Here we consider the equation \[u_t =(u^m u_x)_x \] either as an initial value problem or a two-sided initial/boundary value problem, e.g. \(u(0,t)=g_0(t)\), \(u(1,t)=g_1(t)\), \(u(x,0)=f(x)\). Find relevant values for \(m\) in the literature.

Google Porous medium equation, mesa problem, Barenblatt solution

Challenges Non-linearity, applications.