Critical mass for finite-time chemotactic collapse in the critical dimension via comparison

TL;DR

This paper refines comparison methods to analyze finite-time chemotactic collapse in critical dimensions, establishing precise mass thresholds for blowup and global boundedness in chemotaxis models.

Contribution

It simplifies and extends comparison techniques to determine critical mass for finite-time blowup in chemotaxis systems, including models with indirect signal production.

Findings

Critical mass for blowup in 2D is $8\,\pi$.

Global boundedness for initial mass less than $64\pi^2$ in 4D model.

Existence of initial data leading to finite-time collapse for mass exceeding $64\pi^2$.

Abstract

We study the Neumann initial-boundary value problem for the parabolic-elliptic chemotaxis system, proposed by J\"ager and Luckhaus (1992). We confirm that their comparison methods can be simplified and refined, applicable to seek the critical mass $8\pi$ concerning finite-time blowup in the unit disk. As an application, we deal with a parabolic-elliptic-parabolic chemotaxis model involving indirect signal production in the unit ball of $\mathbb R^4$, proposed by Tao and Winkler (2025). Within the framework of radially symmetric solutions, we prove that if initial mass is less than $64\pi^2$, then solution is globally bounded; for any $m$ exceeding $64\pi^2$, there exist nonnegative initial data with prescribed mass $m$ such that the corresponding classical solutions exhibit a formation of Dirac-delta type singularity in finite time, termed a chemotactic collapse.