A one-dimensional Keller-Segel equation with a drift issued from the boundary

Vincent Calvez (1, 2) , Nicolas Meunier (3)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 NUMED - Numerical Medicine
3 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

We investigate in this note the dynamics of a one-dimensional Keller-Segel type model on the half-line. On the contrary to the classical configuration, the chemical production term is located on the boundary. We prove, under suitable assumptions, the following dichotomy which is reminiscent of the two-dimensional Keller-Segel system. Solutions are global if the mass is below the critical mass, they blow-up in finite time above the critical mass, and they converge to some equilibrium at the critical mass. Entropy techniques are presented which aim at providing quantitative convergence results for the subcritical case. This note is completed with a brief introduction to a more realistic model (still one-dimensional).

Mots clés: chemotaxis; cell polarization; relative entropy; blow-up

Domaines: Equations aux dérivées partielles [math.AP]

Vincent Calvez, Nicolas Meunier. A one-dimensional Keller-Segel equation with a drift issued from the boundary. 2009. ⟨hal-00424649⟩