Stability of stationary states of non-local equations with singular interaction potentials. (English) Zbl 1219.35342
Summary: 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.
MSC:
| 35R09 | Integro-partial differential equations |
| 35B35 | Stability in context of PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
non-local evolution equation; singular interaction potential; stability analysis; numerical simulationReferences:
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