An aggregation equation with degenerate diffusion: qualitative property of solutions
Abstract
We study a nonlocal aggregation equation with degenerate diffusion, set in a periodic domain. This equation represents the generalization to $m > 1$ of the McKeanVlasov equation where here the "diffusive" portion of the dynamics are governed by Porous medium selfinteractions. We focus primarily on $m\in(1,2]$ with particular emphasis on $m = 2$. In general, we establish regularity properties and, for small interaction, exponential decay to the uniform stationary solution. For $m=2$, we obtain essentially sharp results on the rate of decay for the entire regime up to the (sharp) transitional value of the interaction parameter.
 Publication:

arXiv eprints
 Pub Date:
 April 2012
 DOI:
 10.48550/arXiv.1204.3938
 arXiv:
 arXiv:1204.3938
 Bibcode:
 2012arXiv1204.3938C
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematical Physics