Higher-order robust attractors for stochastic retarded degenerate parabolic equations

Abstract

We study the existence, measurability and time-dependent property of pullback random attractors in the higher-order space for stochastic degenerate parabolic equations with variable delay defined on R n . Let X, Z be the square integrable space and the p-times (p > 2) integrable space, respectively. We first prove the existence of a unique pullback ( X ϱ , Z ϱ ) -random attractor A with X ϱ = C ( [ − ϱ , 0 ] ; X ) and Z ϱ = C ( [ − ϱ , 0 ] ; Z ) , and establish the forward compactness, measurability and long-time stability of the bi-spatial attractor A in the higher-order space Z ϱ . We then investigate the higher-order delay-free stability of A in Z ϱ , more precisely, the upper semicontinuity of A under the topology of Z ϱ as the memory time tends to zero is established.