exists if $u_0^{m+2} \in L^{1}(\Omega)$, $u_0 \in L^1(\Omega)$, and $u_0 \ge 0$. The uniqueness of the weak solution is guaranteed if it is further assumed that $u \in L^2(\Omega \times (0,T))$.

PME is degenerate whenever $u = 0$. Due to this degeneracy, PDE has the property of the finite speed of propagation: if compact-supported initially, its solution remains compact-supported at any finite time, with the support monotonically expanding as time evolves. The boundary of the support forms a moving interface $\Gamma(t)$ which is commonly referred to as a free boundary. The velocity of the free boundary is given by Darcy’s law (e.g., see Shmarev [@Shmarev2005]), i.e., $$\Gamma'(t) = - \lim_{{\boldsymbol{x}} \to \Gamma(t)^-} \grad \left( \frac{u^m({\boldsymbol{x}},t)}{m} \right),
    \label{Darcy-law}$$ where the limit is taken from the interior of the support. In addition, PME exhibits a waiting-time phenomenon: for a certain type of initial solutions the free boundary does not move until a finite amount of time has elapsed. Loosely speaking, from (\[Darcy-law\]) we may expect to see this phenomenon for initial solutions having vanishing $\grad (u^m)$ at the initial free boundary.

A few classes of special solutions to IBVP have been known, among which is the Barenblatt-Pattle solution, viz., $$u(r,t) = 
        \begin{cases}
            