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

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 $\mathrm{\Omega }$ 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 $L^p(\Omega) \times W^{1,2}(\Omega)$forany$p \in (1,\frac{2n}{n+2})$.