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.
Highlights
The goal of this work is to investigate properties of the following class of homogeneous functionals, defined for centred probability densities ρ(x), belonging to suitable Lp-spaces, and some interaction strength
The sub-critical case is analysed in scaled variables and we show the existence of global minimisers with the properties above leading to the existence of self-similar solutions in original variables
A full proof of non-criticality involves the analysis of the minimisation problem in scaled variables as for k < 0 showing that global minimisers exist in the right functional spaces for all values of the critical parameter and that they are stationary states
Summary
The goal of this work is to investigate properties of the following class of homogeneous functionals, defined for centred probability densities ρ(x), belonging to suitable Lp-spaces, and some interaction strength. Performing gradient flows of non-convex functionals is much more delicate, and one has to seek compensations Such compensations do exist in our case, and we will observe them at the level of existence of minimisers for the free energy functional Fm,k and stationary states of the family of PDEs (1.2) in particular regimes. There is one related result in [29] for the limiting case in one dimension taking m = 0, corresponding to logarithmic diffusion, and k = 1 They showed that no criticality is present in that case as solutions to (1.2) with (m = 0, k = 1) are globally defined in time for all values of the parameter χ > 0. A full proof of non-criticality involves the analysis of the minimisation problem in scaled variables as for k < 0 showing that global minimisers exist in the right functional spaces for all values of the critical parameter and that they are stationary states. We illustrate these results by numerical experiments in one dimension corroborating the absence of critical behaviour for k > 0
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