Local boundedness of solutions to nonlocal equations modeled on the fractional p-Laplacian

TL;DR

This paper establishes local boundedness estimates for solutions to a class of nonlocal, possibly degenerate, parabolic equations involving the fractional p-Laplacian, extending regularity theory for such integro-differential equations.

Contribution

It provides new estimates for the local boundedness of subsolutions to nonlocal equations modeled on the fractional p-Laplacian, including degenerate cases.

Findings

Proved local boundedness estimates for solutions.

Extended regularity results to degenerate nonlocal equations.

Analyzed equations with kernels comparable to |x-y|^{n+ps}.

Abstract

We state and prove estimates for the local boundedness of subsolutions of non-local, possibly degenerate, parabolic integro-differential equations of the form \begin{equation*} \partial_tu(x,t)+\mbox{P.V.}\int\limits_{\mathbb R^n}K(x,y,t) |u(x,t)-u(y,t) |^{p-2}(u(x,t)-u(y,t))\, dy,\end{equation*} $(x,t)\in\mathbb R^n\times\mathbb R$, where $\mbox{P.V.} $ means in the principle value sense, $p\in (1,\infty)$ and the kernel obeys $K(x,y,t)\approx |x-y |^{n+ps}$ for some $s\in (0,1)$, uniformly in $(x,y,t)\in\mathbb R^n\times \mathbb R^n\times\mathbb R$.