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 $u_t = \mathrm{I}u$, where $\mathrm{I}$ 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 $\mathrm{I}$. We prove that if $u$ is a viscosity solution in $B_1 \times (-1,0]$ which is merely bounded in $\mathbb {R}^n \times (-1,0]$, 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 Dávila—and a blow up and compactness argument.