Stationary states of quadratic diffusion equations with long-range attraction
Burger M, Di Francesco M, Franek M (2013)
Publication Language: English
Publication Type: Journal article
Publication year: 2013
Journal
Book Volume: 11
Pages Range: 709-738
Issue: 3
DOI: 10.4310/CMS.2013.v11.n3.a3
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 ow 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 diusivity 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 x0, compactly supported on sets of the form [x0-L,x0+L], C2 on their support, and strictly decreasing on (x0,x0+L). Moreover, they are global minimizers of the energy functional E. The results are complemented by numerical simulations. © 2013 International Press.
Authors with CRIS profile
Involved external institutions
Autonomous University of Barcelona (UAB) / Universitat Autònoma de Barcelona
Spain (ES)
Westfälische Wilhelms-Universität (WWU) Münster
Germany (DE)
Spain (ES)
Westfälische Wilhelms-Universität (WWU) Münster
Germany (DE)
How to cite
APA:
Burger, M., Di Francesco, M., & Franek, M. (2013). Stationary states of quadratic diffusion equations with long-range attraction. Communications in Mathematical Sciences, 11, 709-738. https://dx.doi.org/10.4310/CMS.2013.v11.n3.a3
MLA:
Burger, Martin, Marco Di Francesco, and Marzena Franek. "Stationary states of quadratic diffusion equations with long-range attraction." Communications in Mathematical Sciences 11 (2013): 709-738.
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