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

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.