Global existence and infinite time blow-up of classical solutions to chemotaxis systems of local sensing in higher dimensions

TL;DR

This paper proves that classical solutions to a higher-dimensional chemotaxis system with local sensing exist globally and do not blow up in finite time, with some cases allowing blow-up at infinite time, using a novel analytical approach.

Contribution

It introduces a new method involving an auxiliary identity to establish global existence and infinite-time blow-up results for chemotaxis systems in higher dimensions.

Findings

No finite-time blow-up occurs in the system.

Solutions can blow up at infinite time under certain conditions.

The new approach simplifies the analysis of complex chemotaxis models.

Abstract

This paper deals with the fully parabolic chemotaxis system of local sensing in higher dimensions. Despite the striking similarity between this system and the Keller--Segel system, we prove the absence of finite-time blow-up phenomenon in this system even in the supercritical case. It means that for any regular initial data, independently of the magnitude of mass, the classical solution exists globally in time in the higher dimensional setting. Moreover, for the exponential decaying motility case, it is established that solutions may blow up at infinite time for any magnitude of mass. In order to prove our theorem, we deal with some auxiliary identity as an evolution equation with a time dependent operator. In view of this new perspective, the direct consequence of the abstract theory is rich enough to establish global existence of the system.