Existence and extinction in finite time for Stratonovich gradient noise porous media equations

We study existence and uniqueness of distributional solutions to the stochastic partial differential equation $dX - ( u Delta X + Delta psi (X) ) dt = sum_{i=1}^N langle b_i, abla X rangle circ dbeta_i$ in $]0,T[ times mathcal{O}$, with $X(0) = x(xi)$ in $mathcal{O}$ and $X = 0$ on $]0,T[ times partial mathcal{O}$. Moreover, we prove extinction in finite time of the solutions in the special case of fast diffusion model and of self-organized criticality model.