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Nonlinear partial differential equation with a dynamical boundary condition

Research Project

Project/Area Number 16K17629
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeMulti-year Fund
Research Field Mathematical analysis
Research InstitutionRyukoku University (2017-2019)
Osaka Prefecture University (2016)

Principal Investigator

Kawakami Tatsuki  龍谷大学, 理工学部, 教授 (20546147)

Project Period (FY) 2016-04-01 – 2020-03-31
Project Status Completed (Fiscal Year 2019)
Budget Amount *help
¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords動的境界条件 / 可解性 / 臨界指数 / 拡散極限 / 外部領域 / 指数型非線形項 / Cartan-Hadamard多様体 / 退化拡散方程式 / 分数冪拡散方程式 / 時間大域可解性 / 高次漸近展開 / 特異拡散方程式 / 非線形楕円型方程式 / 粘性Hamilton-Jacobi方程式 / 指数型非線形 / 半線形楕円型方程式
Outline of Final Research Achievements

We consider some nonlinear partial differential equations with a dynamical boundary condition in unbounded domains.For the nonlinear elliptic equation in the exterior of unit ball, we gave several results on existence, nonexistence and large time behavior of small positive solutions.For the heat equation, we considered two cases, the half space and the exterior of unit ball, and proved the existence of global-in-time solutions. Furthermore, we showed that, if the diffusion coefficient tends to infinity (it called by the large diffusion limit), then the solution converge to solutions of the Laplace equation with same dynamical boundary condition.Moreover, we also considered some diffusion equations, which are related with anomalous diffusion, and obtained the critical exponent for the existence of global-in-time solutions.

Academic Significance and Societal Importance of the Research Achievements

動的境界条件は近年、国内外において領域の有界性にかかわらず活発に研究が行われてきている。その中で解の時間大域可解性や内部の拡散現象の極限を考察した熱方程式の拡散極限は、時間大域挙動に対する内部と外部の拡散現象の影響の考察や、今後の非線形問題への応用に向けて最も基本的かつ重要な問題であり、必要不可欠な研究と言える。本成果が足掛かりとなり、今後非線形問題等に大いに進展していくことが期待される。

Report

(5 results)
  • 2019 Annual Research Report   Final Research Report ( PDF )
  • 2018 Research-status Report
  • 2017 Research-status Report
  • 2016 Research-status Report
  • Research Products

    (56 results)

All 2020 2019 2018 2017 2016 Other

All Journal Article (11 results) (of which Int'l Joint Research: 7 results,  Peer Reviewed: 11 results,  Open Access: 3 results) Presentation (37 results) (of which Int'l Joint Research: 16 results,  Invited: 37 results) Book (1 results) Remarks (2 results) Funded Workshop (5 results)

  • [Journal Article] The large diffusion limit for the heat equation with a dynamical boundary condition2020

    • Author(s)
      M. Fila, K. Ishige and T. Kawakami
    • Journal Title

      Communications in Contemporary Mathematics

      Volume: to appear Issue: 01 Pages: 2050003-2050003

    • DOI

      10.1142/s0219199720500030

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Critical behavior for a semilinear parabolic equation with forcing term depending on time and space2020

    • Author(s)
      M. Jleli, T. Kawakami and B. Samet
    • Journal Title

      Journal of Mathematical Analysis and Applications

      Volume: to appear Issue: 2 Pages: 123931-123931

    • DOI

      10.1016/j.jmaa.2020.123931

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Nonexistence of radial optimal functions for the Sobolev inequality on M^n2020

    • Author(s)
      T. Kawakami and M. Muratori
    • Journal Title

      Springer INdAM Series

      Volume: -

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Critical Fujita exponents for semilinear heat equations with quadratically decaying potential2019

    • Author(s)
      K. Ishige and T. Kawakami
    • Journal Title

      Indiana University Mathematics Journal

      Volume: 印刷中

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Rate of convergence in the large diffusion limit for the heat equation with a dynamical boundary condition2019

    • Author(s)
      M. Fila, K Ishige, T. Kawakami and J. Lankeit
    • Journal Title

      Asymptotic Analysis

      Volume: 印刷中

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Critical exponent for the global existence of solutions to a semilinear heat equation with degenerate coefficients2019

    • Author(s)
      Y. Fujishima, T. Kawakami and Y. Sire
    • Journal Title

      Calculus of Variations and Partial Differential Equations

      Volume: 印刷中

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] An exterior nonlinear elliptic problem with a dynamical boundary condition2017

    • Author(s)
      Marek Fila, Kazuhiro Ishige and Tatsuki Kawakami
    • Journal Title

      Rev. Mat. Complut.

