Stability of stationary states of non-local equations with singular interaction potentials
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AbstractWe 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.
CitationFellner 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.
SponsorsBoth 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.