The mesa problem for the fractional porous medium equation

Abstract

We investigate the behaviour of the solutions $u_m(x,t)$ of the fractional porous medium equation

with initial data $u(x,0)\ge 0$, $x\in {\mathbb{R}}^N$, in the limit $m\to \infty$ with fixed $s\in (0,1)$. We first identify the limit $F_{\infty }$ of the Barenblatt solutions $U_m(x,t)$ \normalcolor as the solution of a stationary fractional obstacle problem, and we observe that, contrary to the case $s=1$, the limit is not compactly supported but exhibits a typical fractional tail with power-like decay. In other words, we do not get a plain mesa in the limit, but a mesa with a tail. This is not the whole story since the limit of $V_m(x,t)=mt{U}_m^m(x,t)$ exists and is compactly supported (in $x$). We then study the limit $m\to \infty$ for a wide class of solutions with nonnegative initial data, and show also in this setting the phenomenon of initial discontinuity, whereby the solution does not take on the prescribed initial data. Finally, we derive counterexamples to expected propagation and comparison properties based on symmetrization and pose a related open problem.