Fast Diffusion leads to partial mass concentration in KellerSegel type stationary solutions
Abstract
We show that partial mass concentration can happen for stationary solutions of aggregationdiffusion 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2020
 arXiv:
 arXiv:2012.08586
 Bibcode:
 2020arXiv201208586C
 Keywords:

 Mathematics  Analysis of PDEs;
 35A23;
 26D15;
 35K55;
 46E35;
 49J40