  As it is well known, the parabolic-elliptic Keller-Segel system of chemotaxis
on the plane has global-in-time regular nonnegative solutions with total mass
below the critical value $8\pi$. Solutions with mass above $8\pi$ blow up in a
finite time. We show that the case of the parabolic-parabolic Keller-Segel is
different: each mass may lead to a global-in-time-solution, even if the initial
data is a finite signed measure.
  These solutions need not be unique, even if we limit ourselves to nonnegative
solutions.
