Global boundedness of weak solutions to a flux-limited Keller--Segel system with superlinear production
Shohei Kohatsu
TL;DR
The paper addresses global boundedness of weak solutions to a flux-limited Keller–Segel system with superlinear production in a bounded domain with Neumann boundary conditions. The authors develop a regularized approximate problem and derive entropy-type differential inequalities for functionals involving $\int_Ω u^q$ and $\int_Ω|∇v|^2$, obtaining uniform-in-$\varepsilon$ bounds. Under a smallness condition on $p$ that depends on dimension and $\theta$ (namely $p∈(1,\min\{2,(2\theta+1)/(2\theta-1)\})$ for $n=1$, and $p∈(1, n\theta/(n\theta-1))$ for $n≥2$), the paper constructs global weak solutions that remain bounded: $\sup_{t>0}(\|u(·,t)\|_{L^∞}+\|v(·,t)\|_{L^∞})<∞$. This extends prior results for sublinear or θ≤1 cases to the superlinear regime in arbitrary dimensions, highlighting how flux limitation and carefully designed a priori estimates yield global regularity in a chemotaxis system with production growth.
Abstract
The flux-limited Keller--Segel system \begin{align*} \begin{cases} u_t = Δu - χ\nabla \cdot (u|\nabla v|^{p-2}\nabla v), \\[] v_t = Δv - v + u^θ \end{cases} \end{align*} is considered under homogeneous Neumann boundary conditions in a bounded domain $Ω\subset \mathbb{R}^n$ $(n \in \mathbb{N})$. In the case that $θ\le 1$, 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 $θ> 1$ with a smallness condition on $p$.