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In this paper, we investigate nonlocal interaction equations withrepulsive-attractive radial potentials. Such equations describe the evolutionof a continuum density of particles in which they repulse each other in theshort range and attract each other in the long range. We prove that under someconditions on the potential, radially symmetric solutions convergeexponentially fast in some transport distance toward a spherical shellstationary state. Otherwise we prove that it is not possible for a radiallysymmetric solution to converge weakly toward the spherical shell stationarystate. We also investigate under which condition it is possible for anon-radially symmetric solution to converge toward a singular stationary statesupported on a general hypersurface. Finally we provide a detailed analysis ofthe specific case of the repulsive-attractive power law potential as well asnumerical results. We point out the the conditions of radial ins/stability aresharp.

Nonlocal interactions by repulsive-attractive potentials: radial ins/stability