  We consider two-dimensional versions of the Keller--Segel model for the
chemotaxis with either classical (Brownian) or fractional (anomalous)
diffusion. Criteria for blowup of solutions in terms of suitable Morrey spaces
norms are derived. Moreover, the impact of the consumption term on the
global-in-time existence of solutions is analyzed for the classical
Keller--Segel system.
