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

Abstract 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>4</mml:mn> </mml:msup> </mml:mrow> </mml:math> , proposed by Tao and Winkler (2025). Within the framework of radially symmetric solutions, we prove that if initial mass is less than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>64</mml:mn> <mml:msup> <mml:mi>π</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> , then solution is globally bounded; for any m exceeding <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>64</mml:mn> <mml:msup> <mml:mi>π</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> , 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.