Porous media equations with nonlinear gradient noise and Dirichlet boundary conditions
Andrea Clini · Stochastic Processes and their Applications · 2023
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.