  The existence of generalised global supersolutions with a control upon the
total mass is established for a wide family of parabolic-parabolic chemotaxis
systems and general integrable initial data in any space dimension. It is
verified that as long as a supersolution of this sort remains smooth, it
coincides with the classical solution. At the same time, the proposed
construction provides solvability beyond a blow-up time. The considered class
of systems includes the basic form of the Keller-Segel model as well as the
case of a chemorepellent.
