  We study the Neumann initial-boundary value problem for the fully parabolic
Keller-Segel system u_t=\Delta u - \nabla \cdot (u\nabla v), \qquad x\in\Omega,
\ t>0, [1mm] v_t=\Delta v-v+u, \qquad x\in\Omega, \ t>0, where $\Omega$ is a
ball in ${\mathbb{R}}^n$ with $n\ge 3$.
  It is proved that for any prescribed $m>0$ there exist radially symmetric
positive initial data $(u_0,v_0) \in C^0(\bar\Omega) \times
W^{1,\infty}(\Omega)$ with $\int_\Omega u_0=m$ such that the corresponding
solution blows up in finite time.
  Moreover, by providing an essentially explicit blow-up criterion it is shown
that within the space of all radial functions, the set of such blow-up
enforcing initial data indeed is large in an appropriate sense; in particular,
this set is dense with respect to the topology of $L^p(\Omega) \times
W^{1,2}(\Omega)$ for any $p \in (1,\frac{2n}{n+2})$.
