Radial symmetry and mass-independent boundedness of stationary states of aggregation-diffusion models

Abstract

General aggregation diffusion equations have been used in a variety of different settings, including the modelling of chemotaxis and the biological aggregation of insects and herding of animals. We consider a non-local aggregation diffusion equation, where the repulsion is modelled by nonlinear diffusion (Laplace operator applied to $ m $th power of the spatial density) and attraction modelled by non-local interaction. The competition between these forces gives rise to characteristic time-independent morphologies. When the attractive interaction kernel is radially symmetric and strictly increasing with respect to the norm in the $ n $-dimensional linear space of the space variable, it is previously known that all stationary solutions are radially symmetric and decreasing up to a translation. We extend this result to attractive kernels with compact support, where a wider variety of time-independent patterns occur. We prove that for compactly supported attractive kernels and for power in the diffusion term $ m>1 $, all stationary states are radially symmetric and decreasing up to a translation on each connected component 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. This result is valid for both attractive kernels with compact support and unbounded support. Finally, we investigate a model that incorporates both non-local attraction and non-local repulsion. We show that this model may be considered as a generalization of the aggregation diffusion equation and we present numerical results showing that $ m=2 $ is a threshold value such that, for $ m>2 $, stationary states of the fully non-local model possess a mass-independent upper-bound.