Boundary regularity for the fractional heat equation

We study the regularity up to the boundary of solutions to fractional heat equation in bounded $$C^{1,1}$$ C1,1 domains. More precisely, we consider solutions to $$\partial _t u + (-\Delta )^su=0 \hbox { in }\Omega ,\ t > 0$$ ∂tu+(-Δ)su=0inΩ,t>0 , with zero Dirichlet conditions in $$\mathbb {R}^n{\setminus } \Omega $$ Rn\Ω and with initial data $$u_0\in L^2(\Omega )$$ u0∈L2(Ω) . Using the results of the second author and Serra for the elliptic problem, we show that for all $$t>0$$ t>0 we have $$u(\cdot , t)\in C^s(\mathbb {R}^n)$$ u(·,t)∈Cs(Rn) and $$u(\cdot , t)/\delta ^s \in C^{s-\epsilon }(\overline{\Omega })$$ u(·,t)/δs∈Cs-ϵ(Ω¯) for any $$\epsilon > 0$$ ϵ>0 and $$\delta (x) = \hbox {dist}(x,\partial \Omega )$$ δ(x)=dist(x,∂Ω) . Our regularity results apply not only to the fractional Laplacian but also to more general integro-differential operators, namely those corresponding to stable Lévy processes. As a consequence of our results, we show that solutions to the fractional heat equation satisfy a Pohozaev-type identity for positive times.