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Research for the Cauchy problem for nonlinear Klein-Gordon equations in homogeneous and isotropic spaces

Research Project

Project/Area Number 17KK0082
Research Category

Fund for the Promotion of Joint International Research (Fostering Joint International Research)

Allocation TypeMulti-year Fund
Research Field Basic analysis
Research InstitutionOsaka University (2022-2023)
Yamagata University (2017-2021)

Principal Investigator

Nakamura Makoto  大阪大学, 大学院情報科学研究科, 教授 (70312634)

Project Period (FY) 2018 – 2023
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥10,010,000 (Direct Cost: ¥7,700,000、Indirect Cost: ¥2,310,000)
Keywords非線形クライン・ゴルドン方程式 / 初期値問題 / 一様等方計量 / 大域解析 / 爆発解析 / クライン・ゴルドン方程式 / 爆発解 / 偏微分方程式論 / 漸近挙動 / 非線形 / 偏微分方程式 / アインシュタイン方程式
Outline of Final Research Achievements

The Cauchy problem for the nonlinear Klein-Gordon equation is mainly considered in the Friedmann-Lemaitre-Robertson-Walker spacetime as an example of nonlinear partial differential equation involved in the cosmology. In visiting Stanford University and The University of Texas Rio Grande Valley, the survey on the trend of research and discussion was done to promote this study. As results, several effects of the spatial expansion and contraction to nonlinear waves have been characterized in the theory of partial differential equations, and a rich structure of dissipation and anti-dissipation was clarified. Those results were announced as papers and talks, and some conferences were organized. Some results are in preparation for publication.

Academic Significance and Societal Importance of the Research Achievements

アインシュタインの重力場方程式の宇宙項は、空間の膨張と収縮を表すが、物理的正体は全く不明である。方程式の解は空間の曲がりの波(重力波)を表す。宇宙項と重力波の数学的性質を調べることには大きな意義がある。本研究では、アインシュタイン方程式を用いて時空計量と非線形場方程式を導出することから始め、その初期値問題の適切性理論の構築まで一貫した研究を行う。一様等方空間(ロバートソン・ウォーカー計量を持つ時空)における非線形クライン・ゴルドン方程式についての研究を、海外共同研究者の先端的方法を取り込み発展させる。膨張・収縮する空間において、非線形波は、どのように伝わるかを詳細に明らかにする。

Report

(7 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • Research Products

    (71 results)

All 2024 2023 2022 2021 2020 2019 2018

All Int'l Joint Research (2 results) Journal Article (21 results) (of which Peer Reviewed: 20 results) Presentation (38 results) (of which Int'l Joint Research: 14 results,  Invited: 27 results) Funded Workshop (10 results)

  • [Int'l Joint Research] テキサス大学リオグランデバリー校(米国)2023

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] スタンフォード大学(米国)2022

    • Related Report
      2023 Annual Research Report
  • [Journal Article] Numerical accuracy and stability of semilinear Klein-Gordon equation in de Sitter spacetime2023

    • Author(s)
      T. Tsuchiya, M. Nakamura
    • Journal Title

      JSIAM Letters

      Volume: 15 Pages: 45-48

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the Cauchy problem for the Hartree type semilinear Schroedinger equation in the de Sitter spacetime2023

    • Author(s)
      M. Ichimiya, M. Nakamura
    • Journal Title

      Evolution Equations and Control Theory

      Volume: 12 Issue: 6 Pages: 1602-1628

    • DOI

      10.3934/eect.2023028

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The Cauchy problem for a semilinear ordinary differential equation in the homogeneous and isotropic spacetime2023

    • Author(s)
      M. Nakamura
    • Journal Title

      Tsukuba Journal of Mathematics

      Volume: 47 Pages: 153-189

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the Cauchy problem for the Klein-Gordon equation with the Hartree type semilinear term in the de Sitter spacetime2021

    • Author(s)
      M. Nakamura, H. Takashima
    • Journal Title

      Differential and Integral Equations

      Volume: 34

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] EXISTENCE AND NON-EXISTENCE OF GLOBAL SOLUTIONS FOR THE SEMILINEAR COMPLEX GINZBURG-LANDAU TYPE EQUATION IN HOMOGENEOUS AND ISOTROPIC SPACETIME2021

    • Author(s)
      NAKAMURA Makoto; SATO Yuya
    • Journal Title

      Kyushu Journal of Mathematics

      Volume: 75 Issue: 2 Pages: 169-209

    • DOI

      10.2206/kyushujm.75.169

    • NAID

      130008101303

    • ISSN
      1340-6116, 1883-2032
    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the Cauchy problem for the semilinear Proca equations in the de Sitter spacetime2021

