Open AccessFast Diffusion leads to partial mass concentration in Keller–Segel type stationary solutions
Author(s) -
José Antonio Ortega Carrillo,
Matías G. Delgadino,
Rupert L. Frank,
Mathieu Lewin
Publication year - 2022
Publication title -
mathematical models and methods in applied sciences/mathematical models and methods in applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.582
H-Index - 85
eISSN - 1793-4060
pISSN - 0218-2025
DOI - 10.1142/s021820252250018x
Subject(s) - exponent , diffusion , quartic function , mathematics , type (biology) , range (aeronautics) , homogeneous , mathematical analysis , statistical physics , space (punctuation) , computation , physics , thermodynamics , combinatorics , pure mathematics , computer science , materials science , ecology , philosophy , linguistics , biology , composite material , algorithm , operating system
We show that partial mass concentration can happen for stationary solutions of aggregation–diffusion equations with homogeneous attractive kernels in the fast diffusion range. More precisely, we prove that the free energy admits a radial global minimizer in the set of probability measures which may have part of its mass concentrated in a Dirac delta at a given point. In the case of the quartic interaction potential, we find the exact range of the diffusion exponent where concentration occurs in space dimensions [Formula: see text]. We then provide numerical computations which suggest the occurrence of mass concentration in all dimensions [Formula: see text], for homogeneous interaction potentials with higher power.