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 $$ u_t+(-Δ)^s (u^m)=0, \quad x\in {\mathbb{R}}^N, \ t>0. $$ with initial data $u(x,0)\ge 0$, $x\in {\mathbb{R}}^N$, in the limit as $m\to\infty$ with fixed $s\in (0,1)$. We first identify the limit of the Barenblatt solutions as the solution of a 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 tails. We then study the limit for a class of nonnegative initial data and derive counterexamples to expected propagation and comparison properties based on symmetrization.