Equilibria of homogeneous functionals in the fair-competition regime
File(s)
Article In Press
Author(s)
Calvez, V
Carrillo de la Plata, J
Hoffmann, F
Type
Journal Article
Abstract
We consider macroscopic descriptions of particles where repulsion is modelled by non-linear power-law diffusion and attraction by a homogeneous singular/non-singular kernel leading to variants of the Keller–Segel model of chemotaxis. We analyse the regime in which both homogeneities scale the same with respect to dilations, that we coin as fair-competition. In the singular kernel case, we show that existence of global equilibria can only happen at a certain critical value and they are characterised as optimisers of a variant of HLS inequalities. We also study the existence of self-similar solutions for the sub-critical case, or equivalently of optimisers of rescaled free energies. These optimisers are shown to be compactly supported radially symmetric and non-increasing stationary solutions of the non-linear Keller–Segel equation. On the other hand, we show that no radially symmetric non-increasing stationary solutions exist in the non-singular kernel case, implying that there is no criticality. However, we show the existence of positive self-similar solutions for all values of the parameter under the condition that diffusion is not too fast. We finally illustrate some of the open problems in the non-singular kernel case by numerical experiments.
Date Issued
2017-04-10
Date Acceptance
2017-03-16
Citation
Nonlinear Analysis, 2017, 159, pp.85-128
ISSN
0362-546X
Publisher
Elsevier
Start Page
85
End Page
128
Journal / Book Title
Nonlinear Analysis
Volume
159
Copyright Statement
© 2017 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
License URL
Sponsor
The Royal Society
Grant Number
WM120001
Subjects
Science & Technology
Physical Sciences
Mathematics, Applied
Mathematics
Aggregation-diffusion equations
Gradient flows
Minimisation
KELLER-SEGEL MODEL
GLOBAL EXISTENCE
CRITICAL MASS
BLOW-UP
DEGENERATE DIFFUSION
ASYMPTOTIC-BEHAVIOR
EQUATION
AGGREGATION
CHEMOTAXIS
SYSTEM
0101 Pure Mathematics
0102 Applied Mathematics
Applied Mathematics
Publication Status
Published