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

TL;DR

This paper studies global solutions for a class of nonlocal aggregation equations with decaying interaction potentials, showing that sufficiently spread initial data lead to spreading solutions, contrasting with blow-up in classical models.

Contribution

It establishes global existence and long-time behavior for L^1-critical nonlocal aggregation equations with fast-decaying potentials, extending understanding beyond classical PKS models.

Findings

Sufficiently spread initial data lead to global solutions.

Long-time asymptotics are self-similar solutions of heat or porous media equations.

Results apply to both linear and nonlinear diffusion cases.

Abstract

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.