  We consider a parabolic-elliptic Keller-Segel type system, which is related
to a simplified model of chemotaxis. Concerning the maximal range of existence
of solutions, there are essentially two kinds of results: either global
existence in time for general subcritical ($\|\rho_0\|_1<8\pi$) initial data,
or blow--up in finite time for suitably chosen supercritical
($\|\rho_0\|_1>8\pi$) initial data with concentration around finitely many
points. As a matter of fact there are no results claiming the existence of
global solutions in the supercritical case. We solve this problem here and
prove that, for a particular set of initial data which share large
supercritical masses, the corresponding solution is global and uniformly
bounded.
