Boundary concentration phenomena for the higher-dimensional Keller-Segel system

TL;DR

This paper investigates how steady states of a higher-dimensional Keller-Segel system can concentrate along specific boundary submanifolds, revealing new boundary concentration phenomena in chemotaxis models.

Contribution

It identifies three types of boundary concentration phenomena for steady states in a higher-dimensional Keller-Segel system with symmetric domains.

Findings

Concentration occurs along (N-2)-dimensional boundary submanifolds.

Steady states can form Dirac measures supported on these submanifolds.

The study extends understanding of boundary phenomena in chemotaxis models.

Abstract

We study the existence of steady states to the Keller-Segel system with linear chemotactical sensitivity function on a smooth bounded domain in $\mathbb R^N,$ $N\ge3,$ having rotational symmetry. We find three different types of chemoattractant concentration which concentrate along suitable $(N-2)-$dimensional minimal submanifolds of the boundary. The corresponding density of the cellular slime molds exhibit in the limit one or more Dirac measures supported on those boundary submanifolds.