Research output: Contribution to journalArticlepeer-review

70 Scopus citations


In this paper, we are interested in the large-time behaviour of a solution to a non-local interaction equation, where a density of particles/individuals evolves subject to an interaction potential and an external potential. It is known that for regular interaction potentials, stable stationary states of these equations are generically finite sums of Dirac masses. For a finite sum of Dirac masses, we give (i) a condition to be a stationary state, (ii) two necessary conditions of linear stability w.r.t. shifts and reallocations of individual Dirac masses, and (iii) show that these linear stability conditions imply local non-linear stability. Finally, we show that for regular repulsive interaction potential Wε converging to a singular repulsive interaction potential W, the Dirac-type stationary states ρ̄ ε approximate weakly a unique stationary state ρ̄ ∈ L∞. We illustrate our results with numerical examples. © 2010 World Scientific Publishing Company.
Original languageEnglish (US)
Pages (from-to)2267-2291
Number of pages25
JournalMathematical Models and Methods in Applied Sciences
Issue number12
StatePublished - Dec 2010
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-I1-007-43
Acknowledgements: Both authors would like to thank Prof. Christian Schmeiser for initiating the research done in this paper and Dr. Marco Di Francesco for many valuable discussions. K.F. has been supported by Award No. KUK-I1-007-43 of Peter A. Markowich, made by King Abdullah University of Science and Technology (KAUST) and by the bilateral Austria-France project (Austria: FR 05/2007 France: Amadeus 13785 UA). G.R. has been partially supported by the DEASE program affiliated at the WPI, Wolfgang Pauli Institute, University of Vienna.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.


Dive into the research topics of 'STABLE STATIONARY STATES OF NON-LOCAL INTERACTION EQUATIONS'. Together they form a unique fingerprint.

Cite this