For aggregation-diffusion equations with attractive potentials, this paper proves steady states of fixed mass are unique iff m >= 2, and constructs non-uniqueness examples for 1 < m < 2.
Uniqueness of stationary states for singular Keller-Segel type models
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abstract
We consider a generalised Keller-Segel model with non-linear porous medium type diffusion and non-local attractive power law interaction, focusing on potentials that are more singular than Newtonian interaction. We show uniqueness of stationary states (if they exist) in any dimension both in the diffusion-dominated regime and in the fair-competition regime when attraction and repulsion are in balance. As stationary states are radially symmetric decreasing, the question of uniqueness reduces to the radial setting. Our key result is a sharp generalised Hardy-Littlewood-Sobolev type functional inequality in the radial setting.
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Uniqueness and non-uniqueness of steady states of aggregation-diffusion equations
For aggregation-diffusion equations with attractive potentials, this paper proves steady states of fixed mass are unique iff m >= 2, and constructs non-uniqueness examples for 1 < m < 2.