A sharp mass threshold for boundedness in a critical quasilinear chemotaxis model with indirect signal production: The radially symmetric case

A critical quasilinear chemotaxis model with indirect signal production in space dimensions higher than two is studied under radial symmetry. Exploiting the variational structure of the system, the existence of a sharp mass threshold M\textsubscr{c} > 0 is established: all solutions with mass below M\textsubscr{c} are global and bounded, while those with mass above M\textsubscr{c} and sufficiently negative energy are global but blow up in infinite time. Moreover, the value of the critical mass M\textsubscr{c} is related to the best constant of a variant of a Hardy-Littlewood-Sobolev inequality.