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Self-similarity and singularity of solutions for nonlinear parabolic PDEs

Research Project

Project/Area Number 20540210
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionEhime University (2009-2011)
Kobe University (2008)

Principal Investigator

NALTO Yuki (NAITO Yuki)  Ehime University, 理工学研究科, 教授 (10231458)

Co-Investigator(Kenkyū-buntansha) ISHII Katsuyuki  神戸大学, 海事科学研究科, 准教授 (40232227)
KUWAMURA Masataka  神戸大学, 人間発達環境学研究科, 准教授 (30270333)
Co-Investigator(Renkei-kenkyūsha) SUZUKI Takashi  大阪大学, 基礎工学研究科, 教授 (40114516)
SENBA Takashi  九州工業大学, 工学研究科, 教授 (30196985)
ISHIGE Kazuhiro  東北大学, 理学研究科, 教授 (90272020)
Project Period (FY) 2008 – 2010
Project Status Completed (Fiscal Year 2011)
Budget Amount *help
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2011: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2010: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2009: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2008: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords非線形偏微分方程式 / 自己相似性 / 解の爆発 / 特異性 / 定常問題 / 非線形放物型偏微分方程式 / 自己相似解 / Sobolev臨界指数 / 非線形楕円型偏微分方程式 / 変分法 / 走化性方程式系 / p-Laplace方程式 / 爆発現象 / 非線形熱方程式
Research Abstract

The structure of self-similar solutions for semilinear heat equations is studied, and the role of self-similar solutions in the problem of asymptotic behavior of time-glabal solutions is investigated. In particular, new results on the structure of self-similar solutions for semilinear heat equations with Sobolev critical nonlinarity are obtained. Furthermore, the applications to the nonlinear problem involving chemotaxis system are studied.

Report

(5 results)
  • 2010 Annual Research Report   Self-evaluation Report ( PDF )   Final Research Report ( PDF )
  • 2009 Annual Research Report
  • 2008 Annual Research Report
  • Research Products

    (66 results)

All 2011 2010 2009 2008 Other

All Journal Article (31 results) (of which Peer Reviewed: 28 results) Presentation (31 results) Book (3 results) Remarks (1 results)

  • [Journal Article] Existence and non-existence of sign-changing solutions for a class of two-point boundary value problems involving one-dimensional p-Laplacian2011

    • Author(s)
      Yuki Naito
    • Journal Title

      Mathematica Bohemia

      Volume: (to appear)

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Non-homogeneous semilinear elliptic equations involving critical Sobolev exponent2011

    • Author(s)
      Yuki Naito, Tokushi Sato
    • Journal Title

      Ann. Mat. Pura Appl

      Volume: (to appear)

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Existence and non-existence of sign-changing solutions for a class of two-point boundary value problems involving one-dimensional p-Laplacian2011

    • Author(s)
      Yuki Naito
    • Journal Title

      Mathematica Bohemia (to appear)

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Non-homogeneous semilinear elliptic equations involving critical Sobolev exponent2011

    • Author(s)
      Yuki Naito, Tokushi Sato
    • Journal Title

      Ann.Mat.Pura Appl. (to appear)

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Existence and non-existence of sign-changing solutions for a class of two-point boundary value problems involving one-dimensional p-Laplacian2011

    • Author(s)
      Yuki Naito
    • Journal Title

      Mathematica Bohemica

      Volume: (to appear)

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Non-homogeneous semilinear elliptic equations involving critical Sobolev exponent2011

    • Author(s)
      Yuki Naito, Tokushi Sato
    • Journal Title

      Ann.Mat.Pura Appl.

      Volume: (to appear)

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Mathematical modelling and experiments for the proliferation and differentiation of Drosophila intestinal stem cells I2010

    • Author(s)
      M. Kuwamura, K. Maeda and T. Adachi-Yamada
    • Journal Title

      J. Biol. Dyns

      Volume: 42 Pages: 248-257

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Mathematical modelling and experiments for the proliferation and differentiation of Drosophila intestinal stem cells I2010

    • Author(s)
      M. Kuwamura, K. Maeda, T. Adachi-Yamada
    • Journal Title

      J.Biol.Dyns. 42

      Pages: 248-257

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Mathematical modelling and experiments for the proliferation and differentiation of Drosophila intestinal stem cells I2010

    • Author(s)
      M.Kuwamura, K.Maeda, T.Adachi-Yamada
    • Journal Title

      J.Biol.Dyns.

