Regularity in $L_p$ Sobolev spaces of solutions to fractional heat equations

Abstract

This work contributes in two areas, with sharp results, to the current investigation of regularity of solutions of heat equations (*) $Pu+\partial_tu=f$ on $\Omega\times I $, where $P$ is a nonlocal operator, and $\Omega \subset R^n$, $I\subset R$. 1) For $P$ a strongly elliptic pseudodifferential operator ($\psi $do) on $R^n$ of order $d\in R_+$, a symbol calculus on $R^{n+1}$ is introduced, that allows showing optimal regularity of solutions in the scale of anisotropic Bessel-potential spaces $H^{(s,s/d)}$, globally over $R^n\times R$, and locally over $\Omega\times I$, for $s\in R$, $1 1/p$ (where $d(x)= \operatorname{dist}(x,\partial\Omega)$). The interior regularity of $u$ is lifted when $f$ is more smooth.