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

Philippe Laurençot

doi:10.3934/dcdsb.2019145

Article Contents

2019, Volume 24, Issue 12: 6419-6444. Doi: 10.3934/dcdsb.2019145

This issue Previous Article A hybrid method for stiff reaction–diffusion equations Next Article Second-order linear structure-preserving modified finite volume schemes for the regularized long wave equation

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

Philippe Laurençot

Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, F–31062 Toulouse Cedex 9, France

Received: October 2018

Revised: February 2019

Early access: July 2019

Published: September 2019

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Abstract

The well-posedness of a chemotaxis system with indirect signal production in a two-dimensional domain is shown, all solutions being global unlike for 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.

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Mathematics Subject Classification: Primary: 35B40; Secondary: 35M33, 35K10, 35Q92.

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