Axisymmetric type II blowup solutions to the three dimensional Keller-Segel system

TL;DR

This paper constructs axisymmetric type II blowup solutions for the 3D Keller-Segel system, revealing finite-time singularities with mass concentration on a ring, advancing understanding of blowup behavior in chemotaxis models.

Contribution

It introduces the first construction of axisymmetric type II blowup solutions with precise blowup rates and mass concentration profiles in the 3D Keller-Segel system.

Findings

Finite-time blowup solutions with axisymmetry.

Mass concentrates along a one-dimensional ring.

Precise blowup rate derived using spectral analysis.

Abstract

We construct axisymmetric solutions to the three-dimensional parabolic-elliptic Keller-Segel system that blows up in finite time. In particular, the singularity is of type II, which admits locally a leading order profile of the rescaled stationary solution of the two-dimensional system. Additionally, mass concentration occurs along a one-dimensional ring in the plane. In the analysis, we rely on an approximate solution of the eigenproblem associated with the linearized operator around the stationary solution as well as the modulation dynamics to control the perturbation function and derive the accurate blowup rate.