      Volume: 4 Pages: 248-257

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Existence and non-existence of sign-changing solutions for a class of two-point boundary value problems involving one-dimensional p-Laplacian2010

    • Author(s)
      Y.Naito
    • Journal Title

      Mathematica Bohemia (印刷中)

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Self-similar blow-up for a chemotaxis system in higher dimensional domains2009

    • Author(s)
      Yuki Naito, Takasi Senba
    • Journal Title

      RIMS Kokyuroku Bessatsu

      Volume: 15 Pages: 87-99

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] A minimum model of prey-predator system with dormancy of predators and the paradox of enrichment2009

    • Author(s)
      M. Kuwamura, T. Nakazawa, T. Ogawa
    • Journal Title

      J. Math. Biol

      Volume: 58 Pages: 459-479

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Mixed-mode oscillations and chaos in a prey-predator system with dormancy of predators2009

    • Author(s)
      M. Kuwamura, H. Chiba
    • Journal Title

      Chaos

      Volume: 19(43121) Pages: 1-10

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Non-homogeneous semilinear elliptic equations involving critical Sobolev exponent2009

    • Author(s)
      Y. Naito, T. Sato
    • Journal Title

      RIMS Kokyuroku

      Volume: 1671 Pages: 91-95

    • Related Report
      2010 Final Research Report
  • [Journal Article] Self-similar blow-up for a chemotaxis system in higher dimensional domains2009

    • Author(s)
      Yuki Naito, Takasi Senba
    • Journal Title

      RIMS Kokyuroku Bessatsu B 15

      Pages: 87-99

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] A minimum model of prey-predator system with dormancy of predators and the paradox of enrichment2009

    • Author(s)
      M. Kuwamura, T. Nakazawa, T. Ogawa
    • Journal Title

      J.Math.Biol. 58

      Pages: 459-479

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Mixed-mode oscillations and chaos in a prey-predator system with dormancy of predators2009

    • Author(s)
      M. Kuwamura, H. Chiba
    • Journal Title

      Chaos 19(43121)

      Pages: 1-10

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Non-homogeneous semilinear elliptic equations involving critical Sobolev exponent2009

    • Author(s)
      Y. Naito, T. Sato
    • Journal Title

      RIMS Kokyuroku 1671

      Pages: 91-95

    • Related Report
      2010 Self-evaluation Report
  • [Journal Article] Self-similar blow-up for a chemotaxis system in higher dimensional domains2009

    • Author(s)
      Y.Naito, T.Senba
    • Journal Title

      RIMS Kokyuroku Bessatsu B15

      Pages: 87-99

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A minimum model of prey-predator system with dormancy of predators and the paradox of enrichment2009

    • Author(s)
      M.Kuwamura, T.Nakazawa, T.Ogawa
    • Journal Title

      J.Math.Biol. 58

      Pages: 459-479

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Mixed-mode oscillations and chaos in a prey-predator system with dormancy of predators2009

    • Author(s)
      M.Kuwamura, H.Chiba
    • Journal Title

      Chaos 19(43121)

      Pages: 1-10

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Non-homogeneous semilinear elliptic equations involving critical Sobolev exponent2009

    • Author(s)
      Y.Naito, T.Sato
    • Journal Title

      RIMS Kokyuroku 1671

      Pages: 91-95

    • Related Report
      2009 Annual Research Report
  • [Journal Article] Sharp conditions for the existence of sign-changing solutions to equations involving the one-dimensional p-Laplacian2008

    • Author(s)
      Yuki Naito, Satoshi Tanaka
    • Journal Title

      Nonlinear Anal. TMA

      Volume: 69 Pages: 3070-3083

    • NAID

      120001384683

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Self-similar solutions for a semilinear heat equation with critical Sobolev exponent2008

    • Author(s)
      Yuki Naito
    • Journal Title

      Indiana Univ. Math. J.

