Blow-up behavior of solutions to a degenerate parabolic–parabolic Keller–Segel system

Kazuhiro Ishige; Philippe Laurençot; Noriko Mizoguchi

doi:10.1007/s00208-016-1400-7

Blow-up behavior of solutions to a degenerate parabolic–parabolic Keller–Segel system

Kazuhiro Ishige, Philippe Laurençot, Noriko Mizoguchi

SCI - Mathematics

Research output: Contribution to journal › Article › peer-review

7 Citations (Scopus)

Abstract

Qualitative properties of solutions blowing up in finite time are obtained for a degenerate parabolic–parabolic Keller–Segel system, the nonlinear diffusion being of porous medium type with an exponent smaller or equal to the critical one m_c: = 2 (N- 1) / N. In both cases, it is shown that only type II blow-up is possible, that is, blow-up at the same rate as backward self-similar solutions never occurs. Further information on the generation of singularities induced by mass concentration are given in the critical case.

Original language	English
Pages (from-to)	461-499
Number of pages	39
Journal	Mathematische Annalen
Volume	367
Issue number	1-2
DOIs	https://doi.org/10.1007/s00208-016-1400-7
Publication status	Published - 2017 Feb 1

Keywords

35B44
35K51
35K65

ASJC Scopus subject areas

Mathematics(all)

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10.1007/s00208-016-1400-7

Cite this

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abstract = "Qualitative properties of solutions blowing up in finite time are obtained for a degenerate parabolic–parabolic Keller–Segel system, the nonlinear diffusion being of porous medium type with an exponent smaller or equal to the critical one mc: = 2 (N- 1) / N. In both cases, it is shown that only type II blow-up is possible, that is, blow-up at the same rate as backward self-similar solutions never occurs. Further information on the generation of singularities induced by mass concentration are given in the critical case.",

keywords = "35B44, 35K51, 35K65",

author = "Kazuhiro Ishige and Philippe Lauren{\c c}ot and Noriko Mizoguchi",

note = "Funding Information: We warmly thank the referee for carefully reading the manuscript and relevant remarks. Ishige was partially supported by the JSPS Grant-in-Aid for Scientific Research (A) (No. 15H02058). Mizoguchi was partially supported by the JSPS Grant-in-Aid for Scientific Research (B) (No. 26287021) and by the JSPS Grant-in-Aid for Scientific Research (S) (No. 23224003). Publisher Copyright: {\textcopyright} 2016, Springer-Verlag Berlin Heidelberg.",

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AU - Ishige, Kazuhiro

AU - Laurençot, Philippe

AU - Mizoguchi, Noriko

N1 - Funding Information: We warmly thank the referee for carefully reading the manuscript and relevant remarks. Ishige was partially supported by the JSPS Grant-in-Aid for Scientific Research (A) (No. 15H02058). Mizoguchi was partially supported by the JSPS Grant-in-Aid for Scientific Research (B) (No. 26287021) and by the JSPS Grant-in-Aid for Scientific Research (S) (No. 23224003). Publisher Copyright: © 2016, Springer-Verlag Berlin Heidelberg.

PY - 2017/2/1

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N2 - Qualitative properties of solutions blowing up in finite time are obtained for a degenerate parabolic–parabolic Keller–Segel system, the nonlinear diffusion being of porous medium type with an exponent smaller or equal to the critical one mc: = 2 (N- 1) / N. In both cases, it is shown that only type II blow-up is possible, that is, blow-up at the same rate as backward self-similar solutions never occurs. Further information on the generation of singularities induced by mass concentration are given in the critical case.

AB - Qualitative properties of solutions blowing up in finite time are obtained for a degenerate parabolic–parabolic Keller–Segel system, the nonlinear diffusion being of porous medium type with an exponent smaller or equal to the critical one mc: = 2 (N- 1) / N. In both cases, it is shown that only type II blow-up is possible, that is, blow-up at the same rate as backward self-similar solutions never occurs. Further information on the generation of singularities induced by mass concentration are given in the critical case.

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KW - 35K65

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Blow-up behavior of solutions to a degenerate parabolic–parabolic Keller–Segel system

Abstract

Keywords

ASJC Scopus subject areas

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