H\"older estimates and large time behavior for a nonlocal doubly nonlinear evolution

TL;DR

This paper studies a nonlinear, nonlocal PDE involving the fractional p-Laplacian, establishing existence, uniqueness, Hölder regularity, and long-term behavior of solutions, linking them to eigenvalue problems.

Contribution

It proves existence and uniqueness of weak and viscosity solutions, provides Hölder estimates, and analyzes asymptotic behavior related to eigenvalues of the fractional p-Laplacian.

Findings

Existence of weak solutions

Uniqueness as viscosity solutions

Hölder regularity of solutions

Abstract

The nonlinear and nonlocal PDE $$ |v_t|^{p-2}v_t+(-\Delta_p)^sv=0 \, , $$ where $$ (-\Delta_p)^s v\, (x,t)=2 \,\text{PV} \int_{\mathbb{R}^n}\frac{|v(x,t)-v(x+y,t)|^{p-2}(v(x,t)-v(x+y,t))}{|y|^{n+sp}}\, dy, $$ has the interesting feature that an associated Rayleigh quotient is non-increasing in time along solutions. We prove the existence of a weak solution of the corresponding initial value problem which is also unique as a viscosity solution. Moreover, we provide H\"older estimates for viscosity solutions and relate the asymptotic behavior of solutions to the eigenvalue problem for the fractional $p$-Laplacian.