Finite-time blow-up for the three dimensional axially symmetric Keller-Segel system

TL;DR

This paper constructs finite-time blow-up solutions with multiple rings in a 3D Keller-Segel system, providing detailed asymptotics for Type II singularities and advancing understanding of blow-up phenomena.

Contribution

It introduces a novel construction of axially symmetric blow-up solutions with multiple rings and refines the asymptotic analysis of Type II singularities in the Keller-Segel system.

Findings

Mass concentrates along multiple rings during blow-up

Derived precise asymptotic expansion for Type II singularities

Extended previous work with a generalized blow-up rate analysis

Abstract

We construct axially symmetric finite-time blow-up solutions to the three-dimensional Keller-Segel system. By adapting gluing techniques, we derive a precise asymptotic expansion for Type II singularities that generalizes the recent work of Hou, Nguyen, and Song. In our construction the mass concentrates along multiple rings and we obtain a refined expansion for the blow-up rate.