Boundedness and finite-time blow-up in a repulsion-consumption system with flux limitation

TL;DR

This paper studies a mathematical model describing cell movement with flux limitation, establishing conditions for global existence or finite-time blow-up of solutions based on parameters and initial data.

Contribution

It provides new criteria for global boundedness or blow-up in a flux-limited chemotaxis system, extending previous results to more general flux functions and boundary conditions.

Findings

Global bounded solutions for certain parameter ranges

Finite-time blow-up under large boundary signals

Comparison with previous chemotaxis models

Abstract

We investigate the following repulsion-consumption system with flux limitation \begin{align}\tag{$\star$} \left\{ \begin{array}{ll} u_t=\Delta u+\nabla \cdot(uf(|\nabla v|^2) \nabla v), & x \in \Omega, t>0, \tau v_t=\Delta v-u v, & x \in \Omega, t>0, \end{array} \right. \end{align} under no-flux/Dirichlet boundary conditions, where $\Omega \subset \mathbb{R}^n$ is a bounded domain and $f(\xi)$ generalizes the prototype given by $f(\xi)=(1+\xi)^{-\alpha}$ ($\xi \geqslant 0$). We are mainly concerned with the global existence and finite time blow-up of system ($\star$). The main results assert that, for $\alpha > \frac{n-2}{2n}$, then when $\tau=1$ and under radial settings, or when $\tau=0$ without radial assumptions, for arbitrary initial data, the problem ($\star$) possesses global bounded classical solutions; for $\alpha<0$, $\tau=0$, $n=2$ and under radial settings, for any initial data, whenever the boundary signal level large enough, the solutions of the corresponding problem blow up in finite time. Our results can be compared respectively with the blow-up phenomenon obtained by Ahn \& Winkler (2023) for the system with nonlinear diffusion and linear chemotactic sensitivity, and by Wang \& Winkler (2023) for the system with nonlinear diffusion and singular sensitivity .