## On the existence of random attractors for a stochastic semilinear degenerate parabolic equation

I’m writing a paper which consider the regularity of random attractors for a class of stochastic degenerate parabolic equations of the form

$du - (div(\sigma(x)\nabla u) - f(u) - \lambda u)dt = gdt + \sum_{j=1}^mh_jdw_j$

where $f(u)$ satisfy the polynomial growth conditions and $\{w_j\}$ are independent two-sided real-valued Wiener processes on a probability space.

The random attractors of this equation has been studied very recently in P. Kloeden et al. In that article, the authors prove the existence of a random attractor for the random dynamical system generated by the equation. Notice that, the attractor is only shown in $L^2$ (even in the case non-degenerate, there is no higher regularity of attractors has been shown). In this paper, we try to improve the result of P. Kloeden et al, and prove that the random attractor is infact compact in $\mathcal D_0^1$ and attracts all orbit of solutions with the topo of $\mathcal D_0^1$.

The method used here is similar to deterministic case (i.e. $h_j = 0$), but we cannot differentiate the equation to get the boundedness of $u_t$. Thus, we have to use the sharp estimates of solutions and idea of splitting solutions. By using egienfunctions, we can divide $\mathcal D_0^1$ intwo two parts, where one part is $H_m = span(e_1,e_2,\ldots,e_m)$. We will prove that the attractor (in $L^2$) $\mathcal A$ is bounded in $H_m$ meanwhile $\|(Id-P_m)u\|$ as small as possible for all $u\in \mathcal A$, where $P_m$ is canonical projector from $\mathcal D_0^1$ onto $H_m$. This will give us the compactness of attractor in $\mathcal D_0^1$.

Result: Under very natural conditions of $f, g$ and $h_j, j =1,\ldots, m$, the random dynamical system generated by stochastic semilinear degenerate parabolic equation possesses a random attractor in $\mathcal D_0^1$.