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

Nikolaos Bournaveas (1) , Vincent Calvez (2, 3)

1 School of Mathematics - University of Edinburgh
2 DMA - Département de Mathématiques et Applications - ENS Paris
3 NUMED - Numerical Medicine

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.

Mots clés: Chemotaxis; Kinetic equation; Blow-up; Virial identity

Domaines: Equations aux dérivées partielles [math.AP]

Nikolaos Bournaveas, Vincent Calvez. Critical mass phenomenon for a chemotaxis kinetic model with spherically symmetric initial data. 2008. ⟨hal-00349928⟩