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 Jäger and Luckhaus (1992). We confirm that their comparison methods can be simplified and refined, applicable to seek the critical mass 8 π concerning finite-time blow-up 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 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 π 2 , then solution is globally bounded; for any m exceeding 64 π 2 , there exist non-negative 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.