# Equilibria of aggregation-diffusion models with nonlinear potentials

## TL;DR

This paper studies equilibrium states in a nonlinear aggregation-diffusion model with a focus on nonlocal potentials, revealing their connection to functional inequalities and analyzing their behavior as the interaction becomes local.

## Contribution

It introduces a novel analysis of radial stationary states linked to Hardy-Littlewood-Sobolev inequalities and explores their asymptotic limits as the fractional parameter approaches zero.

## Key findings

- Radial stationary states relate to extremals of Hardy-Littlewood-Sobolev inequalities.
- Stationary states correspond to minimizers of a free energy functional when aggregation is weaker than diffusion.
- As the fractional parameter s approaches zero, the nonlocal interaction transitions to backward diffusion, and the stationary states' behavior is characterized.

## Abstract

We consider an evolution model with nonlinear diffusion of porous medium type in competition with a nonlocal drift term favoring mass aggregation. The distinguishing trait of the model is the choice of a nonlinear $(s,p)$ Riesz potential for describing the overall aggregation effect. We investigate radial stationary states of the dynamics, showing their relation with extremals of suitable Hardy-Littlewood-Sobolev inequalities. In the case that aggregation does not dominate over diffusion, radial stationary states also relate to global minimizers of a homogeneous free energy functional featuring the $(s,p)$ energy associated to the nonlinear potential. In the limit as the fractional parameter $s$ tends to zero, the nonlocal interaction term becomes a backward diffusion and we describe the asymptotic behavior of the stationary states.