Porous media equations with nonlinear gradient noise and Dirichlet boundary conditions

Abstract

We establish pathwise existence of solutions for stochastic porous media and fast diffusion equations of type (1.1), in the full regime m∈(0,∞) and for any initial data u0∈L2(Q). Moreover, if the initial data is positive, solutions are pathwise unique. In turn, the solution map to (1.1) is a continuous function of the driving noise and it generates an associated random dynamical system. Finally, in the regime m∈{1}∪(2,∞), all the aforementioned results also hold for signed initial data.