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 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.

References in corpus (3)

• Cooling process for inelastic Boltzmann equations for hard spheres, Part I: The Cauchy problem
• Virial theorem and dynamical evolution of self-gravitating Brownian particles and bacterial populations in an unbounded domain
• Tanaka Theorem for Inelastic Maxwell Models

Cited by in corpus (8)

• Stiff-response-induced instability for chemotactic bacteria and flux-limited Keller-Segel equation
• Kinetic chemotaxis tumbling kernel determined from macroscopic quantities
• Formation of singularities in solutions to ideal hydrodynamics of freely cooling inelastic gases
• Derivation of the bacterial run-and-tumble kinetic equation from a model with biochemical pathway
• On the asymptotic behaviour of a run and tumble equation for bacterial chemotaxis
• A High Order Stochastic Asymptotic Preserving Scheme for Chemotaxis Kinetic Models with Random Inputs
• Blow-up, concentration phenomenon and global existence for the Keller-Segel model in high dimension
• Aggregated Steady States of a Kinetic Model for Chemotaxis