Random attractors for stochastic porous media equations perturbed by space–time linear multiplicative noise

Abstract

Unique existence of solutions to porous media equations driven by continuous linear multiplicative space–time rough signals is proven for initial data in $L^{1}(\mathcal{O})$ on bounded domains $\mathcal{O} $. The generation of a continuous, order-preserving random dynamical system on $L^{1}(\mathcal{O})$ and the existence of a random attractor for stochastic porous media equations perturbed by linear multiplicative noise in space and time is obtained. The random attractor is shown to be compact and attracting in $L^{\infty}(\mathcal{O})$ norm. Uniform $L^{\infty}$ bounds and uniform space–time continuity of the solutions is shown. General noise including fractional Brownian motion for all Hurst parameters is treated and a pathwise Wong–Zakai result for driving noise given by a continuous semimartingale is obtained. For fast diffusion equations driven by continuous linear multiplicative space–time rough signals, existence of solutions is proven for initial data in $L^{m+1}(\mathcal{O})$.