  The parabolic-elliptic cross-diffusion system \[
  \left\{ \begin{array}{l}
  u_t = \Delta u - \nabla \cdot \Big(uf(|\nabla v|^2) \nabla v \Big), \\[1mm]
  0 = \Delta v - \mu + u,
  \qquad \int_\Omega v=0,
  \qquad
  \mu:=\frac{1}{|\Omega|} \int_\Omega u dx,
  \end{array} \right. \] is considered along with homogeneous Neumann-type
boundary conditions in a smoothly bounded domain $\Omega\subset R^n$, $n\ge 1$,
where $f$ generalizes the prototype given by \[
  f(\xi) = (1+\xi)^{-\alpha},
  \qquad \xi\ge 0,
  \qquad \mbox{for all } \xi\ge 0, \] with $\alpha\in R$.
  In this framework, the main results assert that if $n\ge 2$, $\Omega$ is a
ball and \[
  \alpha<\frac{n-2}{2(n-1)}, \] then throughout a considerably large set of
radially symmetric initial data, an associated initial value problem admits
solutions blowing up in finite time with respect to the $L^\infty$ norm of
their first components.
  This is complemented by a second statement which ensures that in general and
not necessarily symmetric settings, if either $n=1$ and $\alpha\in R$ is
arbitrary, or $n\ge 2$ and $\alpha>\frac{n-2}{2(n-1)}$, then any explosion is
ruled out in the sense that for arbitrary nonnegative and continuous initial
data, a global bounded classical solution exists.
