Large time behavior for a nonlocal nonlinear gradient flow

Abstract

We study the large time behavior of the nonlinear and nonlocal equation $$v_t+(-{\Delta }_p{)}^{s}v=f,$$ 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.