Stable stationary states of non-local interaction equations

In this paper, we are interested in the large-time behaviour of a solution to a non-local interaction equation, where a density of particles/individuals evolves subject to an interaction potential and an external potential. It is known that for regular interaction potentials, stable stationary states of these 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 imply local non-linear stability. Finally, we show that for regular repulsive interaction potential W_ε converging to a singular repulsive interaction potential W, the Dirac-type stationary states ρ̄ _ε approximate weakly a unique stationary state ρ̄ ∈ L^∞. We illustrate our results with numerical examples.