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On the stochastic porous medium equation. (English) Zbl 1099.35187

The author proves existence and uniqueness for the Cauchy problem and the initial-boundary value problem \[ u_{t}=\sum_{i=1}^{n}\partial_{x_{i}}(| u| ^{p-2}\partial_{x_{i}}u)+\sum_{j=1}^{\infty}f_{j}{dB_{j}\over dt},\quad (t,x)\in (0,T)\times G, \] \(u=0\), \((t,x)\in (0,T)\times\partial G\), \(u(0,x)=u_0(x)\), \(x\in G\), where \(G\) is a bounded domain in \(\mathbb R^{n}\) with smooth boundary \(\partial G\); \(B_{j}\) are standard Brownian motions which are mutually independent. It is also established the existence of invariant measures when the space domain is bounded.

MSC:

35R60 PDEs with randomness, stochastic partial differential equations
76S05 Flows in porous media; filtration; seepage
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References:

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