Regularity for fully nonlinear nonlocal parabolic equations with rough kernels

Abstract

We prove space and time regularity for solutions of fully nonlinear parabolic integro-differential equations with rough kernels. We consider parabolic equations $u_t = \I u$, where $\I$ is translation invariant and elliptic with respect to the class $\mathcal L_0(\sigma)$ of Caffarelli and Silvestre, $\sigma\in(0,2)$ being the order of $\I$. We prove that if $u$ is a viscosity solution in $B_1 \times (-1,0]$ which is merely bounded in $\R^n \times (-1,0]$, then $u$ is $C^\beta$ in space and $C^{\beta/\sigma}$ in time in $\overline{B_{1/2}} \times [-1/2,0]$, for all $\beta< \min\{\sigma, 1+\alpha\}$, where $\alpha>0$. Our proof combines a Liouville type theorem ---relaying on the nonlocal parabolic $C^\alpha$ estimate of Chang and D\'avila--- and a blow up and compactness argument.