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

Xuan Mao, Meng Liu, Yuxiang Li

TL;DR

The paper refines comparison methods for chemotaxis models to identify a critical mass threshold for finite-time blowup in the Jäger–Luckhaus system and extends the analysis to the four-dimensional indirect chemotaxis model with an auxiliary signal. By introducing cumulated densities and a cooperative reduced system, the authors construct explicit exploding subsolutions and apply a comparison principle to prove finite-time blowup for radially symmetric data when the mass exceeds $64\pi^2$, while establishing epsilon-regularity and Lyapunov-based bounds to obtain global boundedness below this threshold. A key contribution is showing that blow-up induces chemotactic collapse, with the density converging to a Dirac delta at the origin carrying mass at least $64\pi^2$. The results unify and sharpen the understanding of critical mass phenomena across dimensions and using indirect signal production, providing a robust framework for predicting and describing collapse in chemotaxis models. The work has implications for the mathematical theory of blow-up in nonlinear parabolic-elliptic systems and informs the modeling of aggregation phenomena with signaling feedbacks in high dimensions.

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 $8π$ 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 $64π^2$, then solution is globally bounded; for any $m$ exceeding $64π^2$, 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.

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

TL;DR

The paper refines comparison methods for chemotaxis models to identify a critical mass threshold for finite-time blowup in the Jäger–Luckhaus system and extends the analysis to the four-dimensional indirect chemotaxis model with an auxiliary signal. By introducing cumulated densities and a cooperative reduced system, the authors construct explicit exploding subsolutions and apply a comparison principle to prove finite-time blowup for radially symmetric data when the mass exceeds , while establishing epsilon-regularity and Lyapunov-based bounds to obtain global boundedness below this threshold. A key contribution is showing that blow-up induces chemotactic collapse, with the density converging to a Dirac delta at the origin carrying mass at least . The results unify and sharpen the understanding of critical mass phenomena across dimensions and using indirect signal production, providing a robust framework for predicting and describing collapse in chemotaxis models. The work has implications for the mathematical theory of blow-up in nonlinear parabolic-elliptic systems and informs the modeling of aggregation phenomena with signaling feedbacks in high dimensions.

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 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 , proposed by Tao and Winkler (2025). Within the framework of radially symmetric solutions, we prove that if initial mass is less than , then solution is globally bounded; for any exceeding , there exist nonnegative initial data with prescribed mass such that the corresponding classical solutions exhibit a formation of Dirac-delta type singularity in finite time, termed a chemotactic collapse.

Paper Structure

This paper contains 12 sections, 19 theorems, 213 equations.

Table of Contents

  1. Introduction
  2. A case study. Proof of Proposition \ref{['prop: finite-time blowup in JL system']}
  3. Finite-time blowup
  4. Preliminaries
  5. Stationary solutions
  6. Construction of exploding subsolutions
  7. Proof of main theorem
  8. Global boundedness
  9. Lyapunov functional and Adams inequality
  10. Boundedness away from the origin
  11. Proof of Proposition \ref{['prop: varepsilon regularity']}
  12. Chemotactic collapse

Key Result

Proposition 1.1

Let $\Omega$ be the unit disk. For any $m>8\pi$, there exist nonnegative, increasing and concave functions $\tilde{M}$ from $C^1([0,1])$ and a time $T^\star\in(0,\infty)$ such that and that whenever with as well as then the corresponding classical solution of sys: JL system blows up in finite time $T_{\max}\leq T^\star$ in the following sense: There exist $T_{\max } \in(0, T^{\star}]$ as well

Theorems & Definitions (35)

  • Proposition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Proposition 1.5
  • Proposition 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof : Proof of Proposition \ref{['prop: finite-time blowup in JL system']}
  • ...and 25 more