    • Author(s)
      Nakamura, Makoto
    • Journal Title

      Journal of Differential Equations

      Volume: 270 Pages: 1218-1257

    • DOI

      10.1016/j.jde.2020.09.015

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] Asymptotic behaviors of global solutions for a semilinear diffusion equation in the de Sitter spacetime2020

    • Author(s)
      Nakamura Makoto; Takeda, Hiroshi
    • Journal Title

      Asymptotic Analysis

      Volume: - Issue: 3-4 Pages: 203-245

    • DOI

      10.3233/asy-201652

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the nonrelativistic limit of a semilinear field equation in a homogeneous and isotropic space,2020

    • Author(s)
      M. Nakamura,
    • Journal Title

      Kyoto Journal of Mathematics

      Volume: 60 Pages: 1333-1359

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] Remarks on global solutions for the semilinear diffusion equation in the de Sitter spacetime2020

    • Author(s)
      M. Nakamura, Y. Sato,
    • Journal Title

      Hokkaido Mathematical Journal

      Volume: 49 Pages: 481-508

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] Remarks on the Navier-Stokes equations and the elastic wave equations in homogeneous and isotropic spacetimes2020

    • Author(s)
      M. Nakamura,
    • Journal Title

      Tsukuba Journal of Mathematics

      Volume: 44 Pages: 271-308

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] On some effects of background metrics for several partial differential equations2020

    • Author(s)
      M. Nakamura,
    • Journal Title

      Advanced Studies in Pure Mathematics, Mathematical Society of Japan, Kinokuniya, Tokyo

      Volume: 85 Pages: 315-324

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] Remarks on the derivation of several second order partial differential equations from a generalization of the Einstein equations2020

    • Author(s)
      M. Nakamura,
    • Journal Title

      Osaka J. Math.

      Volume: 57 Pages: 305-331

    • NAID

      120006846161

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the numerical analysis of the Cauchy problem for semilinear Klein-Gordon equations in the de Sitter spacetime2019

    • Author(s)
      T. Tsuchiya, M. Nakamura,
    • Journal Title

      Journal of Computational and Applied Mathematics

      Volume: 361 Pages: 396-412

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the regularity of the semilinear term on the Cauchy problem for the Schroedinger equation2019

    • Author(s)
      M. Nakamura,
    • Journal Title

      Proceedings of the 11th ISAAC congress 2017

      Volume: - Pages: 369-377

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Global solutions for nonlinear Schroedinger equations in de Sitter spacetime2019

    • Author(s)
      M. Nakamura,
    • Journal Title

      Advanced Studies of Pure Mathematics

      Volume: 81 Pages: 311-322

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Strichartz type estimates in mixed Besov spaces with application to critical nonlinear Shroedinger equations2019

    • Author(s)
      M. Nakamura, T. Wada
    • Journal Title

      Journal of Differential Equations

      Volume: 印刷中

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Remarks on global solutions for the semilinear diffusion equation in the de Sitter spacetime2019

    • Author(s)
      M. Nakamura, Y. Sato
    • Journal Title

      Hokkaido Mathematical Journal

      Volume: 印刷中

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] The Cauchy problem for dissipative wave equations with weighted nonliner terms2019

    • Author(s)
      M. Nakamura, H. Wadade
    • Journal Title

      New Tools for Nonlinear PDEs and Application

      Volume: 印刷中

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the nonrelativistic limit of a semilinear field equation in a homogeneous and isotropic space2019

    • Author(s)
      M. Nakamura
    • Journal Title

      Kyoto Journal of Mathematics

      Volume: 印刷中

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the regularity of the semilinear term on the Cauchy problem for the Schroedinger equation2019

    • Author(s)
      M. Nakamura
    • Journal Title

      Proceedings of the 11th ISAAC congress 2017

      Volume: 印刷中

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the eects of spatial expansion and contraction on several semilinear partial dierential equations2018

    • Author(s)
      M. Nakamura
    • Journal Title

      京都大学数理解析研究所講究録``非線形波動・分散型方程式''

      Volume: 2093 Pages: 27-37

    • Related Report
      2018 Research-status Report
  • [Presentation] Remarks on global solutions on semilinear Klein-Gordon equations in Friedmann-Lemaitre-Robertson-Walker spacetimes2024

    • Author(s)
      M. Nakamura
    • Organizer
      Mathematical Physics Seminar, The University of Texas Rio Grande Valley.
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Global solutions of Klein-Gordon equation under the quartic potential in the de Sitter spacetime2023

    • Author(s)
      M. Nakamura
    • Organizer
      14th International ISAAC Congress, University of Sao Paulo, Campus Ribeirao Preto (Brazil).
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On the Klein-Gordon equation under the quartic potential in the de Sitter spacetime2023

