Boundary regularity for the fractional heat equation

TL;DR

This paper investigates the boundary regularity of solutions to the fractional heat equation in smooth domains, establishing their smoothness properties and extending results to general integro-differential operators, with implications for Pohozaev identities.

Contribution

The paper proves boundary regularity results for fractional heat equations in $C^{1,1}$ domains, extending to stable Lévy processes and deriving Pohozaev-type identities.

Findings

Solutions are in $C^s( ^n)$ for all $t>0$.

Solutions satisfy $u(ullet,t)/ extrm{dist}(ullet,oundary ext{domain})^s ext{ is in } C^{s- ext{epsilon}}$.

Regularity results apply to a broad class of integro-differential operators.

Abstract

We study the regularity up to the boundary of solutions to fractional heat equation in bounded $C^{1,1}$ domains. More precisely, we consider solutions to $\partial_t u + (-\Delta)^s u=0 \textrm{ in }\Omega,\ t > 0$, with zero Dirichlet conditions in $\mathbb{R}^n\setminus \Omega$ and with initial data $u_0\in L^2(\Omega)$. Using the results of the second author and Serra for the elliptic problem, we show that for all $t>0$ we have $u(\cdot, t)\in C^s(\mathbb{R}^n)$ and $u(\cdot, t)/\delta^s \in C^{s-\epsilon}(\overline\Omega)$ for any $\epsilon > 0$ and $\delta(x) = \textrm{dist}(x,\partial\Omega)$. Our regularity results apply not only to the fractional Laplacian but also to more general integro-differential operators, namely those corresponding to stable L\'evy processes. As a consequence of our results, we show that solutions to the fractional heat equation satisfy a Pohozaev-type identity for positive times.