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

Large mass global solutions for a class of L1-critical nonlocal aggregation equations and parabolic-elliptic Patlak-Keller-Segel models