Infinite speed of propagation and regularity of solutions to the fractional porous medium equation in general domains

TL;DR

This paper demonstrates that solutions to the fractional porous medium equation exhibit infinite speed of propagation and regularity properties, contrasting with the finite speed in the local case, and provides quantitative bounds and regularity estimates.

Contribution

It establishes a global Harnack principle, proves classical regularity in the interior, and derives sharp boundary regularity estimates for solutions to the fractional porous medium equation.

Findings

Solutions are comparable to the distance to the boundary raised to the power s/m.

Solutions are classical in the interior, with smoothness in space and time.

Sharp boundary regularity estimates are established.

Abstract

We study the positivity and regularity of solutions to the fractional porous medium equations $u_t+(-\Delta)^su^m=0$ in $(0,\infty)\times\Omega$, for $m>1$ and $s\in (0,1)$ and with Dirichlet boundary data $u=0$ in $(0,\infty)\times({\mathbb R}^N\setminus\Omega)$, and nonnegative initial condition $u(0,\cdot)=u_0\geq0$. Our first result is a quantitative lower bound for solutions which holds for all positive times $t>0$. As a consequence, we find a global Harnack principle stating that for any $t>0$ solutions are comparable to $d^{s/m}$, where $d$ is the distance to $\partial\Omega$. This is in sharp contrast with the local case $s=1$, in which the equation has finite speed of propagation. After this, we study the regularity of solutions. We prove that solutions are classical in the interior ($C^\infty$ in $x$ and $C^{1,\alpha}$ in $t$) and establish a sharp $C^{s/m}_x$ regularity estimate up to the boundary. Our methods are quite general, and can be applied to a wider class of nonlocal parabolic equations of the form $u_t-\mathcal L F(u)=0$ in $\Omega$, both in bounded or unbounded domains.