      Volume: 57 Pages: 1283-1315

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Self-Similarity in Chemotaxis Systems2008

    • Author(s)
      Yuki Naito, Takashi Suzuki
    • Journal Title

      Colloq. Math

      Volume: 111 Pages: 11-34

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Sharp conditions for the existence of sign-changing solutions to equations involving the one-dimensional p-Laplacian2008

    • Author(s)
      Yuki Naito, Satoshi Tanaka
    • Journal Title

      Nonlinear Anal.TMA 69

      Pages: 3070-3083

    • NAID

      120001384683

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Self-similar solutions for a semilinear heat equation with critical Sobolev exponent2008

    • Author(s)
      Yuki Naito
    • Journal Title

      Indiana Univ.Math.J. 57

      Pages: 1283-1315

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Self-Similarity in Chemotaxis Systems2008

    • Author(s)
      Yuki Naito, Takashi Suzuki
    • Journal Title

      Colloq.Math. 111

      Pages: 11-34

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Sharp Conditions for the ex istence of sign-changing solutions to equations involving the one-dimensionalp-Laplacian2008

    • Author(s)
      Yuki Nai to, Satoshi Tanaka
    • Journal Title

      Nonlinear Anal. TMA 69

      Pages: 3070-3080

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Self-similar solutions for a semilinear heat equation with critical Sobolev exponent2008

    • Author(s)
      Yuki Naito
    • Journal Title

      Indiana Univ. Math. J. 57

      Pages: 1283-1315

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Self-Similarity in Chemotaxis Systems2008

    • Author(s)
      Yuki Naito, Takashi Suzuki
    • Journal Title

      Colloq. Math. Ill

      Pages: 11-34

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Presentation] Non-homogeneous semilinear elliptic equations involving critical Sobolev exponent2011

    • Author(s)
      Yuki Naito
    • Organizer
      首都大・変分問題セミナー
    • Place of Presentation
      首都大学東京
    • Year and Date
      2011-01-14
    • Related Report
      2010 Annual Research Report 2010 Final Research Report
  • [Presentation] 臨界Sobolev指数をもつ非済次半線形楕円型偏微分方程式2010

    • Author(s)
      佐藤得志、内藤雄基
    • Organizer
      日本数学会2010年度秋期総合分科会
    • Place of Presentation
      名古屋大学
    • Year and Date
      2010-09-24
    • Related Report
      2010 Annual Research Report 2010 Final Research Report
  • [Presentation] Dormancy of predators dependent on the rate of variation in the prey density2010

    • Author(s)
      桑村雅隆, 仲澤剛史
    • Organizer
      第20回日本数理生物学会大会
    • Place of Presentation
      北海道大学
    • Year and Date
      2010-09-15
    • Related Report
      2010 Final Research Report
  • [Presentation] Dormancy of predators dependent on the rate of variation in the preydensity2010

    • Author(s)
      桑村雅隆、仲澤剛史
    • Organizer
      第20回日本数理生物学会大会
    • Place of Presentation
      北海道大学(北海道)
    • Year and Date
      2010-09-15
    • Related Report
      2010 Annual Research Report
  • [Presentation] The role of self-similarity in the Cauchy problems for semilinear heat equations2010

    • Author(s)
      Yuki Naito
    • Organizer
      The 4th Euro-Japanese Workshop on Blow-up
    • Place of Presentation
      Lotentz Center,ライデン、オランダ
    • Year and Date
      2010-09-09
    • Related Report
      2010 Final Research Report
  • [Presentation] The role of self-similarity in the Cauchy problems for semilinear heat equations2010

    • Author(s)
      内藤雄基
    • Organizer
      The 4th Euro-Japanese Workshop on Blow-up
    • Place of Presentation
      オランダ、ライデン、Lotentz Center
    • Year and Date
      2010-09-09
    • Related Report
      2010 Annual Research Report 2010 Self-evaluation Report
  • [Presentation] Asymptotic behavior of solutions to the Cauchy problem for a semilinear heat equation2010

    • Author(s)
      Yuki Naito
    • Organizer
      Nonlinear PDE seminar(Pusan University)
    • Place of Presentation
      Pusan, Koera
    • Year and Date
      2010-08-30
    • Related Report
      2010 Final Research Report
  • [Presentation] Asymptotic behavior of solutions to the Cauchy problem for a semilinear heat equation2010

    • Author(s)
      内藤雄基
    • Organizer
      Nonlinear PDE seminar(Pusan University)
    • Place of Presentation
      韓国、釜山、Pusan Univ.
    • Year and Date
      2010-08-30
    • Related Report
      2010 Annual Research Report
  • [Presentation] A time discretization to the American option pricing2010