    • Author(s)
      M. Nakamura
    • Organizer
      Workshop on Nonlinear Hyperbolic PDEs, On the occasion of 60th birthday of Professor Yi Zhou, Tohoku University, Sendai, Japan.
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On small global solutions of Klein-Gordon equation under the quartic potential in the de Sitter spacetime2023

    • Author(s)
      M. Nakamura
    • Organizer
      神戸大学解析セミナー, 神戸大学.
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] Global solutions of Klein-Gordon equation under the quartic potential in the de Sitter spacetime2023

    • Author(s)
      M. Nakamura
    • Organizer
      大阪大学理学部数学教室談話会, 大阪大学.
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] On the Klein-Gordon equation under the quartic potential in the de Sitter spacetime2023

    • Author(s)
      M. Nakamura
    • Organizer
      Japanese Academic Seminars at Stanford, Stanford University
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] The Cauchy problem of Klein-Gordon equation under the quartic potential in the de Sitter spacetime2023

    • Author(s)
      M. Nakamura
    • Organizer
      Distinguished Colloquium Series, The University of Texas Rio Grande Valley
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Global solutions and blow-up solutions of power-type semilinear Klein-Gordon equations in the de Sitter spacetime2022

    • Author(s)
      M. Nakamura
    • Organizer
      微分方程式セミナー, 大阪大学
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] The Cauchy problem for the Klein-Gordon equation under the quartic potential in the de Sitter spacetime2022

    • Author(s)
      M. Nakamura
    • Organizer
      応用解析研究会, 早稲田大学
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Global solutions for semilinear Proca equations in the de Sitter spacetime2022

    • Author(s)
      M. Nakamura
    • Organizer
      NLPDEセミナー, 京都大学
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Global solutions for the Klein-Gordon equation under the quartic potential in the de Sitter spacetime2022

    • Author(s)
      M. Nakamura
    • Organizer
      熊本大学応用解析セミナー, 熊本大学
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Existence and non-existence of global solutions for the Klein-Gordon equation in the de Sitter spacetime2022

    • Author(s)
      M. Nakamura
    • Organizer
      第65回南大阪応用数学セミナー, 大阪公立大学
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Semilinear Proca equations in the de Sitter spacetime2022

    • Author(s)
      M. Nakamura
    • Organizer
      第18回非線型の諸問題(オンライン)
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] On the Cauchy problem for the Klein-Gordon equation under the quartic potential in the de Sitter spacetime2022

    • Author(s)
      M. Nakamura
    • Organizer
      Analysis & PDE seminar, Stanford University
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] On the Cauchy problem for the semilinear Proca equations in the de Sitter spacetime2021

    • Author(s)
      M. Nakamura
    • Organizer
      The 13th International ISAAC Congress
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research
  • [Presentation] On the Klein-Gordon equation with the Hartree type semilinear term in the de Sitter spacetime2021

    • Author(s)
      M. Nakamura, H. Takashima
    • Organizer
      日本数学会年会,2021年3月15日~18日,慶應義塾大学.
    • Related Report
      2020 Research-status Report
  • [Presentation] Global solutions for the semilinear diffusion equation in the de Sitter spacetime2019

    • Author(s)
      M. Nakamura
    • Organizer
      The 12th International ISAAC Congress
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Remarks on the Navier-Stokes equations in homogeneous and isotropic spacetimes2019

    • Author(s)
      M. Nakamura
    • Organizer
      The 12th International ISAAC Congress
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] On the semilinear partial differential equations in homogeneous and isotropic spacetimes2019

    • Author(s)
      M. Nakamura
    • Organizer
      Geometric Analysis and General Relativity
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On the Cauchy problem for a semilinear ordinary differential equation in homogeneous and isotropic spaces2019

    • Author(s)
      M. Nakamura
    • Organizer
      日本数学会
    • Related Report
      2019 Research-status Report
  • [Presentation] On global solutions for the semilinear complex Ginzburg-Landau type equation in homogeneous and isotropic spaces2019

    • Author(s)
      M. Nakamura
    • Organizer
      日本数学会
    • Related Report
      2019 Research-status Report
  • [Presentation] On the Cauchy problem for the Navier-Stokes equations in the de Sitter spacetime2019

    • Author(s)
      M. Nakamura
    • Organizer
      日本数学会
    • Related Report
      2019 Research-status Report
  • [Presentation] Partial differential equations in homogeneous and isotropic spaces2019

    • Author(s)
      M. Nakamura
    • Organizer
      日本数学会
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] On the Cauchy problem for the semilinear Proca equations in the de Sitter spacetime2019

