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自然現象を記述する偏微分方程式の解の漸近挙動に関する研究

Research Project

Project/Area Number 14J01884
Research Category

Grant-in-Aid for JSPS Fellows

Allocation TypeSingle-year Grants
Section国内
Research Field Mathematical analysis
Research InstitutionKyoto University

Principal Investigator

池田 正弘  京都大学, 理学研究科, 特別研究員(PD)

Project Period (FY) 2014-04-25 – 2017-03-31
Project Status Completed (Fiscal Year 2016)
Budget Amount *help
¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000)
Fiscal Year 2016: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2014: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Keywords非線形消散型波動方程式 / 藤田指数 / 解の有限時間爆発 / ライフスパン / Blow-up rate / 常微分不等式 / 時間依存 / シュレディンガー方程式 / 解の分類 / 質量超臨界 / ディラックのデルタ関数 / 非線形シュレディンガー方程式 / 解の爆発 / ライフスパンの評価 / 非線形消散型クライン・ゴルドン方程式 / 臨界指数 / 局所解の非存在 / 絶対値冪乗
Outline of Annual Research Achievements

本年度は, 冪乗型非線形項と時間変数に依存した係数の摩擦項を持つ波動方程式の初期値問題に対する解の爆発に関する研究を行った. 定数係数の摩擦項を持つ非線形波動方程式の初期値問題に対する大域解は, 十分に時間が経過すると対応する熱方程式の解で近似できることはよく知られている. それにも関わらず解の爆発に関する研究は, 熱方程式のそれに比べて進んでいるとは言えない. 2015年に私は, 若杉氏と共同で, 絶対値冪型の非線形項に対して, その次数が藤田臨界指数よりも小さい場合に, 解の最大存在時刻に関するほぼ最良の評価を導出した. また, この結果を摩擦項の係数が効果的に働く時間変数か空間変数のどちらか一方に依存する場合に拡張することに成功した. しかし, 最適な評価は得られておらず, また解の発散速度も不明のままであった. そこで私は, 藤原氏と若杉氏と共同で, この問題を再度異なるアプローチを用いて研究することにより, 定数係数を含んだ効果的に働く時間変数に依存した摩擦項を持つ絶対値冪乗型の非線形項に対して, 解の爆発時刻の最良の評価と解のある積分量の発散速度を導出した. この結果は, 国際誌に投稿しており, 国内外の研究集会では既に口頭発表を行った.

Research Progress Status

28年度が最終年度であるため、記入しない。

Strategy for Future Research Activity

28年度が最終年度であるため、記入しない。

Report

(3 results)
  • 2016 Annual Research Report
  • 2015 Annual Research Report
  • 2014 Annual Research Report
  • Research Products

    (24 results)

All 2017 2016 2015 2014

All Journal Article (7 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 7 results,  Acknowledgement Compliant: 5 results) Presentation (17 results) (of which Int'l Joint Research: 3 results,  Invited: 16 results)

  • [Journal Article] The Cauchy problem for the nonlinear damped wave equation with slowly decaying data2017

    • Author(s)
      Masahiro Ikeda, Takahisa Inui, Yuta Wakasugi
    • Journal Title

      Nonlinear Differential Equations and Applications NoDEA

      Volume: 24 Issue: 2

    • DOI

      10.1007/s00030-017-0434-1

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Global Dynamics below the standing waves for the focusing semi-linear Schrodinger equation with a repulsive Dirac delta potential2017

    • Author(s)
      Masahiro Ikeda, Takahisa Inui,
    • Journal Title

      Analysis and PDE

      Volume: 10 Issue: 2 Pages: 481-512

    • DOI

      10.2140/apde.2017.10.481

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Asymptotics for the Schr\"dinger equation with a repulsive delta potential and a long-range dissipative nonlinearity2017

    • Author(s)
      Masahiro Ikeda
    • Journal Title

      Proceedings of the conference "Asymptotic Analysis for Nonlinear Dispersive and Wave Equations"

      Volume: 未定

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] A remark on non-existence for the semi-linear damped Klein-Gordon equations2016

    • Author(s)
      Masahiro Ikeda Takahisa Inui
    • Journal Title

      RIMS Kokyuroku Bessatsu

      Volume: 未定

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Lifespan of solutions to the damped wave equation with a critical nonlinearity2016

    • Author(s)
      Masahiro Ikeda Takayoshi Ogawa
    • Journal Title

      Journal of Differential Equations

      Volume: 未定

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Some non-existence results for the semi-linear Schrodinger equation without gauge invariance2015

    • Author(s)
      Masahiro Ikeda and Takahisa Inui
    • Journal Title

      Journal of Mathematical Analysis and Applications

      Volume: 425 Issue: 2 Pages: 758-773

    • DOI

      10.1016/j.jmaa.2015.01.003

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Small data blow-up of L^2 or H^1-solution for the semilinear Schrodinger equation without gauge invariance2015

    • Author(s)
      Masahiro Ikeda and Takahisa Inui
    • Journal Title

      Journal of Evolution Equations

      Volume: 未定 Issue: 3 Pages: 571-581

    • DOI

      10.1007/s00028-015-0273-7

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Presentation] Sharp Lifespan Estimates and Blow-up Rates for the Semilinear Wave Equation with Time-Dependent Damping and Subcritical Nonlinearities2016