    • Author(s)
      石井克幸
    • Organizer
      Nonlinear PDE seminar
    • Place of Presentation
      Federica II Univ. Napoli、イタリア
    • Year and Date
      2010-04-15
    • Related Report
      2010 Final Research Report
  • [Presentation] Atime discretization to the American option pricing2010

    • Author(s)
      石井克幸
    • Organizer
      Nonlinear PDE seminar
    • Place of Presentation
      Federica II Univ.Napoli、イタリア
    • Year and Date
      2010-04-15
    • Related Report
      2010 Annual Research Report
  • [Presentation] A time discretization to the American option pricing2010

    • Author(s)
      石井克幸
    • Organizer
      Nonlinear PDE seminar
    • Place of Presentation
      Sapienza Univ. Roma、イタリア
    • Year and Date
      2010-04-14
    • Related Report
      2010 Final Research Report
  • [Presentation] Atime discretization to the American option pricing2010

    • Author(s)
      石井克幸
    • Organizer
      Nonlinear PDE seminar
    • Place of Presentation
      Sapienza Univ.Roma、イタリア
    • Year and Date
      2010-04-14
    • Related Report
      2010 Annual Research Report
  • [Presentation] Critical phenomena in linear elliptic problem with Hardy potential2009

    • Author(s)
      内藤雄基
    • Organizer
      京都大学数理解析研究所研究集会「現象解析と関数方程式の新展望」
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2009-11-18
    • Related Report
      2010 Final Research Report 2009 Annual Research Report
  • [Presentation] A variational approach for self-similar solutions to nonlinear heat equations2009

    • Author(s)
      Yuki Naito
    • Organizer
      POSTEC Analysis seminar
    • Place of Presentation
      POSTEC, Pohang, Korea
    • Year and Date
      2009-10-24
    • Related Report
      2010 Final Research Report
  • [Presentation] A variational approach for self-similar solutions to nonlinear heat equations2009

    • Author(s)
      内藤雄基
    • Organizer
      POSTEC Analysis seminar
    • Place of Presentation
      POSTEC, Korea
    • Year and Date
      2009-10-24
    • Related Report
      2009 Annual Research Report
  • [Presentation] 高次元領域における走化性方程式系の解の自己相似爆発2009

    • Author(s)
      仙葉隆、内藤雄基
    • Organizer
      日本数学会函数方程式分科会
    • Place of Presentation
      大阪大学
    • Year and Date
      2009-09-26
    • Related Report
      2010 Self-evaluation Report 2010 Final Research Report 2009 Annual Research Report
  • [Presentation] Blowup rate for a parabolic-elliptic system in higher dimensional domains2009

    • Author(s)
      Yuki Naito
    • Organizer
      Equadiff 12(2009)
    • Place of Presentation
      Brno, Czech Republic
    • Year and Date
      2009-07-21
    • Related Report
      2010 Final Research Report 2009 Annual Research Report
  • [Presentation] Blowup rate for a parabolic-elliptic system in higher dimensional domains2009

    • Author(s)
      Yuki Naito
    • Organizer
      Equadiff 12
    • Place of Presentation
      Brno, Czech Republic
    • Year and Date
      2009-07-21
    • Related Report
      2010 Self-evaluation Report
  • [Presentation] Self-similar blowup for a chemotaxis system2009

    • Author(s)
      Yuki Naito
    • Organizer
      09 International Workshop on Nonlinear PDE and Applications
    • Place of Presentation
      Puan National University, Pusan, Korea
    • Year and Date
      2009-06-29
    • Related Report
      2010 Final Research Report
  • [Presentation] Self-similar blowup for a chemotaxis system2009

    • Author(s)
      内藤雄基
    • Organizer
      09 International Workshop on Nonlinear PDE and Applications
    • Place of Presentation
      Puan National Univ.Korea
    • Year and Date
      2009-06-29
    • Related Report
      2009 Annual Research Report
  • [Presentation] Non-homogeneous semilinear elliptic equations involving critical Sobolev exponent2009

    • Author(s)
      内藤雄基
    • Organizer
      京都大学数理解析研究所研究集会「変分問題とその周辺」
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2009-06-09
    • Related Report
      2010 Final Research Report 2009 Annual Research Report
  • [Presentation] Non-homogeneous semilinear elliptic equations involving critical Sobolev exponent2008

