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We analyze under which conditions equilibration between two competingeffects, repulsion modeled by nonlinear diffusion and attraction modeled bynonlocal interaction, occurs. This balance leads to continuous compactlysupported radially decreasing equilibrium configurations for all masses. Allstationary states with suitable regularity are shown to be radially symmetricby means of continuous Steiner symmetrization techniques. Calculus ofvariations tools allow us to show the existence of global minimizers amongthese equilibria. Finally, in the particular case of Newtonian interaction intwo dimensions they lead to uniqueness of equilibria for any given mass up totranslation and to the convergence of solutions of the associated nonlinearaggregation-diffusion equations towards this unique equilibrium profile up totranslations as $t\to\infty$.

Nonlinear Aggregation-Diffusion Equations: Radial Symmetry and Long Time Asymptotics