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 σ which many be non-smooth or unbounded. Without any restrictions on the upper growth order p of the nonlinearity, except that p⩾2, we show that the associated random dynamical system admits a unique compact random attractor in the space D01,2(DN,σ)∩Lϖ(DN) for any ϖ∈[2,2p-2], where DN is an arbitrary (bounded or unbounded) domain in RN,N⩾2.