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<p<\infty $. Similar results hold in anisotropic Besov spaces $B^{(s,s/d)}$. 2) Let $\Omega $ be smooth bounded, and let $P$ equal $(-\Delta)^a$ ($0<a<1$), or its generalizations to singular integral operators with regular kernels, that are infinitesimal generators of stable L\'evy processes. With the Dirichlet condition $u=0$ on $R^n\setminus\Omega$, the initial condition $u|_{t=0}=0$, and $f\in L_p(\Omega \times I)$, (*) has a unique solution $u\in L_p(I, H_p^{a(2a)}(\bar\Omega))$ with $\partial_tu\in L_p(\Omega \times I)$. Here $H_p^{a(2a)}(\bar\Omega)$ equals $\dot H_p^{2a}(\bar\Omega)$ if $a<1/p$, and is contained in $\dot H_p^{2a-\varepsilon}(\bar\Omega)$ if $a=1/p$, but contains nontrivial elements from $ d^a \bar H_p^{a}(\Omega)$ if $a>1/p$ (where $d(x)= \operatorname{dist}(x,\partial\Omega)$). The interior regularity of $u$ is lifted when $f$ is more smooth.