2014/03/17 by Jacob Bedrossian, Bedrossian, Jacob
Mathematics · Medicine · #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.1403.4124
openalex publication_date 2014/03/17 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
We consider a class of L1 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 L1 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 Lp estimates similar to those used in small-data and local theory for PKS-type equations.