Large time behavior for a nonlocal nonlinear gradient flow

Feng Li; Erik Lindgren

doi:10.3934/dcds.2022079

Article Contents

2023, Volume 43, Issue 3&4: 1516-1546. Doi: 10.3934/dcds.2022079

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Large time behavior for a nonlocal nonlinear gradient flow

Feng Li and
Erik Lindgren^,

Department of Mathematics, Uppsala University, Box 480,751 06 Uppsala, Sweden

^*Corresponding author: Erik Lindgren
^*Corresponding author: Erik Lindgren

In honour of Juan Luis Vázquez's 75th birthday

Received: January 2022

Revised: May 2022

Early access: June 2022

Published: March 2023

Erik Lindgren was supported by the Swedish Research Council, 2017-03736

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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 \, {\rm{P.V.}} \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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Mathematics Subject Classification: Primary: 35K55, 35K65, 35K67, 35R11, 35B40.

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