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Study on variational problems, optimization problems and nonlinear partial differential equations

Research Project

Project/Area Number 16K05240
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

KURATA Kazuhiro  首都大学東京, 理学研究科, 教授 (10186489)

Research Collaborator JIMBO shuichi  
TANAKA kazunaga  
SHIBATA tetsutaro  
SHIBATA masataka  
Project Period (FY) 2016-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000)
Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Keywords変分問題 / 最適化問題 / 固有値問題 / 非線形楕円型偏微分方程式 / パターン形成 / 楕円型偏微分方程式 / 漸近解析 / FitzHugh-Nagumo反応拡散系 / 非線形シュレディンガー方程式 / パターン形成問題 / Quantum Waveguide問題 / 偏微分方程式 / 数理物理
Outline of Final Research Achievements

In the research of pattern formations from mathematical viewpoint, we studied the structure of energy minimizers of the associated nonlinear variational problem for the 3-component FitzHugh-Nagumo reaction-diffusion system, especially the micro-structure, precise energy estimates and their stability. We also established the precise asymptotic energy formula of heteroclinic solutions of the FitzHugh-Nagumo system in the singular perturbation problem. Moreover, in the study of the Schnakenberg mathematical model describing a chemical reaction phenomenon, we constructed one-peak stationary solution and revealed the effect of the heterogeneous coefficient to its stability by using asymptotic analysis.

Academic Significance and Societal Importance of the Research Achievements

生命現象や物理現象等における興味深いパターン形成の起こる数理的メカニズムを解明することは、数値シミュレーションによる解析も含め、様々な研究アプローチがあり学際的な研究分野である。その中で、本研究は、特に非線形数理モデルの厳密な数学的解析による定量的評価を行いパターン形成のメカニズムを明らかにするもので、学術的意義が高い。また、現象における環境効果は数理モデルである非線形微分方程式の係数の空間非一様性として現われ、その環境効果がパターン形成及びその安定性を制御するメカニズムを解明することにもなり、社会的意義が大きい。

Report

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

    (17 results)

All 2019 2018 2017 2016 Other

All Journal Article (4 results) (of which Peer Reviewed: 4 results,  Acknowledgement Compliant: 1 results) Presentation (12 results) (of which Int'l Joint Research: 2 results,  Invited: 11 results) Remarks (1 results)

  • [Journal Article] Existence and stability of one-peak symmetric stationary solutions for the Schnakenberg model with heterogeneity2019

    • Author(s)
      Yuta Ishii and Kazuhiro Kurata
    • Journal Title

      Discrete and Conti. Dynamical System, Series-A.

      Volume: 35 Issue: 5 Pages: 2807-2875

    • DOI

      10.3934/dcds.2019118

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On a Variational Problem Arising from the Three-component FitzHugh-Nagumo Type Reaction-Diffusion Systems2018

    • Author(s)
      Takashi Kajiwara and Kazuhiro Kurata
    • Journal Title

      Tokyo J. Mathematics

      Volume: 41 Issue: 1

    • DOI

      10.3836/tjm/1502179257

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Three-component FitzHugh-Nagumo Type Reaction-Diffusion Systems2017

    • Author(s)
      Takashi Kajiwara and Kazuhiro Kurata
    • Journal Title

      Tokyo J. Mathematics

      Volume: 40

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Asymptotic behavior of eigenvalues of the Laplacian on a thin domain under the mixed boundary condition2016

    • Author(s)
      S. Jimbo and K. Kurata
    • Journal Title

      Indiana. Univ. Math. J.

      Volume: 65 Issue: 3 Pages: 867-898

    • DOI

      10.1512/iumj.2016.65.5831

    • Related Report
      2016 Research-status Report
    • Peer Reviewed
  • [Presentation] Precise asymptotic formula for the ground state energy of a quantum waveguide problem with Robin boundary condition2018

    • Author(s)
      倉田 和浩
    • Organizer
      「信州大学偏微分方程式研究集会」
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Existence and stability of one-peak symmetric stationary solutions for the Schnakenberg model with heterogeneity,2018

    • Author(s)
      倉田 和浩
    • Organizer
      数理研研究集会「偏微分方程式の解の幾何」
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Construction and stability analysis of one-peak symmetric stationary solutions to the Schnakenberg model with heterogeneity,2018

    • Author(s)
      Kazuhiro Kurata
    • Organizer
      The 12th AIMS international conference on Dynamical System and Differential equations
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Variational problems for Nonlinear Schr\"{o}dinger and Allen-Cahn equations on non-compact metric graphs2018

    • Author(s)
      倉田 和浩
    • Organizer
      RACMaS Workshop on Differential Equations and Networks
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] 交差拡散効果をもつあるprey-predator反応拡散系の非定数定常解について2017

    • Author(s)
      倉田 和浩
    • Organizer
      研究集会「非線形現象の数値シミュレーションと解析2017」
    • Place of Presentation
      北海道大学
    • Year and Date
      2017-03-07
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] 非有界network上のAllen-Cahn方程式のヘテロクリニック解について2017

    • Author(s)
      倉田 和浩
    • Organizer
      第11回非線形偏微分方程式と変分問題
    • Place of Presentation
      首都大学東京
    • Year and Date
      2017-02-12
    • Related Report
      2016 Research-status Report
  • [Presentation] ある非コンパクトな距離グラフ上での非線型シュレディンガー方程式に 付随する変分問題の解の漸近挙動について2017

    • Author(s)
      倉田 和浩
    • Organizer
      RIMS研究集会「保存則をもつ偏微分方程式に対する解の正則性, 特異性および漸近挙動について」
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] On an asymptotic behavior of minimizers of the variational problem arising from a certain FitzHugh-Nagumo system with an inhomogeneous diffusion coefficient2017

    • Author(s)
      倉田 和浩
    • Organizer
      研究集会「第13回非線型の諸問題」
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] On an optimization problem of the first eigenvalue of the Laplacian on a thin domain with Neumann windows2017

    • Author(s)
      倉田 和浩
    • Organizer
      研究集会「Potential Analysis and Related Fields 2017」
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] 空間2次元の異種成分間の斥力相互作用をもつ 2成分Bose-Einstein凝縮体に関する変分問題について2016

    • Author(s)
      倉田 和浩
    • Organizer
      東北大学・談話会
    • Place of Presentation
      東北大学
    • Year and Date
      2016-11-28
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Existence and non-existence of non-constant stationary patterns for certain prey-predator type reaction-diffusion systems with a cross-diffusion effect2016

    • Author(s)
      倉田 和浩
    • Organizer
      研究集会「ミクロな振る舞いと集団的パターン形成に係る階層的構造の解明」
    • Place of Presentation
      京都数理解析研究所
    • Year and Date
      2016-09-14
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Existence and non-existence of non-constant stationary patterns for certain prey-predator type reaction-diffusion systems with a cross-diffusion effect2016

    • Author(s)
      Kazuhiro Kurata
    • Organizer
      The 11th AIMS Conference on Dynamical Systems, Differential Equations and Applications
    • Place of Presentation
      Orlando, USA
    • Year and Date
      2016-07-02
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Remarks] Kurata's Home Page

    • URL

      http://www.comp.tmu.ac.jp/tmu-kurata/index.html

    • Related Report
      2018 Annual Research Report

URL: 

Published: 2016-04-21   Modified: 2020-03-30  

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