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.
Highlights
We consider a family of partial differential equations modelling self-attracting diffusive particles at the macroscopic scale,{ ∂tρ = ∆ρm + ∇ · (ρ∇S), t > 0, x ∈ RN, ρ(t = 0, x) = ρ0(x), (1.1)where the diffusion exponent m > 1 is of porous medium type [31]
Our goal here is to extend the results on the uniqueness of stationary states of system (1.3) to more singular k, higher dimensions N and any m ≥ mc by building on the techniques employed in [12]
An important point to make is that due to the results in [16,17], any stationary solution in the sense of Definition 1 in all the cases for m, W and μresc discussed in the previous paragraphs are radially symmetric decreasing about their centre of mass and compactly supported. This means that the question of uniqueness for stationary states is reduced to the radial setting
Summary
We consider a family of partial differential equations modelling self-attracting diffusive particles at the macroscopic scale,. The energy functional is homogeneous if attraction and repulsion are in balance, so that the two terms of the energy scale with the same power, that is, if m = mc for k mc := 1 − N This motivates the definition of three different regimes: the diffusion-dominated regime m > mc, the fair-competition regime m = mc, and the attraction-dominated regime 0 < m < mc. We prove that any radial critical point of the energy functional is a global minimiser (as if the functional would be convex), and we control the equality cases This amounts to estimate precisely the balance between the convex part (non-linear diffusion) and the non-convex part (non-local attraction) in order to show that convexity is strong enough to discard any other critical point than the global minimum. Our methodology strongly relies on radial symmetry, so that [16,17] is a prerequisite to our result
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