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We study the Neumann initial-boundary value problem for the fully parabolicKeller-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 aball in ${\mathbb{R}}^n$ with $n\ge 3$.  It is proved that for any prescribed $m>0$ there exist radially symmetricpositive initial data $(u_0,v_0) \in C^0(\bar\Omega) \timesW^{1,\infty}(\Omega)$ with $\int_\Omega u_0=m$ such that the correspondingsolution blows up in finite time.  Moreover, by providing an essentially explicit blow-up criterion it is shownthat within the space of all radial functions, the set of such blow-upenforcing initial data indeed is large in an appropriate sense; in particular,this set is dense with respect to the topology of $L^p(\Omega) \timesW^{1,2}(\Omega)$ for any $p \in (1,\frac{2n}{n+2})$.

Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system