Regularity of random attractors for a stochastic degenerate parabolic equations driven by multiplicative noise

We study the regularity of random attractors for a class of degenerate parabolic equations with leading term div(σ(x)∇u) and multiplicative noises. Under some mild conditions on the diffusion variable σ(x) and without any restriction on the upper growth p of nonlinearity, except that p > 2, we show the existences of random attractor in D1,20 (DN, σ) ∩ Lϖ(DN) (ϖ ∈ [2, 2p – 2]) space, where DN is an arbitrary (bounded or unbounded) domain in N, N ≥ 2. For this purpose, some abstract results based on the omega-limit compactness are established.