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

Carrillo, J. A. and Delgadino, M. G. and Frank, R. L. and Lewin, M. (2020) Fast Diffusion leads to partial mass concentration in Keller-Segel type stationary solutions. . (Unpublished)

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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.

Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription Paper
Carrillo, J. A.0000-0001-8819-4660
Frank, R. L.0000-0001-7973-4688
Lewin, M.0000-0002-1755-0207
Additional Information:Attribution 4.0 International (CC BY 4.0) The authors would like to thank Jean Dolbeault, David Gómez-Castro and Juan Luis Vázquez for pointing us out reference [8] and fruitful comments. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Advanced Grant Nonlocal-CPD 883363 of J.A.C. and Consolidator Grant MDFT 725528 of M.L.). M.G.D. was partially supported by CNPq-Brazil (#308800/2019-2) and Instituto Serrapilheira. R.L.F. was partially supported by the U.S. National Science Foundation through grants DMS-1363432 and DMS-1954995.
Funding AgencyGrant Number
European Research Council (ERC)883363
European Research Council (ERC)725528
Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)308800/2019-2
Instituto SerrapilheiraUNSPECIFIED
Record Number:CaltechAUTHORS:20211004-222702303
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:111199
Deposited By: George Porter
Deposited On:04 Oct 2021 22:53
Last Modified:04 Oct 2021 22:53

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