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arXiv · 2012.08586

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

J. A. Carrillo·M. G. Delgadino·R. L. Frank·M. Lewin
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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.

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J. A. Carrillo, M. G. Delgadino, R. L. Frank, M. Lewin. 2020-12-15. Fast Diffusion leads to partial mass concentration in Keller-Segel type stationary solutions. https://arxiv.org/abs/2012.08586

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