Large time behavior for a nonlocal nonlinear gradient flow

2022/02/09 by Feng Li, Erik Lindgren, Li, Feng +1
Mathematics · Engineering · #Advanced Mathematical Physics Problems #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2202.04398

Abstract

We study the large time behavior of the nonlinear and nonlocal equation v_t+(-Δ_p)^sv=f , where p∈ (1,2)∪ (2,∞), s∈ (0,1) and (-Δ_p)^s v (x,t)=2 pv ∫_ℝⁿ\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→∞. 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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