  We study existence and uniqueness of distributional solutions to the
stochastic partial differential equation $dX - ( \nu \Delta X + \Delta \psi (X)
) dt = \sum_{i=1}^N \langle b_i, \nabla X \rangle \circ d\beta_i$ 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.
