Large time behavior for a nonlocal nonlinear gradient flow

Feng Li; Erik Lindgren

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arXiv · 2202.04398

Large time behavior for a nonlocal nonlinear gradient flow

Feng Li·Erik Lindgren

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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.

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Feng Li, Erik Lindgren. 2022-02-09. Large time behavior for a nonlocal nonlinear gradient flow. https://arxiv.org/abs/2202.04398

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