Fractional $p$-caloric functions are Lipschitz

Abstract

We study the parabolic fractional $p-$Laplace equation ${\partial }_{t}u+(-\hat{I}{”}_p{)}^{s}u=0$ in the degenerate range $2<p<2/(1-s)$. We show that weak solutions are Lipschitz continuous in space and, if $p>1/(1-s)$, also in time. We also prove a comparison principle for both weak and viscosity solutions, and establish the equivalence between the two notions of solution.