Strong damping effect of chemo-repulsion prevents blow-up

Abstract

In this paper, we study Keller–Segel type chemotaxis systems with power-like nonlinear sensitivity, production of signals, and switching chemotaxis mechanism. We establish explicit relations to ensure local- and global-in-time boundedness of classical solutions. In the chemo-attractive setting, our results cover and unify separate cases and they are critical to the quite known blow-up results in the existing literature, while, in the chemo-repulsive setting, we find that much wider regimes compared to the attraction case can ensure global existence and boundedness. In comparison to the known results on the dichotomy between global solvability and blow-up for the associated chemo-attraction system, our findings reveal that the strong damping effect of chemo-repulsion prevents blow-up. Furthermore, our 3D local-in-time boundedness moves one step further toward the yet-to-be-proven popular saying that no blow-up would occur in the 3D minimal chemo-repulsion model.