Intermediate asymptotics for critical and supercritical aggregation equations and Patlak-Keller-Segel models

TL;DR

This paper investigates the long-term behavior of solutions to aggregation and Patlak-Keller-Segel models, identifying conditions under which solutions approach self-similar spreading solutions, with quantitative convergence rates derived.

Contribution

It introduces a method combining decay estimates and entropy techniques to determine when solutions decay to self-similar forms and provides explicit convergence rates.

Findings

Solutions decay to self-similar spreading solutions under certain conditions.

Quantitative convergence rates depend on diffusion nonlinearity and interaction kernel strength.

Entropy-entropy dissipation methods effectively analyze long-term asymptotics.

Abstract

We examine the long-term asymptotic behavior of dissipating solutions to aggregation equations and Patlak-Keller-Segel models with degenerate power-law and linear diffusion. The purpose of this work is to identify when solutions decay to the self-similar spreading solutions of the homogeneous diffusion equations. Combined with strong decay estimates, entropy-entropy dissipation methods provide a natural solution to this question and make it possible to derive quantitative convergence rates in $L^1$. The estimated rate depends only on the nonlinearity of the diffusion and the strength of the interaction kernel at long range.