Large time behavior for a nonlocal nonlinear gradient flow

TL;DR

This paper investigates the long-term decay behavior of solutions to a nonlocal, nonlinear gradient flow equation involving fractional p-Laplacians, providing sharp decay estimates using an iterative method inspired by Moser's technique.

Contribution

It introduces sharp decay estimates for solutions of a nonlocal nonlinear gradient flow, employing a novel iterative approach based on Moser's method.

Findings

Established sharp decay rates for large time behavior.

Developed an iterative proof technique inspired by Moser's method.

Extended understanding of fractional p-Laplacian gradient flows.

Abstract

We study the large time behavior of the nonlinear and nonlocal equation $$ v_t+(-\Delta_p)^sv=f \, , $$ where $p\in (1,2)\cup (2,\infty)$, $s\in (0,1)$ and $$ (-\Delta_p)^s v\, (x,t)=2 \,\text{pv} \int_{\mathbb{R}^n}\frac{|v(x,t)-v(x+y,t)|^{p-2}(v(x,t)-v(x+y,t))}{|y|^{n+sp}}\, dy. $$ This equation arises as a gradient flow in fractional Sobolev spaces. We obtain sharp decay estimates as $t\to\infty$. The proofs are based on an iteration method in the spirit of J. Moser previously used by P. Juutinen and P. Lindqvist.