  It is known that classical solutions to the one-dimensional quasilinear
Smoluchowski-Poisson system with nonlinear diffusion $a(u)=(1+u)^{-p}$ may blow
up in finite time if $p>1$ and exist globally if $p<1$. The case $p=1$ thus
appears to be critical but it turns out that all solutions are global also in
that case. Two classes of diffusion coefficients are actually identified in
this paper, one for which all solutions to the corresponding quasilinear
Smoluchowski-Poisson system are global and the other one leading to finite time
blow-up for sufficiently concentrated initial data. The cornerstone of the
proof are an alternative formulation of the Smoluchowski-Poisson system which
relies on a novel change of variables and a virial identity.
