Stationary states of an aggregation equation with degenerate diffusion and bounded attractive potential

TL;DR

This paper studies stationary solutions of a non-local aggregation equation with degenerate diffusion and bounded attractive potential, characterizing their properties and existence in various dimensions.

Contribution

It provides a comprehensive analysis of stationary solutions, including existence, uniqueness, and support properties, for a broad class of degenerate diffusion exponents.

Findings

Global minimizers are compactly supported.

Unique stationary solution for quadratic diffusion up to translation.

Existence of stationary solutions depends on diffusion parameters.

Abstract

We investigate stationary solutions of a non-local aggregation equation with degenerate power-law diffusion and bounded attractive potential in arbitrary dimensions. Compact stationary solutions are characterized and compactness considerations are used to derive the existence of global minimizers of the corresponding energy depending on the prefactor of the degenerate diffusion for all exponents of the degenerate diffusion greater than one. We show that a global minimizer is compactly supported and, in case of quadratic diffusion, we prove that it is the unique stationary solution up to a translation. The existence of stationary solutions being only local minimizers is discussed.