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Weak solutions for a doubly degenerate quasilinear parabolic equation with random forcing

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  • We investigate the problem of existence of a probabilistic weak solution for the initial boundary value problem for the model doubly degenerate stochastic quasilinear parabolic equation

    $d(|y|^{\alpha -2}y) - [ \sum_{i=1}^{n} \frac{\partial }{\partial x_{i}}( |\frac{\partial y}{\partial x}|^{p-2}\frac{\partial y}{\partial x_{i}}) -c_{1}\|y| ^{2\mu -2}y] dt=fdW$

    where $W$ is a $d$-dimensional Wiener process defined on a complete probability space, $f$ is a vector-function, $p$, $\alpha $, $\mu $ are some non negative numbers satisfying appropriate restrictions. The equation arises from a suitable stochastic perturbation of the Darcy Law in the motion of an ideal barotropic gas.

    Mathematics Subject Classification: Primary: 60H15; Secondary: 35R60.

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