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 N ≥ 6. We then provide numerical computations which suggest the occurrence of mass concentration in all dimensions N ≥ 3, for homogeneous interaction potentials with higher power.