TY - JOUR
T1 - Stability of stationary states of non-local equations with singular interaction potentials
AU - Fellner, Klemens
AU - Raoul, Gaël
N1 - KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: 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.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.
PY - 2011/4
Y1 - 2011/4
N2 - 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.
AB - 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.
UR - http://hdl.handle.net/10754/599711
UR - https://linkinghub.elsevier.com/retrieve/pii/S0895717710001391
UR - http://www.scopus.com/inward/record.url?scp=79951580992&partnerID=8YFLogxK
U2 - 10.1016/j.mcm.2010.03.021
DO - 10.1016/j.mcm.2010.03.021
M3 - Article
SN - 0895-7177
VL - 53
SP - 1436
EP - 1450
JO - Mathematical and Computer Modelling
JF - Mathematical and Computer Modelling
IS - 7-8
ER -