Critical mass phenomenon for a chemotaxis kinetic model with spherically symmetric initial data

TL;DR

This paper demonstrates a critical mass phenomenon in a chemotaxis kinetic model with spherical symmetry, showing that large initial mass leads to finite-time blow-up, while small mass does not, linking microscopic and macroscopic behaviors.

Contribution

It establishes the critical mass phenomenon at the kinetic level for a chemotaxis model, bridging microscopic features with the classical Keller-Segel system.

Findings

Large initial mass solutions blow up in finite time.

Small initial mass solutions do not blow up.

The kinetic model's blow-up criterion aligns with Keller-Segel predictions.

Abstract

The goal of this paper is to exhibit a critical mass phenomenon occuring in a model for cell self-organization via chemotaxis. The very well known dichotomy arising in the behavior of the macroscopic Keller-Segel system is derived at the kinetic level, being closer to microscopic features. Indeed, under the assumption of spherical symmetry, we prove that solutions with initial data of large mass blow-up in finite time, whereas solutions with initial data of small mass do not. Blow-up is the consequence of a virial identity and the existence part is derived from a comparison argument. Spherical symmetry is crucial within the two approaches. We also briefly investigate the drift-diffusion limit of such a kinetic model. We recover partially at the limit the Keller-Segel criterion for blow-up, thus arguing in favour of a global link between the two models.