Stochastic Porous Media Equations

Abstract

This survey is devoted to the presentation of a few recent results concerning the existence, longtime behaviour and localization of solutions to stochastic porous media equations with linearly multiplicative Gaussian noise. Some of these results are given without proof, or with a sketch of proof only. In Section 1, we present a few basic results on existence of solutions and some models described by deterministic porous media equations. Section 2 is devoted to an existence and uniqueness result for porous media equations perturbed by a linear multiplicative Wiener process. In Section 3, we present two results pertaining to extinction in finite time of solutions to fast diffusion porous media equations with linear multiplicative Gaussian noise and to propagation with finite speed in the low diffusion case. Section 4 is devoted to some asymptotic results for the solutions of stochastic self-organized criticality equations.