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    Stability of stationary states of non-local equations with singular interaction potentials

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    Type
    Article
    Authors
    Fellner, Klemens
    Raoul, Gaël
    Date
    2011-04
    Permanent link to this record
    http://hdl.handle.net/10754/599711
    
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    Abstract
    We study the large-time behaviour of a non-local evolution equation for the density of particles or individuals subject to an external and an interaction potential. In particular, we consider interaction potentials which are singular in the sense that their first derivative is discontinuous at the origin.For locally attractive singular interaction potentials we prove under a linear stability condition local non-linear stability of stationary states consisting of a finite sum of Dirac masses. For singular repulsive interaction potentials we show the stability of stationary states of uniformly bounded solutions under a convexity condition.Finally, we present numerical simulations to illustrate our results. © 2010 Elsevier Ltd.
    Citation
    Fellner K, Raoul G (2011) Stability of stationary states of non-local equations with singular interaction potentials. Mathematical and Computer Modelling 53: 1436–1450. Available: http://dx.doi.org/10.1016/j.mcm.2010.03.021.
    Sponsors
    Both authors would like to thank Prof. Jose A. Carrillo, Dr. Marco Di Francesco and Prof. Christian Schmeiser for many valuable discussions. KF 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). GR has partially been supported by the DEASE program affiliated at the WPI, Wolfgang Pauli Institute, University of Vienna, and by the ANR grant CBDif.
    Publisher
    Elsevier BV
    Journal
    Mathematical and Computer Modelling
    DOI
    10.1016/j.mcm.2010.03.021
    ae974a485f413a2113503eed53cd6c53
    10.1016/j.mcm.2010.03.021
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