  We consider a Keller-Segel model with non-linear porous medium type diffusion
and nonlocal attractive power law interaction, focusing on potentials that are
less singular than Newtonian interaction. Here, the nonlinear diffusion is
chosen to be $\frac{2d}{d+2s}<m<2-\frac{2s}{d}$ in which case the steady states
are compactly supported. We analyse under which conditions on the initial data
the regime that attractive forces are stronger than diffusion occurs and
classify the global existence and finite time blow-up of solutions. It is shown
that there is a threshold value which is characterized by the optimal constant
of a variant of Hardy-Littlewood-Sobolev inequality such that the solution will
exist globally if the initial data is below the threshold, while the solution
blows up in finite time when the initial data is above the threshold.
