Stationary states of quadratic diffusion equations with long-range attraction

M. Burger, Marco Di Francesco, M. Franek

Research output: Contribution to journalArticlepeer-review

25 Citations (Scopus)
103 Downloads (Pure)

Abstract

We study the existence and uniqueness of nontrivial stationary solutions to a nonlocal aggregation equation with quadratic diffusion arising in many contexts in
population dynamics. The equation is the Wasserstein gradient flow generated by the
energy E, which is the sum of a quadratic free energy and the interaction energy. The
interaction kernel is taken radial and attractive, nonnegative and integrable, with further technical smoothness assumptions. The existence vs. nonexistence of such solutions is ruled by a threshold phenomenon, namely nontrivial steady states exist if and only if the diffusivity constant is strictly smaller than the total mass of the interaction kernel.
In the one dimensional case we prove that steady states are unique up to translations and mass constraint. The strategy is based on a strong version of the Krein-Rutman theorem. The steady states are symmetric with respect to their center of mass, supported on compact intervals, strictly
decreasing on the right half interval. Moreover, they are global minimizers of the energy functional E. The results are complemented by numerical simulations.
Original languageEnglish
Pages (from-to)709-738
Number of pages30
JournalCommunications in Mathematical Sciences
Volume11
Issue number3
Publication statusPublished - 2013

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