Hölder regularity of doubly nonlinear nonlocal quasilinear parabolic equations in some mixed singular-degenerate regime

Abstract

We study local Hölder regularity of bounded, weak solutions for the nonlocal quasilinear equations of the form \[ (|u|^{q-2}u)_t + \text{P.V.} \int_{\mathbb{R}^n} \frac{|u(x,t) - u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{n+sp}} dy = 0, \] with $p\in (1,\infty )$, $q\in (1,\infty )$ and $s\in (0,1)$. Analogous Hölder continuity result in the local case is known in the purely singular case $\{1<p<2,p<q\}$, purely degenerate case $\{2<p,q<p\}$, scale invariant case $\{p=q\}$ and translation invariant case $\{q=2,1<p<\infty \}$. In the nonlocal setting, Hölder regularity is known when the equation is either translation invariant $\{q=2,1<p<\infty \}$ or scale invariant $\{q=p,1<p<\infty \}$ or purely degenerate case $\{2<p,q<p\}$. Similar strategy can be used to obtain Hölder regularity in the purely singular case $\{1<p<2,p<q\}$. In this paper, we adapt several ideas developed over the past few years and combine it with a new intrinsic scaling to prove Hölder regularity in the mixed singular-degenerate range $\max \{p,q,2\}<\min \left\{q+\frac{p-1}{1+\frac{n}{sp}},2+\frac{p-1}{1+\frac{n}{sp}}\right\}$. The proof explicitly makes use of the nonlocal nature of the problem and as a consequence, our estimates are not stable at $s\to 0$. We note that the analogous regularity in the local problem remains open.

43 pages