Finite time blow-up for a one-dimensional quasilinear parabolic–parabolic chemotaxis system

Finite time blow-up is shown to occur for solutions to a one-dimensional quasilinear parabolic–parabolic chemotaxis system as soon as the mean value of the initial condition exceeds some threshold value. The proof combines a novel identity of virial type with the boundedness from below of the Liapunov functional associated to the system, the latter being peculiar to the one-dimensional setting.