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 $\max{p,q,2} < \min\left{q + \tfrac{p-1}{1+\frac{n}{sp}}, 2 + \tfrac{p-1}{1+\frac{n}{sp}}\right}$. The proof explicitly makes use of the nonlocal nature of the problem and as a consequence, our estimates are not stable at $s \rightarrow 0$. We note that the analogous regularity in the local problem remains open.