Collapsing-ring blowup solutions for the Keller-Segel system in three dimensions and higher

TL;DR

This paper constructs and analyzes a finite-time blowup solution for the Keller-Segel system in three or more dimensions, featuring a ring-shaped concentration of mass that collapses non-self-similarly, with proven stability in symmetric cases.

Contribution

It introduces the first stability analysis of a ring-shaped blowup in the Keller-Segel system, using a novel framework to handle complex interactions between different blowup zones.

Findings

Constructed a finite-time blowup solution with ring concentration

Proved stability of the blowup dynamics among symmetric solutions

Developed a new analytical framework for multi-zone interactions

Abstract

We consider the parabolic-elliptic Keller-Segel system in three dimensions and higher, corresponding to the mass supercritical case. We construct rigorously a solution which blows up in finite time by having its mass concentrating near a ring that shrinks to a point. In particular, the singularity is of type II, non self-similar. We show the stability of this dynamics among spherically symmetric solutions. In renormalised variables, the solution ressembles a traveling wave imploding at the origin, and this, to our knowledge, is the first stability result for such phenomenon for an evolution PDE. We develop a framework to handle the interactions between the two blowup zones contributing to the mechanism: a thin inner zone around the ring where viscosity effects occur, and an outer zone where the evolution is mostly inviscid.