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

Abstract

We investigate the problem of existence of a probabilistic weak solution forthe initial boundary value problem for the model doubly degeneratestochastic quasilinear parabolic equation$d(|y|^{\alpha -2}y) - [ \sum_{i=1}^{n}\frac{\partial }{\partial x_{i}}( |\frac{\partial y}{\partialx}|^{p-2}\frac{\partial y}{\partial x_{i}})-c_{1}\|y| ^{2\mu -2}y] dt=fdW$where $W$ is a $d$-dimensional Wiener process defined on a completeprobability space, $f$ is a vector-function, $p$, $\alpha $, $\mu $ are somenon negative numbers satisfying appropriate restrictions. The equationarises from a suitable stochastic perturbation of the Darcy Law in themotion of an ideal barotropic gas.