    • Author(s)
      Yuki Naito
    • Organizer
      仙台楕円型・放物型微分方程式研究集会
    • Place of Presentation
      東北大学
    • Year and Date
      2008-11-08
    • Related Report
      2010 Final Research Report
  • [Presentation] Non-homogeneous semilinear elliptic equat ions involving critical Sobolev exponent2008

    • Author(s)
      Yuki Naito
    • Organizer
      仙台楕円型・放物型微分方程式研究集会
    • Place of Presentation
      東北大学
    • Year and Date
      2008-11-08
    • Related Report
      2008 Annual Research Report
  • [Presentation] A variational approach for self-similar solutions to anonlinear heat equation with critical Sobolev exponent2008

    • Author(s)
      Yuki Naito
    • Organizer
      熊本における偏微分方程式研究集会
    • Place of Presentation
      熊本大学
    • Year and Date
      2008-10-24
    • Related Report
      2010 Final Research Report
  • [Presentation] A variational approach for self-similar solutions to a nonlinear heat equat ion with critical Sobolev exponent2008

    • Author(s)
      Yuki Nai to
    • Organizer
      熊本における偏微分方程式研究集会
    • Place of Presentation
      熊本大学
    • Year and Date
      2008-10-24
    • Related Report
      2008 Annual Research Report
  • [Presentation] Self-similar blowup for a chemotaxis system in higher dimensional domains2008

    • Author(s)
      Yuki Naito
    • Organizer
      Mathematical Analysis on the self-organization and the self-similarity
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2008-09-17
    • Related Report
      2010 Final Research Report
  • [Presentation] Self-similar blowup for a chemotaxis system in higher dimensional domains2008

    • Author(s)
      Yuki Nai to
    • Organizer
      Mathematical Analysis on theself-organization and theself-similarity
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2008-09-17
    • Related Report
      2008 Annual Research Report
  • [Presentation] Threshold solutions for a seimilinear heat equation with Sobolev critical exponent2008

    • Author(s)
      Yuki Naito
    • Organizer
      Third Euro-Japanese Workshop on Blow-up
    • Place of Presentation
      東北大学
    • Year and Date
      2008-09-12
    • Related Report
      2010 Final Research Report
  • [Presentation] Threshold solutions for a seimilinear heat equation with Sobolev critical exponent2008

    • Author(s)
      Yuki Nai to
    • Organizer
      Third Euro-Japanese Workshopon Blow-up
    • Place of Presentation
      東北大学
    • Year and Date
      2008-09-12
    • Related Report
      2008 Annual Research Report
  • [Presentation] Non-homogeneous semilinear elliptic equations involvingcritical Sobolev exponent2008

    • Author(s)
      Yuki Naito
    • Organizer
      International Conference on Nonlinear PDE and Applications
    • Place of Presentation
      Pusan National University, Pusan, Korea
    • Year and Date
      2008-06-11
    • Related Report
      2010 Self-evaluation Report 2010 Final Research Report
  • [Presentation] Non-homogeneous semilinear elliptic equations involving critical Sobolev exponent2008

    • Author(s)
      Yuki Naito
    • Organizer
      International Conference on Nonlinear PDE and Applications
    • Place of Presentation
      Pusan National University, Pusan, Korea
    • Year and Date
      2008-06-11
    • Related Report
      2008 Annual Research Report
  • [Book] 画像処理と曲率流方程式(「現代数学はいかに使われているか[解析編]」)数理科学2008年4月号特集2008

    • Author(s)
      石井克幸
    • Publisher
      サイエンス社
    • Related Report
      2010 Self-evaluation Report
  • [Book] 画像処理と曲率流方程式, 数理科学2008年4月号特集「現代数学はいかに使われているか[解析編]」2008

    • Author(s)
      石井 克幸
    • Total Pages
      280
    • Publisher
      サイエンス社
    • Related Report
      2008 Annual Research Report
  • [Book] 「現代数学はいかに使われているか[解析編]」,画像処理と曲率流方程式,数理科学2008年4月号特集

    • Author(s)
      石井克幸
    • Publisher
      サイエンス社
    • Related Report
      2010 Final Research Report
  • [Remarks] ホームページ

    • URL

      http://continuum.math.sci.ehime-u.ac.jp/~ynaito/

    • Related Report
      2010 Final Research Report

URL: 

Published: 2008-04-01   Modified: 2017-05-19  

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