Mass threshold for global existence in chemotaxis systems with critical flux limitation

TL;DR

This study determines the precise initial mass threshold for global bounded solutions in a radially symmetric chemotaxis system with critical flux limitation, extending understanding of blow-up phenomena in such models.

Contribution

It establishes the exact mass threshold for global existence of solutions in the critical flux chemotaxis system under radial symmetry, which was previously unknown.

Findings

Global solutions exist if initial mass is below the threshold.

Solutions blow up if initial mass exceeds the threshold.

Asymptotic behavior of solutions is analyzed.

Abstract

This paper investigates the flux-limited chemotaxis system, proposed by Kohatsu and Senba~(2025), \begin{equation*} \begin{cases} u_t = \Delta u -\nabla\cdot(u|\nabla v|^{\alpha-2}\nabla v),\\ \:\:0=\Delta v + u, \end{cases} \end{equation*} posed in the unit ball of $\mathbb{R}^N$ for some $N\geq2$, subject to no-flux and homogeneous Dirichlet boundary conditions. Due to precedents, e.g., Tello (2022) and Winkler (2022), the exponent $\alpha = \frac{N}{N-1}$ is the threshold for finite-time blow-up under symmetry assumptions. We further find that under the framework of radially symmetric solutions, the system with critical flux limitation admits a globally bounded weak solution if and only if initial mass is strictly less than $\omega_N \big(\frac{N^2}{N-1}\big)^{N-1}$, where $\omega_N$ denotes the measure of the unit sphere $\mathbb{S}^{N-1}$. Asymptotic behaviors are also considered.