Unstable radial equilibria as sharp concentration thresholds for the Jäger--Luckhaus model on the unit disk
Abstract
We identify unstable radial equilibria as sharp concentration thresholds for the Jäger--Luckhaus chemotaxis system on the unit disk with no-flux boundary conditions. Concentration is compared through cumulative mass. At fixed total mass, every radial initial profile more concentrated than an unstable equilibrium produces finite-time blow-up, whereas every nonstationary profile less concentrated than that equilibrium generates a global, uniformly bounded solution converging to the least concentrated radial equilibrium. We also classify the radial equilibria and determine their nonlinear stability. Let Λ= j1,12, where j1,1 is the first positive zero of the Bessel function J1. For M≠ Λπ, the least concentrated equilibrium is the unique locally asymptotically stable equilibrium under mass-preserving radial perturbations. It is spatially homogeneous for M<Λπ and strictly increasing for M>Λπ. Every other radial equilibrium, including the homogeneous state at M=Λπ, is nonlinearly unstable. These results clarify how total population and initial spatial aggregation jointly determine collapse or bounded redistribution, and how stable peripheral enrichment can occur within the radial model.
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BibTeX
@misc{mao2026unstable,
title = {{Unstable radial equilibria as sharp concentration thresholds for the Jäger--Luckhaus model on the unit disk}},
author = {Xuan Mao and Wanwan Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39209},
eprint = {2610.39209},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39209},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Unstable radial equilibria as sharp concentration thresholds for the Jäger--Luckhaus model on the unit disk
AU - Xuan Mao
AU - Wanwan Wang
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39209
UR - https://arxiv.org/abs/2610.39209
AB - We identify unstable radial equilibria as sharp concentration thresholds for the Jäger--Luckhaus chemotaxis system on the unit disk with no-flux boundary conditions. Concentration is compared through cumulative mass. At fixed total mass, every radial initial profile more concentrated than an unstable equilibrium produces finite-time blow-up, whereas every nonstationary profile less concentrated than that equilibrium generates a global, uniformly bounded solution converging to the least concentrated radial equilibrium. We also classify the radial equilibria and determine their nonlinear stability. Let Λ= j_1,1^2, where j_1,1 is the first positive zero of the Bessel function J_1. For M≠ Λπ, the least concentrated equilibrium is the unique locally asymptotically stable equilibrium under mass-preserving radial perturbations. It is spatially homogeneous for M<Λπ and strictly increasing for M>Λπ. Every other radial equilibrium, including the homogeneous state at M=Λπ, is nonlinearly unstable. These results clarify how total population and initial spatial aggregation jointly determine collapse or bounded redistribution, and how stable peripheral enrichment can occur within the radial model.
ER -