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A study of asymptotic behavior of threshold orbit arising in gradient systems with noncompact energy structure

Research Project

Project/Area Number 17K05323
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Mathematical analysis
Research InstitutionOsaka University

Principal Investigator

Ishiwata Michinori  大阪大学, 基礎工学研究科, 教授 (30350458)

Project Period (FY) 2017-04-01 – 2020-03-31
Project Status Completed (Fiscal Year 2019)
Budget Amount *help
¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2019: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2018: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2017: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Keywords臨界型偏微分方程式 / プロファイル分解 / コンパクト性の破れ / 時間大域解 / 変分問題 / 臨界放物型方程式 / ソボレフ臨界指数 / 時間大域的有界性 / エネルギー臨界 / 熱方程式 / 非コンパクト性 / バブリング / 変分解析
Outline of Final Research Achievements

We treat the time-global asymptotic behavior of some partial differential equations with non-compact solution orbits in phase space. We also deal with the solubility of the maximization problem in order to analyze the non-compactness situation for the functional inequalities associated with these partial differential equations.
For semi-linear parabolic equations, it is shown that the norm of the solution is bounded along the time-global solution, and that the solution has an asymptotic behavior over time, represented by a superposition of "bubbles" in which a finite number of equilibrium solutions in the whole space are deformed under the non-compact group action associated with each of them; for maximization problems associated with Trudinger-Moser-type inequalities, it is shown that the solubility of the maximization problem varies sensitively depending on the size of the lower-order term, which acts to break the compactness of the function, and the size of the constraint parameters.

Academic Significance and Societal Importance of the Research Achievements

従来の力学系理論は、解軌道が相空間の中で相対コンパクトな場合、解のオメガ極限集合が力学系の平衡点の集合の部分集合になることを主張するものであったが、本来偏微分方程式論など具体的な理論での興味は「有界な解軌道」の漸近挙動である。従って、従来の力学系理論における軌道のコンパクト性の仮定は、強すぎる仮定とみなすことができる。本研究によって扱われたいくつかの非コンパクト性を内包する偏微分方程式についての結果から、「有界性+プロファイル分解」パラダイムに基づく解析が有効であることが示された。これにより、既存の抽象力学系の理論を、解軌道が非コンパクトだが有界な場合に拡張する足掛かりを得ることができた。

Report

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

    (25 results)

All 2019 2018 2017 Other

All Int'l Joint Research (7 results) Journal Article (4 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 3 results,  Open Access: 2 results) Presentation (13 results) (of which Int'l Joint Research: 9 results,  Invited: 11 results) Funded Workshop (1 results)

  • [Int'l Joint Research] ミラノ大学(イタリア)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] ボルドー大学(フランス)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] ミラノ大学/フィレンツェ大学(イタリア)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] ボルドー大学(フランス)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] University of Milan(Italy)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] University of Bordeaux(France)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] University of Florence(Italy)

    • Related Report
      2017 Research-status Report
  • [Journal Article] On the maximizing problem associated with Sobolev-type embeddings under inhomogeneous constraints2019

    • Author(s)
      Ishiwata, Michinori; Wadade, Hidemitsu
    • Journal Title

      Appl. Anal.

      Volume: 98 Pages: 1916-1934

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Existence and non-existence of maximizers for the Moser-Trudinger type inequalities under inhomogeneous constraints2019

    • Author(s)
      Ikoma, Norihisa; Ishiwata, Michinori; Wadade, Hidemitsu
    • Journal Title

      Math. Ann.

      Volume: 373 Pages: 831-851

    • Related Report
      2019 Annual Research Report
  • [Journal Article] A natural approach to the asymptotic mean value property for the p-Laplacian2017

    • Author(s)
      Ishiwata Michinori、Magnanini Rolando、Wadade Hidemitsu
    • Journal Title

      Calculus of Variations and Partial Differential Equations

      Volume: 56 Issue: 4 Pages: 56-97

    • DOI

      10.1007/s00526-017-1188-7

    • NAID

      120006343950

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Hardy type inequalities in $L_p$ with sharp remainders2017

    • Author(s)
      N. Ioku, M. Ishiwata, T. Ozawa
    • Journal Title

      Journal of Inequalities and Applications

      Volume: - Issue: 1 Pages: 1-7

    • DOI

      10.1186/s13660-016-1271-1

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Presentation] An application of the profile-decomposition method to parabolic problems2019

    • Author(s)
      石渡 通徳
    • Organizer
      第 699 回「応用解析」研究会
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] On the profile decomposition of a volume functional2019

    • Author(s)
      Michinori Ishiwata
    • Organizer
      Variational analysis of critical problems on nonlinear partial differantial equation
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Bounds of Sobolev norm for solutions of semiliner parabolic equations with critical Sobolev exponent2018

    • Author(s)
      Michinori Ishiwata
    • Organizer
      UK-Japan Workshop on Analysis of Nonlinear Partial Differential Equations
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On the bounds of the Sobolev norm for solutions of semilinear parabolic problems with critical Sobolev exponent2018

    • Author(s)
      Michinori Ishiwata
    • Organizer
      Eighth Euro-Japanese Workshop on Blow-up
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On the soliton-resolution of time-global solutions for the energy critical heat equation2018

    • Author(s)
      Michinori Ishiwata
    • Organizer
      京都大学NLPDEセミナー
    • Related Report
      2018 Research-status Report
  • [Presentation] On global bounds for Sobolev norms of time global solutions of parabolic problem involving critical Sobolev exponent2018

    • Author(s)
      Michinori Ishiwata
    • Organizer
      12th bi-annual Conference on Dynamical Systems, Differential Equations and Applications
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On the compactness of the Sobolev embedding involving variable exponent2018

    • Author(s)
      Michinori Ishiwata
    • Organizer
      12th bi-annual Conference on Dynamical Systems, Differential Equations and Applications
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On the soliton decomposition associated with the energy critical heat equation2018

    • Author(s)
      石渡通徳
    • Organizer
      社会現象と生命動態に関する偏微分方程式
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] On the soliton decomposition of solutions for the energy critical parabolic equation2018

    • Author(s)
      Michinori Ishiwata
    • Organizer
      Theoretical Developments to Phenomenon Analyses based on Nonlinear Evolution Equations
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On global bounds of time-global solutions for semilinear parabolic problems with critical Sobolev exponent2018

    • Author(s)
      Michinori Ishiwata
    • Organizer
      Workshop Critical exponent and nonlinear evolution equations 2019
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On time-global bounds for solutions for semilinear parabolic equation with variable critical exponent2018

    • Author(s)
      石渡通徳
    • Organizer
      第26 回応用解析研究会シンポジウム
    • Related Report
      2018 Research-status Report
  • [Presentation] On the existence of global bounds of Sobolev norms for time global solutions of parabolic equation involving critical Sobolev exponent2017

    • Author(s)
      Michinori Ishiwata
    • Organizer
      GEOMETRIC ASPECTS OF PDE'S
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On the time-global bounds of Sobolev norms of global solutions of parabolic equation involving critical Sobolev exponent2017

    • Author(s)
      Michinori Ishiwata
    • Organizer
      X Workshop on Nonlinear Differential Equations
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Funded Workshop] AMS Spring Central and Western Joint Sectional Meeting2018

    • Related Report
      2018 Research-status Report

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Published: 2017-04-28   Modified: 2021-02-19  

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