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 $$u_t=\sum^N_{i=1}\frac{\partial}{\partial x_i}(a(x,t)|u|^{\alpha_i}u_{x_i})+\sum^N_{i=1}\frac{\partial f_i(u,x,t)}{\partial x_i}$$ 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 $∂Ω×(0,T),$ instead of the boundedness of the gradient $|∇u|$ for the usual porous medium, we can only show that $∇u∈ L^∞(0,T;L^2_{ {\rm loc}}(Ω)).$ 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.