  Unboundedness of solutions is shown to occur in a one-dimensional quasilinear
parabolicparabolic chemotaxis system for any initial mass. Our result is also
independent of the relation between the speeds of the diffusion of cells and
chemoattractant. The proof is achieved by contradiction. It uses a virial type
inequality together with a boundedness from below of the Liapunov functional
associated to the system. Moreover, the same method is applied also in higher
dimensions and an unboundedness result is shown in the case of critical
quasilinear fully parabolic Keller-Segel system for mass large enough in
dimensions n\geq 3. Key words: chemotaxis, infinite-time blowup.
