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

Abstract

The goal of this paper is to exhibit a critical mass phenomenon occurring 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 momentum computation 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.