Boundary regularity for the fractional heat equation

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 $\text{ in }$\Omega,\ t > 0$,withzeroDirichletconditionsin$$\mathbb{R}$^n\setminus \Omega$andwithinitialdata$u_0\in L^2(\Omega)$. 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 $u(\cdot ,t)\mathrm{/}{\delta }^s\in C^{s-ϵ}(\bar{\mathrm{\Omega }})$ for any $\epsilon > 0$and$\delta(x) = $\text{dist}$(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.