June  2007, 7(4): 885-905. doi: 10.3934/dcdsb.2007.7.885

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

1. 

Department of Mathematics, University of Pretoria, Pretoria 0002, South Africa

Received  June 2006 Revised  November 2006 Published  March 2007

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.

Citation: M. Sango. Weak solutions for a doubly degenerate quasilinear parabolic equation with random forcing. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 885-905. doi: 10.3934/dcdsb.2007.7.885
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