I use Porous medium equation
$$\frac{\partial u}{\partial t}=\Delta(u^m)$$
to model gas permeation through membranes. Deep down in the rabbit hole, using FVM on 1D system, I managed to derive something similar to a state representation of the system.
$$K\frac{\partial}{\partial t}\vec{\rho}=\mathbb{A}\cdot \vec{\rho^{\gamma+1}}+\mathbb{B}\cdot \vec{\rho_{I/O}^{\gamma+1}}$$
The vector $\vec{\rho}$ represents densities in equidistant control volumes, the exponentiation is to be done element-wise. Is there some theory developed around such systems? I intend to do at least two things with the system - nonlinear regression of $(K,\gamma)$ against some measurements and simulating the system numerically.
