Fast Diffusion leads to partial mass concentration in Keller–Segel type stationary solutions

Abstract

We show that partial mass concentration can happen for stationary solutions of aggregation–diffusion equations with homogeneous attractive kernels in the fast diffusion range. More precisely, we prove that the free energy admits a radial global minimizer in the set of probability measures which may have part of its mass concentrated in a Dirac delta at a given point. In the case of the quartic interaction potential, we find the exact range of the diffusion exponent where concentration occurs in space dimensions [Formula: see text]. We then provide numerical computations which suggest the occurrence of mass concentration in all dimensions [Formula: see text], for homogeneous interaction potentials with higher power.