-
Abstract
The aim of this paper is to establish rigorous results on the
large time behavior of nonlocal models for aggregation, including
the possible presence of nonlinear diffusion terms modeling local
repulsions. We show that, as expected from the practical
motivation as well as from numerical simulations, one obtains
concentrated densities (Dirac $\delta$ distributions) as
stationary solutions and large time limits in the absence of
diffusion. In addition, we provide a comparison for aggregation
kernels with infinite respectively finite support. In the first
case, there is a unique stationary solution corresponding to
concentration at the center of mass, and all solutions of the
evolution problem converge to the stationary solution for large
time. The speed of convergence in this case is just determined by
the behavior of the aggregation kernels at zero, yielding either
algebraic or exponential decay or even finite time extinction. For
kernels with finite support, we show that an infinite number of
stationary solutions exist, and solutions of the evolution problem
converge only in a measure-valued sense to the set of stationary
solutions, 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 a
quite general proof of a rather natural assertion for such models, namely that there exist stationary solutions
that have the form of a local peak around the center of mass. Our approach even yields the order of the size of
the support in terms of the diffusion coefficients.
All these results are obtained via a reformulation of the
equations considered using the Wasserstein metric for probability
measures, and are carried out in the case of a single spatial
dimension.
Mathematics Subject Classification: 35B30, 35B40, 35B41, 45K05, 92D25.
\begin{equation} \\ \end{equation}
-
Access History
-