  In this paper we prove finite-time blowup of radially symmetric solutions to
the quasilinear parabolic-parabolic two-dimensional Keller-Segel system for any
positive mass. This is done in case of nonlinear diffusion and also in the case
of nonlinear cross-diffusion provided the nonlinear chemosensitivity term is
assumed not to decay. Moreover, it is shown that the above-mentioned lack of
non-decay assumption is essential with respect to keeping the dichotomy
finite-time blowup against boundedness of solutions. Namely, we prove that
without the non-decay assumption possible asymptotic behaviour of solutions
includes also infinite-time blowup.
