Lipschitz regularity for parabolic fractional $p$-Laplace equations

We prove that local weak solutions to nonlocal parabolic $p$-Laplace equations are locally Lipschitz continuous in space, uniformly in time for every $1<p<\mathrm{\infty }$ and $s\in (0,1)$ whenever $sp>p-1$. Our results hold for symmetric, translation-invariant kernels satisfying standard ellipticity bounds, including kernels that may be discontinuous and require only that the tail of the solution be bounded. In the linear case, our proof provides a different route avoiding blow up arguments and Liouville theorems.