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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=Δu - \nabla \cdot (u\nabla v), \qquad x\inΩ, \ t>0, [1mm] v_t=Δv-v+u, \qquad x\inΩ, \ t>0, where $Ω$ 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 $(u_0,v_0) \in C^0(\barΩ) \times W^{1,\infty}(Ω)$ with $\int_Ωu_0=m$ 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 $L^p(Ω) \times W^{1,2}(Ω)$ for any $p \in (1,\frac{2n}{n+2})$.