      Volume: - Issue: 2 Pages: 281-312

    • DOI

      10.1007/s13163-017-0225-6

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Asymptotic expansions of solutions of fractional diffusion equations2017

    • Author(s)
      K. Ishige, T. Kawakami and H. Michihisa
    • Journal Title

      SIAM Journal on Mathematical Analysis

      Volume: 49 Issue: 3 Pages: 2167-2190

    • DOI

      10.1137/16m1101428

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Asymptotic behavior and decay estimates of the solutions for a nonlinear parabolic equation with exponential nonlinearity2017

    • Author(s)
      Giulia Furioli, Tatsuki Kawakami, Bernhard Ruf and Elide Terraneo
    • Journal Title

      J. Differential Equations

      Volume: 262 Issue: 1 Pages: 145-180

    • DOI

      10.1016/j.jde.2016.09.024

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Existence of mild solutions for a Hamilton-Jacobi equation with critical fractional viscosity in the Besov spaces2017

    • Author(s)
      Tsukasa Iwabuchi and Tatsuki Kawakami
    • Journal Title

      J. Math. Pures Appl.

      Volume: 107 Issue: 4 Pages: 464-489

    • DOI

      10.1016/j.matpur.2016.07.007

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Higher order asymptotic expansions to the solutions for a nonlinear damped wave equation2016

    • Author(s)
      Tatsuki Kawakami and Hiroshi Takeda
    • Journal Title

      Nonlinear Differ. Equ. Appl.

      Volume: 23 Issue: 5 Pages: 1-30

    • DOI

      10.1007/s00030-016-0408-8

    • Related Report
      2016 Research-status Report
    • Peer Reviewed
  • [Presentation] The large diffusion limit for the heat equation with a dynamical boundary condition2019

    • Author(s)
      Tatsuki Kawakami
    • Organizer
      VI Italian-Japanese workshop on Geometric Properties for Parabolic and Elliptic PDE's
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Critical Fujita exponents for semilinear heat equations with quadratically decaying potential2019

    • Author(s)
      Tatsuki Kawakami
    • Organizer
      Workshop on Nonlinear parabolic PDEs and related fields
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Critical exponent for the global existence of solutions to a semilinear heat equation with degenerate coefficients2019

    • Author(s)
      Tatsuki Kawakami
    • Organizer
      The 44th Sapporo Symposium on Partial Differential Equations
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Critical exponent for the global existence of solutions to a semilinear heat equation with degenerate coefficients2019

    • Author(s)
      Tatsuki Kawakami
    • Organizer
      4th Swiss-Japanese PDE Seminar
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Critical exponent for the global existence of solutions to a semilinear heat equation with degenerate coefficients2019

    • Author(s)
      Tatsuki Kawakami
    • Organizer
      New development in the theory of evolution equations: theory, phenomena and technology
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Critical Fujita exponents for semilinear heat equations with quadratically decaying potential2019

    • Author(s)
      Tatsuki Kawakami
    • Organizer
      Nonlinear Analysis Workshop
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Critical exponent for the global existence of solutions to a semilinear heat equation with degenerate coefficients2019

    • Author(s)
      川上竜樹
    • Organizer
      応用数理解析セミナー
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] Critical Fujita exponents for semilinear heat equations with quadratically decaying potential2019

    • Author(s)
      川上竜樹
    • Organizer
      大阪大学微分方程式セミナー
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] 動的境界条件を有する線形熱方程式の拡散極限2019

