  We study the blow-up asymptotics of radially decreasing solutions of the
parabolic-elliptic Keller-Segel-Patlak system in space dimensions $n\ge 3$. In
view of the biological background of this system and of its mass conservation
property, blowup is usually interpreted as a phenomenon of concentration or
aggregation of the bacterial population. Understanding the asymptotic behavior
of solutions at the blowup time is thus meaningful for the interpretation of
the model.
  Under mild assumptions on the initial data, for $n\ge 3$, we show that the
final profile satisfies $C_1|x|^{-2}\le u(x,T)\le C_2|x|^{-2}$, with
convergence in $L^1$ as $t\to T$. This is in sharp contrast with the
two-dimensional case, where solutions are known to concentrate to a Dirac mass
at the origin (plus an integrable part). We also obtain refined space-time
estimates of the form $u(x,t)\le C(T-t+|x|^2)^{-1}$ for type~I blowup
solutions. Previous work had shown that radial, self-similar blowup solutions
(which satisfy the above estimates) exist in dimensions $n\ge 3$ and do not
exist in dimension $2$. Our results thus reveal that the final profile
displayed by these special solutions actually corresponds to a much more
general phenomenon.
