Large time behavior of nonlocal aggregation models with nonlinear diffusion

Abstract

The aim of this paper is to establish rigorous results on thelarge time behavior of nonlocal models for aggregation, includingthe possible presence of nonlinear diffusion terms modeling localrepulsions. We show that, as expected from the practicalmotivation as well as from numerical simulations, one obtainsconcentrated densities (Dirac $\delta$ distributions) asstationary solutions and large time limits in the absence ofdiffusion. In addition, we provide a comparison for aggregationkernels with infinite respectively finite support. In the firstcase, there is a unique stationary solution corresponding toconcentration at the center of mass, and all solutions of theevolution problem converge to the stationary solution for largetime. The speed of convergence in this case is just determined bythe behavior of the aggregation kernels at zero, yielding eitheralgebraic or exponential decay or even finite time extinction. Forkernels with finite support, we show that an infinite number ofstationary solutions exist, and solutions of the evolution problemconverge only in a measure-valued sense to the set of stationarysolutions, which we characterize in detail.Moreover, we also consider the behavior in the presence of nonlinear diffusion terms,the most interesting case being the one of small diffusion coefficients. Via the implicit function theorem we give aquite general proof of a rather natural assertion for such models, namely that there exist stationary solutionsthat have the form of a local peak around the center of mass. Our approach even yields the order of the size ofthe support in terms of the diffusion coefficients.All these results are obtained via a reformulation of theequations considered using the Wasserstein metric for probabilitymeasures, and are carried out in the case of a single spatialdimension.