    • Author(s)
      川上竜樹
    • Organizer
      基盤研究(S)キックオフシンポジウム 発展方程式における系統的形状解析および漸近解析
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] 動的境界条件を有する線形熱方程式の拡散極限2019

    • Author(s)
      川上竜樹
    • Organizer
      数学と現象 : Mathematics and Phenomena in Miyazaki 2019
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] Critical Fujita exponents for semilinear heat equations with quadratically decaying potential2019

    • Author(s)
      川上竜樹
    • Organizer
      九州大学関数方程式セミナー
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] 動的境界条件付き熱方程式の拡散極限2019

    • Author(s)
      川上竜樹
    • Organizer
      武蔵野大学・龍谷大学連携シンポジウム
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] A semilinear elliptic equation with a dynamical boundary condition2018

    • Author(s)
      Tatsuki Kawakami
    • Organizer
      138th AMS Meeting, Special Session on Partuial Differential Equations and New Perspective of Variational Methods
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Critical Fujita exponents for semilinear heat equations with quadratically decaying potential2018

    • Author(s)
      Tatsuki Kawakami
    • Organizer
      K-Japan Workshop on Analysis of Nonlinear Partial Differential Equations
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The large diffusion limit for the heat equation with a dynamical boundary condition,2018

    • Author(s)
      Tatsuki Kawakami
    • Organizer
      8th Euro-Japanses Workshop On Blow-up
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Positive solutions of a semilinear elliptic equation with singular Dirichlet boundary data2018

    • Author(s)
      Tatsuki Kawakami
    • Organizer
      2th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Heat equation with a dynamical boundary condition2018

    • Author(s)
      Tatsuki Kawakami
    • Organizer
      2th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Critical Fujita exponents for semilinear heat equations with quadratically decaying potential2018

    • Author(s)
      川上竜樹
    • Organizer
      京都大学NLPDEセミナー
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] 動的境界条件付き半線形楕円型方程式2018

    • Author(s)
      川上竜樹
    • Organizer
      日本数学会2018年度秋季総合分科会 函数方程式論分科会 特別講演
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] An exterior nonlinear elliptic problem with a dynamical boundary condition.2017

    • Author(s)
      Tatsuki Kawakami
    • Organizer
      2017 NCTS Workshop on Applied Mathematics at Tainan
    • Place of Presentation
      NCTS, Tainan University
    • Year and Date
      2017-03-06
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] 指数型非線形項を持つ非線形放物型方程式の時間大域解について2017

    • Author(s)
      川上 竜樹
    • Organizer
      武蔵野大学・龍谷大学連携シンポジウム
    • Place of Presentation
      武蔵野大学
    • Year and Date
      2017-02-11
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] 外部領域における動的境界条件付き半線形楕円型方程式の可解性2017

    • Author(s)
      川上 竜樹
    • Organizer
      応用数学研究会
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Asymptotic expansion of solutions of fractional diffusion equations2017

    • Author(s)
      Tatsuki Kawakami
    • Organizer
      Singularity and asymptotic behavior of solutions for partial differential equations with conservation law
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Existence of mild solutions for the Hamilton-Jacobi equation with critical fractional viscosity in the Besov spaces2017

    • Author(s)
      川上 竜樹
    • Organizer
      名古屋大学微分方程式セミナー
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] 外部領域における動的境界条件付き半線形楕円型方程式の可解性2017

    • Author(s)
      川上 竜樹
    • Organizer
      明治大学非線型数理セミナー
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Decay estimates of the solutions for a nonlinear parabolic equation2017

    • Author(s)
      Tatsuki Kawakami
    • Organizer
      Equadiff 2017
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] An exterior nonlinear elliptic problem wirh a dynamical boundary condition2017

    • Author(s)
      Tatsuki Kawakami
    • Organizer
      Analysis and Partial Differential Equations Seminar
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Asymptotic expansion of solutions of fractional diffusion equations2017

    • Author(s)
      Tatsuki Kawakami
    • Organizer
      Seminario di Analisi Nonlineare
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] A semilinear elliptic equation with a dynamical boundary condition2017

