Asymptotic Behavior for Critical Patlak-Keller-Segel model and an Repulsive-Attractive Aggregation Equation

TL;DR

This paper investigates the long-term behavior of diffusion-aggregation equations, showing convergence to stationary or self-similar solutions depending on initial conditions and symmetry, using maximum principle and energy methods.

Contribution

It provides new results on the asymptotic behavior of radial and non-radial solutions for critical Patlak-Keller-Segel and repulsive-attractive aggregation models.

Findings

Radial solutions with critical mass converge to stationary solutions.

Subcritical radial solutions decay algebraically to self-similar solutions.

Small non-radial solutions converge to self-similar solutions.

Abstract

In this paper we study the long time asymptotic behavior for a class of diffusion-aggregation equations. Most results except the ones in Section 3.3 concern radial solutions. The main tools used in the paper are maximum-principle type arguments on mass concentration of solutions, as well as energy method. For the Patlak-Keller-Segel problem with critical power $m=2-2/d$, we prove that all radial solutions with critical mass would converge to a family of stationary solutions, while all radial solutions with subcritical mass converge to a self-similar dissipating solution algebraically fast. For non-radial solutions, we obtain convergence towards the self-similar dissipating solution when the mass is sufficiently small. We also apply the mass comparison method to another aggregation model with repulsive-attractive interaction, and prove that radial solutions converge to the stationary solution exponentially fast.