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

Abstract

This paper investigates the flux-limited chemotaxis system, proposed by Kohatsu and Senba~(2025), \begin{equation*} \begin{cases} u_t = Δu -\nabla\cdot(u|\nabla v|^{α-2}\nabla v),\\ \:\:0=Δv + u, \end{cases} \end{equation*} posed in the unit ball of ${\mathbb{R}}^N$ for some $N\ge 2$, 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 ${\omega }_N(\frac{N^2}{N-1}{)}^{N-1}$, where ${\omega }_N$ denotes the measure of the unit sphere ${\mathbb{S}}^{N-1}$. Asymptotic behaviors are also considered.