    • Author(s)
      Tatsuki Kawakami
    • Organizer
      Seminario di Analisi matematica
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Asymptotic behavior and decay estimates of the solutions for a nonlinear parabolic equation with exponential nonlinearity2016

    • Author(s)
      Tatsuki Kawakami
    • Organizer
      International Workshop on Nonlinear PDEs 2016 in Osaka
    • Place of Presentation
      Osaka Prefecture University
    • Year and Date
      2016-12-08
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] An exterior nonlinear elliptic problem with a dynamical boundary condition2016

    • Author(s)
      Tatsuki Kawakami
    • Organizer
      Analysis on Shapes of Solutions to Partial Differential Equations
    • Place of Presentation
      RIMS, Kyoto
    • Year and Date
      2016-11-09
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Existence of mild solutions for the Hamilton-Jacobi equation with critical fractional viscosity in the Besov spaces2016

    • Author(s)
      川上 竜樹
    • Organizer
      なかもず解析セミナー
    • Place of Presentation
      大阪府立大学
    • Year and Date
      2016-10-21
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] An exterior nonlinear elliptic problem with a dynamical boundary condition2016

    • Author(s)
      Tatsuki Kawakami
    • Organizer
      Geometry of solutions of PDE’s and its related inverse problems
    • Place of Presentation
      Tohoku University
    • Year and Date
      2016-10-05
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] A semilinear elliptic equation with a dynamical boundary condition2016

    • Author(s)
      Tatsuki Kawakami
    • Organizer
      7th Euro-Japanese workshop on Blow-up
    • Place of Presentation
      Bedlewo, Poland
    • Year and Date
      2016-09-07
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Minimal solutions of a semilinear elliptic equation with a dynamical boundary con- dition2016

    • Author(s)
      Tatsuki Kawakami
    • Organizer
      XXXVI Convegno di Analisi Armonica
    • Place of Presentation
      Gargnano, Italy
    • Year and Date
      2016-06-02
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Minimal solutions of a semilinear elliptic equation with a dynamical boundary con- dition2016

    • Author(s)
      Tatsuki Kawakami
    • Organizer
      9th European conference on elliptic and parabolic problems
    • Place of Presentation
      Gaeta, Italy
    • Year and Date
      2016-05-23
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Higher order asymptotic expansions to the solutions for a nonlinear damped wave equation2016

    • Author(s)
      Tatsuki Kawakami
    • Organizer
      Qualitative Theory of Differential Equation
    • Place of Presentation
      Comenius University, Bratislava, Slovakia
    • Year and Date
      2016-05-19
    • Related Report
      2016 Research-status Report
    • Invited
  • [Book] ベクトル解析入門2019

    • Author(s)
      壁谷 喜継、川上 竜樹
    • Total Pages
      240
    • Publisher
      共立出版
    • ISBN
      4320113756
    • Related Report
      2019 Annual Research Report
  • [Remarks] 川上竜樹のホームページ

    • URL

      https://whale2.math.ryukoku.ac.jp/~kawakami/kawakami.html

    • Related Report
      2018 Research-status Report
  • [Remarks] 川上 竜樹 -研究者- researchmap

    • URL

      https://researchmap.jp/k-tatsuki?lang=japanese

    • Related Report
      2018 Research-status Report
  • [Funded Workshop] Qualitative properties for nonlinear diffusion equations2020

    • Related Report
      2019 Annual Research Report
  • [Funded Workshop] Workshop on Nonlinear parabolic PDEs and related fields -in honor of the 60th birthday of Marek Fila and Peter Polacik-2019

    • Related Report
      2019 Annual Research Report
  • [Funded Workshop] Workshop on Elliptic & Parabolic PDEs 20192019

    • Related Report
      2019 Annual Research Report
  • [Funded Workshop] The 10th Taiwan-Japan Joint Workshop for Young Scholars in Applied Mathematics2019

    • Related Report
      2018 Research-status Report
  • [Funded Workshop] Workshop on Nonlinear Partial Differential Equations -Japan-China Joint Project for Young Mathematicians 2018-2018

    • Related Report
      2018 Research-status Report

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Published: 2016-04-21   Modified: 2021-02-19  

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