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

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 (O) on bounded domains O. The generation of a continuous, order-preserving random dynamical system (RDS) on L 1 (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 ∞ (O) norm. Uniform L ∞ bounds and uniform space-time continuity of the solutions is shown. General noise including fractional Brownian motion for all Hurst parameters is treated. 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 (O).

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