On the Well-Posedness Problem of the Anisotropic Porous Medium Equation with a Variable Diffusion Coefficient

Abstract

The initial-boundary value problem of an anisotropic porous medium equation

$u_{t}=\sum_{i=1}^{N} \frac{\partial}{\partial x_{i}}\left(a(x, t)|u|^{\alpha_{i}} u_{x_{i}}\right)+\sum_{i=1}^{N} \frac{\partial f_{i}(u, x, t)}{\partial x_{i}}$

is studied. Compared with the usual porous medium equation, there are two different characteristics in this equation. One lies in its anisotropic property, another one is that there is a nonnegative variable diffusion coefficient a(x,t) additionally. Since a(x,t) may be degenerate on the parabolic boundary $ \partial \Omega \times(0, T)$, instead of the boundedness of the gradient $ |\nabla u|$ for the usual porous medium, we can only show that $ \nabla u \in L^{\infty}\left(0, T ; L_{\mathrm{loc}}^{2}(\Omega)\right)$. Based on this property, the partial boundary value conditions matching up with the anisotropic porous medium equation are discovered and two stability theorems of weak solutions can be proved naturally.

Keywords

Anisotropic porous medium equation / variable diffusion coefficient / stability / partial boundary condition