Stable stationary states of non-local interaction equations

analysis, numerical simulation In this article, we are interested in the large-time behaviour of a solution to a non-local interaction equation, where a density of par-ticles/individuals evolves subject to an interaction potential and an external potential. It is known that for regular interaction potentials, stable stationary states of this 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 implies local non-linear stability. Fi-nally, we show that for regular repulsive interaction potential Wε con-verging to a singular repulsive interaction potentialW, the Dirac-type stationary states ρ̄ε approximate weakly a unique stationary state ρ ̄ ∈ L∞. We illustrate our results with numerical examples.