  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.

Analysis of PDEs