  Typically, aggregation-diffusion is modeled by parabolic equations that
combine linear or nonlinear diffusion with a Fokker-Planck convection term.
Under very general suitable assumptions, we prove that radial solutions of the
evolution process converge asymptotically in time towards a stationary state
representing the balance between the two effects. Our parabolic system is the
gradient flow of an energy functional, and in fact we show that the stationary
states are minimizers of a relaxed energy. Here, we study radial solutions of
an aggregation-diffusion model that combines nonlinear fast diffusion with a
convection term driven by the gradient of a potential, both in balls and the
whole space. We show that, depending on the exponent of fast diffusion and the
potential, the steady state is given by the sum of an explicit integrable
function, plus a Dirac delta at the origin containing the rest of the mass of
the initial datum. Furthermore, it is a global minimizer of the relaxed energy.
This splitting phenomenon is an uncommon example of blow-up in infinite time.
