Higher H\"older regularity for a subquadratic nonlocal parabolic equation

TL;DR

This paper proves H"older regularity for solutions of a nonlocal parabolic equation involving the fractional p-Laplacian with subquadratic growth, providing explicit exponents and extending previous superquadratic results.

Contribution

It establishes the first explicit H"older regularity results for subquadratic nonlocal parabolic equations with fractional p-Laplacian, including inhomogeneous cases.

Findings

H"older regularity with explicit exponents for solutions

Almost sharp H"older exponents in some cases

Extension of regularity results to inhomogeneous equations

Abstract

In this paper, we are concerned with the H\"older regularity for solutions of the nonlocal evolutionary equation $$ \partial_t u+(-\Delta_p)^s u = 0. $$ Here, $(-\Delta_p)^s$ is the fractional $p$-Laplacian, $0<s<1$ and $1<p<2$. We establish H\"older regularity with explicit H\"older exponents. We also include the inhomogeneous equation with a bounded inhomogeneity. In some cases, the obtained H\"older exponents are almost sharp. Our results complement the previous results for the superquadratic case when $p\geq 2$.