On the Well-Posedness Problem of the Anisotropic Porous Medium Equation with a Variable Diffusion Coefficient (Journal of Partial Differential Equations, 2024)
The initial-boundary value problem of an anisotropic porous medium equation $$u_t=\sum^N_{i=1}\frac{\partial}{\partial x_i}(a(x,t)|u|^{α_i}u_{x_i})+\sum^N_{i=1}\frac{\partial f_i(u,x,t)}{\partial x_i}$$ is studied. Compared with the usual porous medium equation, there are two different characteristics in this equation. One lies in its anisotropic property, another oneis 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 conditionsmatching up with the anisotropic porous medium equation are discovered and twostability theorems of weak solutions can be proved naturally.
Keywords: Anisotropic porous medium equation; variable diffusion coefficient; stability; partial boundary condition.