Best initial criterion for a degenerate Keller–Segel system with rotational flux terms in even-dimensional spaces

Abstract

This paper presents a rigorous analysis for a degenerate diffusion Keller–Segel type system featuring rotational flux terms in even-dimensional spaces. The proposed model synthesizes fundamental principles from mathematical biology, fluid dynamics, and electrokinetic phenomena. Through discussion of the diffusion exponent $m\in [\frac{2n}{n+2},2-\frac{2}{n})$ and rotation angle $\alpha \in (2k\pi -\pi ,2k\pi +\pi ]$, $k\in \mathbb{Z}$, we obtain results on the existence and blow-up of solutions. In particular, we derive a sharp initial criterion to distinguish the global existence and the finite-time blow-up of solutions for the case $\alpha \in (2k\pi -\frac{\pi }{2},2k\pi +\frac{\pi }{2})$, $k\in \mathbb{Z}$ and $m\in (\frac{2n}{n+2},2-\frac{2}{n})$, and show the existence of solutions under any initial condition for the case $\alpha \in (2k\pi -\pi ,2k\pi -\frac{\pi }{2}]\cup [2k\pi +\frac{\pi }{2},2k\pi +\pi ]$, $k\in \mathbb{Z}$. This result reveals that repulsive effects caused by rotation effectively prevent solution blow-up.