Asymptotic behavior for critical Patlak–Keller–Segel model and a repulsive–attractive aggregation equation

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.