H\"older regularity of weak solutions to nonlocal doubly degenerate parabolic equations

Abstract

We study local regularity for nonlocal doubly degenerate parabolic equations. The model equation is \begin{equation*}\begin{split} \partial_t(|u|^{q-1}u)+\mathrm{P}.\mathrm{V}.\int_{\mathbb{R}^n}\frac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{n+sp}}\,\mathrm{d}y=0, \end{split} \end{equation*} where $0<s<1$, $p>2$ and $0<q<p-1$. Under a parabolic tail condition, we show that any locally bounded and sign-changing solution is locally H\"older continuous. Our proof is based on a nonlocal version of De Giorgi technique and the method of intrinsic scaling.