| Abstract: |
| We are concerned with stochastic porous media equations with nonlinear conservative noise and show that for super-linear and sub-critical noise parameters, kinetic solutions have finite speed of propagation. In particular, we propose a novel iteration technique which allows us to obtain a Stampacchia-type inequality involving one single integral quantity, despite the possibly different scaling behaviors of the porous media and the noise term. This allows us to apply directly the stochastic filtering argument developed in [SIAM J. Math. Anal. 47 (2015), 825--854]. Using related ideas, we identify flatness conditions on initial data which guarantee locally the occurrence of a waiting time phenomenon, i.e., the onset of forward propagation of the solution`s support is locally delayed. The condition for the latter matches the one for $B=0$ up to a logarithmic correction if $n=(m+1)/2$, but it requires more and more flatness of the initial data as $n\to 1$. This is in line with the expected behavior of $u$: For $n=1$ an instantaneous forward motion is possible due to the effects of stochastic transport, no matter how flat the initial profile is. |
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