Global bounded and unbounded solutions to a chemotaxis system with indirect signal production

TL;DR

This paper proves the global existence of solutions to a chemotaxis system with indirect signal production in 2D, identifying a critical mass that determines whether solutions remain bounded or become unbounded, extending previous radial symmetry results.

Contribution

It establishes the global well-posedness of the chemotaxis system with indirect signal production in 2D and introduces a new approach using a Liapunov functional to analyze solution behavior.

Findings

Solutions are global in 2D unlike classical models.

Existence of a critical mass $M_c$ separating bounded and unbounded solutions.

Extension of previous radial symmetry results to arbitrary 2D domains.

Abstract

The well-posedness of a chemotaxis system with indirect signal production in a two-dimensional domain is shown, all solutions being global unlike the classical Keller-Segel chemotaxis system. Nevertheless, there is a threshold value $M_c$ of the mass of the first component which separates two different behaviours: solutions are bounded when the mass is below $M_c$ while there are unbounded solutions starting from initial conditions having a mass exceeding $M_c$. This result extends to arbitrary two-dimensional domains a previous result of Tao \& Winkler (2017) obtained for radially symmetric solutions to a simplified version of the model in a ball and relies on a different approach involving a Liapunov functional.