Stationary states of aggregation-diffusion equations with compactly supported attraction kernels: radial symmetry and mass-independent boundedness

TL;DR

This paper investigates stationary solutions of aggregation-diffusion equations with radially symmetric, non-decreasing attraction kernels, establishing their radial symmetry, boundedness, and properties for different diffusion coefficients.

Contribution

It extends known symmetry results to non-decreasing kernels with bounded support and proves mass-independent bounds for stationary states when the diffusion coefficient exceeds 2.

Findings

Stationary states are radially symmetric and decreasing on each connected support component.

For diffusion coefficient m>2, stationary states have an upper-bound independent of initial data.

Results confirm and extend previous numerical findings.

Abstract

We consider a nonlocal aggregation diffusion equation incorporating repulsion modelled by nonlinear diffusion and attraction modelled by nonlocal interaction. When the attractive interaction kernel is radially symmetric and strictly increasing on its domain it is previously known that all stationary solutions are radially symmetric and decreasing up to a translation; however, this result has not been extended to accommodate attractive kernels that are non-decreasing, for instance, attractive kernels with bounded support. For the diffusion coefficient $m>1$, we show that, for attractive kernels that are radially symmetric and non-decreasing, all stationary states are radially symmetric and decreasing up to a translation on each connected subset of their support. Furthermore, for $m>2$, we prove analytically that stationary states have an upper-bound independent of the initial data, confirming previous numerical results given in the literature.