Compactly supported stationary states of the degenerate Keller-Segel system in the diffusion-dominated regime

TL;DR

This paper proves the existence and uniqueness of compactly supported stationary states for a diffusion-dominated Keller-Segel model, using variational methods and approximation techniques.

Contribution

It establishes the existence and uniqueness of global minimizers for the free energy in a nonlinear diffusion Keller-Segel system, with a new approximation approach.

Findings

Existence of a unique global minimizer for the free energy.

Stationary states coincide with the minimizer up to translations.

Stationary states are compactly supported, radially decreasing, and smooth inside the support.

Abstract

We first show the existence of unique global minimizer of the free energy for all masses associated to a nonlinear diffusion version of the classical Keller-Segel model when the diffusion dominates over the attractive force of the chemoattractant. The strategy uses an approximation of the variational problem in the whole space by the minimization problem posed on bounded balls with large radii. We show that all stationary states in a wide class coincide up to translations with the unique compactly supported radially decreasing and smooth inside its support global minimizer of the free energy. Our results complement and show alternative proofs with respect to \cite{Strohmer,LY,Kim-Yao}.