  We consider a class of $L^1$ critical nonlocal aggregation equations with
linear or nonlinear porous media-type diffusion which are characterized by a
long-range interaction potential that decays faster than the Newtonian
potential at infinity. The fast decay breaks the $L^1$ scaling symmetry and we
prove that `sufficiently spread out' initial data, regardless of the mass,
result in global spreading solutions. This is in contrast to the classical
parabolic-elliptic PKS for which essentially all solutions with more than
critical mass are known to blow up in finite time. In all cases, the long-time
asymptotics are given by the self-similar solution to the linear heat equation
or by the Barenblatt solutions of the porous media equation. The results with
linear diffusion are proved using properties of the Fokker-Planck semi-group
whereas the results with nonlinear diffusion are proved using a more
interesting bootstrap argument coupling the entropy-entropy dissipation methods
of the porous media equation together with higher $L^p$ estimates similar to
those used in small-data and local theory for PKS-type equations.

Analysis of PDEs