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

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 \begin{document}$ M_c $\end{document} of the mass of the first component which separates two different behaviours: solutions are bounded when the mass is below \begin{document}$ M_c $\end{document} while there are unbounded solutions starting from initial conditions having a mass exceeding \begin{document}$ M_c $\end{document} . 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.