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 D 0 1 , 2 ( D N , σ ) ∩ L ϖ ( D N ) for any ϖ ∈ [ 2 , 2 p - 2 ] , where D N is an arbitrary (bounded or unbounded) domain in R N , N ⩾ 2 .