Blow-up prevention by sub-logistic sources in 2d Keller-Segel system

Abstract

This paper investigates the global existence of solutions to Keller-Segel systems with sub-logistic sources using the test function method. Prior work demonstrated that sub-logistic sources $f(u)=ru-\mu \frac{u^2}{{\ln }^p(u+e)}$ with $p\in (0,1)$ can prevent blow-up solutions for the 2D minimal Keller-Segel chemotaxis model. Our study extends this result by showing that when $p=1$, sub-logistic sources can still prevent the occurrence of finite time blow-up solutions. Additionally, we provide a concise proof for a known result that the equi-integrability of ${\left\{{\int }_{\Omega }{u}^{\frac{n}{2}}(\cdot ,t)\right\}}_{t\in (0,T_{\max })}$ can avoid blow-up.