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

TL;DR

This paper studies a one-dimensional Keller-Segel model with boundary chemical production, revealing a mass-dependent dichotomy between global existence, finite-time blow-up, and equilibrium convergence, using entropy methods.

Contribution

It introduces a Keller-Segel model with boundary chemical production and characterizes its solution behavior based on mass, extending classical results to boundary-driven dynamics.

Findings

Solutions are global if mass is below critical

Solutions blow up in finite time if mass exceeds critical

Solutions converge to equilibrium at critical mass

Abstract

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).