Nonlocal porous medium equation

The nonlocal porous medium equation of order $\sigma$ is the name currently given to two very different equations, namely

\[ u_t = \nabla \cdot \left ( u \nabla \mathcal{K_\alpha} (u) \right )\]

\[\mbox{ where } \mathcal{K}_\alpha(u) := C_{n,\alpha}\; u * |x|^{-n+\alpha},\;\; \alpha+2=\sigma \]

and

\[ u_t +(-\Delta)^{s}(u^m) = 0 \]

These equations agree when $s=1$ and $m=2$, otherwise they are not only different superficially, they also exhibit extremely different behaviors. They are both fractional order Quasilinear equations. The first of the two has the remarkable property (for nonlocal equations at least) that any initial data with compact support remains with compact support for all later times, the opposite is true of the second equation, for which instantaneous speed of propagation holds, thus, at least from the point of view of free boundary problems, the first equation is the "truer" nonlocal porous medium equation.