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

TL;DR

This paper demonstrates that in certain aggregation-diffusion equations with fast diffusion, stationary solutions can exhibit partial mass concentration, with specific conditions identified for the quartic potential and numerical evidence for higher powers.

Contribution

It establishes the existence of stationary solutions with partial mass concentration for a class of aggregation-diffusion equations, including explicit results for the quartic potential.

Findings

Partial mass concentration occurs in stationary solutions.

Explicit range of diffusion exponents for concentration in N≥6.

Numerical evidence suggests concentration in all N≥3 for higher powers.

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 $N\geq6$. We then provide numerical computations which suggest the occurrence of mass concentration in all dimensions $N\geq3$, for homogeneous interaction potentials with higher power.