Porous medium equation

Remarks

Example: the Barenblatt solution $$U(x, t) = t^{-\alpha} \Big( C - k \abs{x}^2 t^{-2\alpha/n} \Big)_+^{1/(m-1)}, \qquad \alpha = \frac{n}{n(m - 1) + 2}, \quad k = \frac{\alpha (m - 1)}{2 m n},$$ with a constant $C > 0$ and $s_+ = \max(s, 0)$, solves the equation on $\R^n \times (0, \infty)$ in the weak sense. For each $t$ its support is the ball $\abs{x} \le (C/k)^{1/2}\, t^{\alpha/n}$, so it spreads at finite speed. Non-example: for $m = 1$ the equation is the heat equation, which is excluded; its solutions with nonnegative nonzero data are positive everywhere for $t > 0$. Writing $\Delta(u^m) = \nabla \cdot (m u^{m-1} \nabla u)$ shows that the equation is degenerate parabolic where $u = 0$. The case $0 < m < 1$ is called the fast diffusion equation.