Regularity of random attractors for a degenerate parabolic equations driven by additive noises

We study the regularity of random attractors for a class of stochastic degenerate parabolic equations with the leading term involving a diffusion variable sigma which many be non-smooth or unbounded. Without any restrictions on the upper growth order p of the non-linearity, except that p >= 2, we show that the associated random dynamical system admits a unique compact random attractor in the space D-0(1,2) (D-N, sigma) boolean AND L-pi(D-N) for any pi is an element of[2, 2p - 2], where D-N is an arbitrary (bounded or unbounded) domain in R-N, N >= 2. (C) 2014 Elsevier Inc. All rights reserved.