  This paper is concerned with radially symmetric solutions of the
parabolic-elliptic version of the Keller-Segel system with flux limitation, as
given by
  \begin{equation}
  \left\{ \begin{array}{l} \displaystyle u_t=\nabla \cdot \Big(\frac{u\nabla
u}{\sqrt{u^2+|\nabla u|^2}}\Big) - \chi \, \nabla \cdot \Big(\frac{u\nabla
v}{\sqrt{1+|\nabla v|^2}}\Big), \\[1mm]
  0=\Delta v - \mu + u,
  \end{array} \right.
  \qquad \qquad (\star)
  \end{equation} under the initial condition $u|_{t=0}=u_0>0$ and no-flux
boundary conditions in a ball $\Omega\subset R^n$, where $\chi>0$ and
$\mu:=\frac{1}{|\Omega|} \int_\Omega u_0$.
  A previous result [3] has asserted global existence of bounded classical
solutions for arbitrary positive radial initial data $u_0\in C^3(\bar\Omega)$
when either $n\ge 2$ and $\chi<1$, or $n=1$ and $ \int_\Omega
u_0<\frac{1}{\sqrt{(\chi^2-1)_+}}$.
  This present paper shows that these conditions are essentially optimal:
Indeed, it is shown that if the taxis coefficient is large enough in the sense
that $\chi>1$, then for any choice of
  \begin{equation}
  \left\{ \begin{array}{ll} m>\frac{1}{\sqrt{\chi^2-1}} \quad & \mbox{if } n=1,
\\[2mm] m>0 \mbox{ is arbitrary } \quad & \mbox{if } n\ge 2,
  \end{array} \right.
  \end{equation} there exist positive initial data $u_0\in C^3(\bar\Omega)$
satisfying $ \int_\Omega u_0=m$ which are such that for some $T>0$, ($\star$)
possesses a uniquely determined classical solution $(u,v)$ in $\Omega\times
(0,T)$ blowing up at time $T$ in the sense that $\limsup_{t\nearrow T}
\|u(\cdot,t)\|_{L^\infty(\Omega)}=\infty$.\abs
  This result is derived by means of a comparison argument applied to the
doubly degenerate scalar parabolic equation satisfied by the mass accumulation
function associated with ($\star$).