    • Author(s)
      M. Nakamura
    • Organizer
      日本数学会
    • Related Report
      2019 Research-status Report
  • [Presentation] Asymptotic profiles of global solutions for the semilinear diffusion equation in the de Sitter spacetime2019

    • Author(s)
      M. Nakamura, H. Takeda
    • Organizer
      日本数学会
    • Related Report
      2019 Research-status Report
  • [Presentation] Some dissipative and anti-dissipative effects on semilinear PDEs in homogeneous and isotropic spaces2019

    • Author(s)
      M. Nakamura
    • Organizer
      三重偏微分方程式研究集会
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] Remarks on the Navier-Stokes equations in homogeneous and isotropic spacetimes2019

    • Author(s)
      M. Nakamura
    • Organizer
      第21回「特異点と時空、および関連する物理」研究会
    • Related Report
      2019 Research-status Report
  • [Presentation] Remarks on a semilinear diffusion equation in homogeneous and isotropic spaces2019

    • Author(s)
      M. Nakamura
    • Organizer
      Top global university project, Waseda workshop on partial differential equations 2019
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Global solutions for the semilinear diffusion equation in the de Sitter spacetime2019

    • Author(s)
      M. Nakamura
    • Organizer
      Nonlinear Dispersive Equations in Kumamoto, 2019
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Partial dierential equations in uniform and isotropic spaces2019

    • Author(s)
      M. Nakamura
    • Organizer
      2018年度日本数学会東北支部会
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Global solutions for a semilinear diusion equation in expanding or contracting spaces2019

    • Author(s)
      M. Nakamura and Y. Sato
    • Organizer
      日本数学会年会
    • Related Report
      2018 Research-status Report
  • [Presentation] On the Navier-Stokes equations in homogeneous and isotropic spacetimes with a constant density of mass2019

    • Author(s)
      M. Nakamura
    • Organizer
      日本数学会年会
    • Related Report
      2018 Research-status Report
  • [Presentation] On some effects of background metrics for several partial diffential equations2018

    • Author(s)
      M. Nakamura
    • Organizer
      The 11th Mathematical Society of Japan, Seasonal Institute
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On the dissipative effect of the spatial expansion for the semilinear diffusion equation2018

    • Author(s)
      M. Nakamura
    • Organizer
      East Asian Conference in Harmonic Analysis and its Applications 2018
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On the Navier-Stokes equation and the elastic wave equations in uniform and isotropic spaces2018

    • Author(s)
      M. Nakamura
    • Organizer
      Wayamba International Conference
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On the global solutions for the semi-linear diffusion equation in de Sitter spacetime2018

    • Author(s)
      M. Nakamura and Y. Sato
    • Organizer
      Wayamba International Conference
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The Cauchy problem for Navier-Stokes equations in de Sitter spacetime2018

    • Author(s)
      M. Nakamura
    • Organizer
      Workshop on Mathematical Sciences
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The asymptotic behavior of the semi-linear diffusion equation in de Sitter spacetime2018

    • Author(s)
      M. Nakamura and Y. Sato
    • Organizer
      Workshop on Mathematical Sciences
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Funded Workshop] ``The 14th International ISAAC Congress,'' Session ``Partial Differential Equations on Curved Spacetimes,"2023

    • Related Report
      2023 Annual Research Report
  • [Funded Workshop] ``Beijing-Osaka joint workshop for PDE and related topics,"2023

    • Related Report
      2023 Annual Research Report
  • [Funded Workshop] Mathematical Analysis of Nonlinear Dispersive and Wave Equations2022

    • Related Report
      2022 Research-status Report
  • [Funded Workshop] The 13th International ISAAC Congress2021

    • Related Report
      2021 Research-status Report
  • [Funded Workshop] 国際研究集会``General Relativity and Partial Differential Equation''2019

    • Related Report
      2019 Research-status Report
  • [Funded Workshop] 国際研究集会``The 12th International ISAAC Congress''2019

    • Related Report
      2019 Research-status Report
  • [Funded Workshop] Lectures “A novel integral transform approach to PDEs in the curved spacetimes”2018

    • Related Report
      2018 Research-status Report
  • [Funded Workshop] Seminar “Analysis on Partial Differential Equations”2018

    • Related Report
      2018 Research-status Report
  • [Funded Workshop] 研究会“Partial Differential Equation and General Relativity” (Yamagata)2018

    • Related Report
      2018 Research-status Report
  • [Funded Workshop] 研究会“Partial Differential Equation and General Relativity” (Nagoya)2018

    • Related Report
      2018 Research-status Report

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Published: 2018-01-25   Modified: 2025-01-30  

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