics of stochastic delay nonclassical diffusion equations on unbounded domains

The goal of this paper is to investigate three kinds of stability of pullback random attractors (PRAs) for stochastic nonclassical diffusion equations with distributed delay and constant delay perturbed by operator-type noise defined on R-n. We first prove the existence, uniqueness, backward compactness and backward longtime stability of PRAs for this equation. We then establish the zero-memory stability of PRAs. Finally, we study the asymptotically autonomous stability of PRAs. Due to the problem of the non-compactness of Sobolev embeddings on R-n, we use the backward tail-estimates method and spectrum decomposition technique to prove the backward asymptotic compactness of solutions.