Uniqueness of stationary states for singular Keller-Segel type models

2019/05/19 by Calvez, Vincent, Carrillo, Jose Antonio, Hoffmann, Franca · 2 citations
#26D10 #35B38 #35B40 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1905.07788

Abstract

We consider a generalised Keller-Segel model with non-linear porous medium type diffusion and non-local attractive power law interaction, focusing on potentials that are more singular than Newtonian interaction. We show uniqueness of stationary states (if they exist) in any dimension both in the diffusion-dominated regime and in the fair-competition regime when attraction and repulsion are in balance. As stationary states are radially symmetric decreasing, the question of uniqueness reduces to the radial setting. Our key result is a sharp generalised Hardy-Littlewood-Sobolev type functional inequality in the radial setting.

Cited by

• Critical threshold for a two-species chemotaxis system with the energy critical exponent
• Equilibria of aggregation-diffusion models with nonlinear potentials

Related

• The Keller-Segel model with mass critical exponent
• Compactly supported stationary states of the degenerate Keller-Segel system in the diffusion-dominated regime
• Ground States in the Diffusion-Dominated Regime
• Equilibria of homogeneous functionals in the fair-competition regime
• Stationary states of reaction-diffusion and Schrödinger systems with inhomogeneous or controlled diffusion