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 ${\mathbb{R}}^n\times (-1,0]$, 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.

Some typos fixed and proof of Proposition 4.5 simplified

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