    • Author(s)
      Masahiro Ikeda
    • Organizer
      2016 Taiwan-Japan Workshop on Dispersion, Navier Stokes, Kinetic and Inverse Problems
    • Place of Presentation
      National Cheng Kung University
    • Year and Date
      2016-12-24
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] ディラックのデルタ関数をポテンシャルに持つ非線形シュレディンガー方程式の解の分類について2016

    • Author(s)
      池田 正弘
    • Organizer
      線形及び非線形分散型方程式に関する最近の進展
    • Place of Presentation
      京都大学
    • Year and Date
      2016-05-16
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] Diracのデルタ関数をポテンシャルに持つ非線形シュレディンガー方程式の解の分類について2016

    • Author(s)
      池田 正弘, 戌亥 隆恭
    • Organizer
      日本数学会年会
    • Place of Presentation
      筑波大学
    • Year and Date
      2016-03-16
    • Related Report
      2015 Annual Research Report
  • [Presentation] ディラックのデルタ関数をポテンシャルに持つ非線形シュレディンガー方程式の解の分類について2016

    • Author(s)
      池田 正弘
    • Organizer
      数理モデルにおける非線形消散・分散構造の臨界性の未開領域解明
    • Place of Presentation
      東北大学
    • Year and Date
      2016-01-22
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] ディラックのデルタ関数をポテンシャルに持つ非線形シュレディンガー方程式の解の分類について2016

    • Author(s)
      池田 正弘
    • Organizer
      HMA seminar
    • Place of Presentation
      広島大学
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] Lifespan of solutions to the damped wave equation with the Fujita exponent2016

    • Author(s)
      Masahiro Ikeda
    • Organizer
      Seminar on Nonlinear Wave Equations in Hakodate
    • Place of Presentation
      はこだて未来大学
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] ディラックのデルタ関数をポテンシャルに持つ非線形シュレディンガー方程式の解の分類について2016

    • Author(s)
      池田 正弘
    • Organizer
      談話会
    • Place of Presentation
      名古屋大学
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] Sharp estimate of lifespan and d blow-up rate of solutions to the damped wave equation2016

    • Author(s)
      池田 正弘
    • Organizer
      第6回弘前非線形方程式研究会
    • Place of Presentation
      弘前大学
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] Global dynamics below the ground state for the mass-supercritical focusing nonlinear Schr\"dinger equation with a repulsive delta potential2015

    • Author(s)
      池田 正弘
    • Organizer
      第9回実解析と関数解析による微分方程式セミナー
    • Place of Presentation
      ヒルズサンピア山形
    • Year and Date
      2015-12-23
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] Global dynamics for the focusing nonlinear Schr\"dingier equation with a repulsive delta potential in the mass-supercritical case2015

    • Author(s)
      池田 正弘
    • Organizer
      微分方程式の総合的研究
    • Place of Presentation
      東京大学
    • Year and Date
      2015-12-19
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] Estimate on the lifespan of solutions to the damped wave equation with a critical nonlinearity2015

    • Author(s)
      池田 正弘
    • Organizer
      The 3rd CAU-Kyoto U. Joint Workshop on Nonlinear PDEs
    • Place of Presentation
      Seo-gwi-po KAL hotel, Jejudo, Korea
    • Year and Date
      2015-09-21
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Null structure in a system of quadratic derivative nonlinear Schrodinger equations2015

    • Author(s)
      Masahiro Ikeda
    • Organizer
      14th New Mexico Analysis Seminar
    • Place of Presentation
      New Mexico State University
    • Year and Date
      2015-03-27
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] 絶対値べき乗型非線形シュレディンガー方程式の解のライフスパンと局所解の非存在について2014

    • Author(s)
      池田 正弘, 戌亥 隆恭
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      広島大学
    • Year and Date
      2014-09-28
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Non-existence results for a semi-linear Schrodinger equation2014

    • Author(s)
      Masahiro Ikeda
    • Organizer
      Asymptotic Analysis for Nonlinear Dispersive and Wave Equations
    • Place of Presentation
      大阪大学
    • Year and Date
      2014-09-11
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Non-existence results for the semilinear Schrodinger equation2014

    • Author(s)
      Masahiro Ikeda
    • Organizer
      6th Euro-Japanese workshop on blow-up
    • Place of Presentation
      東京工業大学
    • Year and Date
      2014-09-01
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Well-posedness and scattering for a Schrodinger system with null forms at the scaling critical regularity2014

    • Author(s)
      Masahiro Ikeda
    • Organizer
      調和解析と偏微分方程式
    • Place of Presentation
      京都大学
    • Year and Date
      2014-06-30
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] On small data blow-up for the nonlinear damped wave equation and Schrodinger equation2014

    • Author(s)
      池田 正弘
    • Organizer
      微分方程式セミナー
    • Place of Presentation
      大阪大学
    • Year and Date
      2014-05-23
    • Related Report
      2014 Annual Research Report
    • Invited

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Published: 2015-01-22   Modified: 2024-03-26  

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