I’m writing a paper which consider the regularity of random attractors for a class of stochastic degenerate parabolic equations of the form
where satisfy the polynomial growth conditions and 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 (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 and attracts all orbit of solutions with the topo of .
The method used here is similar to deterministic case (i.e. ), but we cannot differentiate the equation to get the boundedness of . Thus, we have to use the sharp estimates of solutions and idea of splitting solutions. By using egienfunctions, we can divide intwo two parts, where one part is . We will prove that the attractor (in ) is bounded in meanwhile as small as possible for all , where is canonical projector from onto . This will give us the compactness of attractor in .
Result: Under very natural conditions of and , the random dynamical system generated by stochastic semilinear degenerate parabolic equation possesses a random attractor in .