Global boundedness of weak solutions to a flux-limited Keller--Segel system with superlinear production

TL;DR

This paper extends the existence of global bounded weak solutions to a flux-limited Keller--Segel system with superlinear production, under certain smallness conditions on the flux limitation parameter, broadening previous results.

Contribution

It proves the existence of global bounded weak solutions for the Keller--Segel system with superlinear production when the flux limitation parameter is sufficiently small.

Findings

Global bounded weak solutions exist for  > 1 under small p.

The results extend previous work for   1.

Conditions on p ensure boundedness of solutions.

Abstract

The flux-limited Keller--Segel system \begin{align*} \begin{cases} u_t = \Delta u - \chi \nabla \cdot (u|\nabla v|^{p-2}\nabla v), \\[] v_t = \Delta v - v + u^{\theta} \end{cases} \end{align*} is considered under homogeneous Neumann boundary conditions in a bounded domain $\Omega \subset \mathbb{R}^n$ $(n \in \mathbb{N})$. In the case that $\theta \le 1$, existence of global bounded weak solutions was established in the previous work (arXiv:2501.04370 ; to be appear in Proceedings of the conference "Critical Phenomena in Nonlinear Partial Differential Equations, Harmonic Analysis, and Functional Inequalities."). The purpose of this paper is to prove that global bounded weak solutions can also be constructed in the case $\theta > 1$ with a smallness condition on $p$.