We study existence and uniqueness of distributional solutions to the stochastic partial differential equation $ dX - \bigl( \nu \Delta X + \Delta \psi (X) \bigr) 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.
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