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 Davila—and a blow up and compactness argument.