# American Institute of Mathematical Sciences

December  2019, 24(12): 6419-6444. doi: 10.3934/dcdsb.2019145

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

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

Received  October 2018 Revised  February 2019 Published  December 2019 Early access  July 2019

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.

Citation: Philippe Laurençot. Global bounded and unbounded solutions to a chemotaxis system with indirect signal production. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6419-6444. doi: 10.3934/dcdsb.2019145
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