Published Paper

Inserted: 11 mar 2016
Last Updated: 19 aug 2024

Journal: Comm. Pure Appl. Math.
Year: 2017

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 $(0,\infty)\times(\R^N\setminus\Omega)$, 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 $s=1$, where 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 wider classes of nonlocal parabolic equations of the form $u_t+\mathcal L F(u)=0$ in $\Omega$, both in bounded or unbounded domains.

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• BFR-Reg-FPME-Arxiv-Rev1.pdf