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Expansion of diffusion equations with a dynamic boundary condition to nonlinear problems

Research Project

Project/Area Number 20K03689
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 12020:Mathematical analysis-related
Research InstitutionRyukoku University

Principal Investigator

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

Project Period (FY) 2020-04-01 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2023: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2022: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2021: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2020: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords動的境界条件 / 可解性 / 分数冪拡散方程式 / 非線形放物型方程式 / 拡散方程式 / 基本解 / 退化放物型方程式 / 臨界指数 / 重み付き空間 / 指数型非線形項 / 非線形境界条件 / 高次漸近展開 / 外部領域 / 分数冪 Hardy-Henon 方程式 / Joseph-Lundgren 指数 / 高階放物型方程式 / 分数冪Laplacian
Outline of Research at the Start

これまで半空間や単位球の外部領域における動的境界条件を有する楕円型方程式の研究によって用いてきた手法を応用・発展させることで、具体的な解表示とそれに基づく様々な評価の導出を行う。また、付随した研究として、半空間の非線形動的境界条件の定常問題として現れる分数冪Laplacianを有する非線形楕円型方程式について,その解構造の解明を目指す。さらに、より一般の動的境界条件への展開として、一般の分数冪拡散方程式に付随する特異または退化係数を有する放物型方程式の可解性や、高次の微分を含む境界条件に関連する高階放物型方程式への分数冪拡散方程式からのアプローチを試みる。

Outline of Final Research Achievements

In this research, our aim is to expand of diffusion equations with a dynamical boundary condition in half-space to nonlinear problems, and we studied related issues. For diffusion equations with a dynamical boundary condition, although only when the initial data on the boundary is zero, we extended solvability of solutions, which was previously obtained only for bounded initial data, for a wider class of internal initial data belongs to an appropriate weighted space. Furthermore, we also obtained related results as the solvability and structure of fractional Hardy-Henon equations, the solvability and asymptotic behavior of the heat equation with an exponential nonlinear boundary condition, the solvability of higher-order parabolic equations, and refined asymptotic expansions for fractional diffusion equations.

Academic Significance and Societal Importance of the Research Achievements

動的境界条件を有する拡散方程式は近年、純粋数学のみならず応用数理や環境工学、生態学など様々な分野において活発に研究されてきている。現象の多くは非線形問題で記述されることからも、その基礎となる線形問題における可解性や付随する楕円型・放物型方程式の考察は、非線形問題への応用上欠かすことのできないものであり、今回の研究成果は今後の非線形問題への展開に向けて大変示唆に富んだものであり、今後の進展が大きに期待される。

Report

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

    (23 results)

All 2023 2022 2021 2020 Other

All Int'l Joint Research (4 results) Journal Article (6 results) (of which Int'l Joint Research: 3 results,  Peer Reviewed: 6 results,  Open Access: 1 results) Presentation (10 results) (of which Int'l Joint Research: 4 results,  Invited: 10 results) Remarks (2 results) Funded Workshop (1 results)

  • [Int'l Joint Research] Comenius University(スロバキア)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] Johns Hopkins University(米国)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] Comenius University(スロバキア)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] Universita di Bergamo/Universita degli Studi di Milano(イタリア)

    • Related Report
      2021 Research-status Report
  • [Journal Article] On weak solutions to a fractional Hardy-Henon equation, Part II: Existence2023

    • Author(s)
      Hasegawa Shoichi、Ikoma Norihisa、Kawakami Tatsuki
    • Journal Title

      Nonlinear Analysis

      Volume: 227 Pages: 113165-113165

    • DOI

      10.1016/j.na.2022.113165

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Solvability of the heat equation on a half-space with a dynamical boundary condition and unbounded initial data2023

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

      Zeitschrift fur angewandte Mathematik und Physik

      Volume: 74 Issue: 4

    • DOI

      10.1007/s00033-023-02040-7

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Heat equation with an exponential nonlinear boundary condition in the half space2022

    • Author(s)
      G. Furioli, T. Kawakami, and E. Terraneo
    • Journal Title

      Partial Differ. Equ. Appl.

      Volume: 3 Issue: 3 Pages: 1-44

    • DOI

      10.1007/s42985-022-00170-7

    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Existence of solutions to nonlinear parabolic equations via majorant integral kernel2022

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

      Nonlinear Anal

      Volume: 223 Pages: 113025-113025

    • DOI

      10.1016/j.na.2022.113025

    • Related Report
      2022 Research-status Report
    • Peer Reviewed
  • [Journal Article] On weak solutions to a fractional Hardy?H?non equation: Part I: Nonexistence2021

    • Author(s)
      Hasegawa Shoichi、Ikoma Norihisa、Kawakami Tatsuki
    • Journal Title

      Communications on Pure & Applied Analysis

      Volume: 0 Issue: 4 Pages: 1559-1600

    • DOI

      10.3934/cpaa.2021033

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] The large diffusion limit for the heat equation in the exterior of the unit ball with a dynamical boundary condition2020

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

      Discrete and Continuous Dynamical Systems

      Volume: 40 Issue: 11 Pages: 6529-6546

    • DOI

      10.3934/dcds.2020289

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Solvability of the heat equation on a half-space with a dynamical boundary condition2023

    • Author(s)
      Tatsuki Kawakami
    • Organizer
      Euro-Japanese Conference on Nonlinear diffusions
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Solvability of the heat equation on a half-space with a dynamical boundary condition2023

    • Author(s)
      Tatsuki Kawakami
    • Organizer
      Evolution Equations and Related Topics - Energy Structures and Quantitative Analysis -
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Existence of solutions to nonlinear parabolic equations via majorant integral kernel2022

    • Author(s)
      Tatsuki Kawakami
    • Organizer
      Seminar on Qualitative Theory of Differential Equations
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Existence of solutions to nonlinear parabolic equations via majorant integral kernel2022

    • Author(s)
      川上竜樹
    • Organizer
      HMAセミナー・冬の研究会2022
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] 非整数階時間微分を含む移流拡散方程式について2022

    • Author(s)
      川上竜樹
    • Organizer
      非線形現象の数値シミュレーションと解析2022
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] 半空間における指数型非線形境界条件を有する熱方程式について2022

    • Author(s)
      川上竜樹
    • Organizer
      北陸応用数理研究会2022
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] The large diffusion limit for the heat equation with a dynamical boundary condition2021

    • Author(s)
      Tatsuki Kawakami
    • Organizer
      BIRS-CMO Workshop "New Trends in Nonlinear Diffusion: a Bridge between PDEs, Analysis and Geometry"
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 時空間非斉次項を有する半線形拡散方程式の臨界指数2021

    • Author(s)
      川上竜樹
    • Organizer
      第15回応用数理研究会
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] Existence of solutions to nonlinear parabolic equations via majorant integral kernel2021

    • Author(s)
      川上竜樹
    • Organizer
      鳥取PDE研究集会2021
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] 動的境界条件を有する半線形楕円型方程式及び線形熱方程式の解析2021

    • Author(s)
      川上竜樹
    • Organizer
      応用数理勉強会2021
    • Related Report
      2021 Research-status Report
    • Invited
  • [Remarks] 川上竜樹のホームページ

    • URL

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

    • Related Report
      2020 Research-status Report
  • [Remarks] 川上 竜樹 -研究者- researchmap

    • URL

      https://researchmap.jp/k-tatsuki

    • Related Report
      2020 Research-status Report
  • [Funded Workshop] Workshop on Elliptic & Parabolic PDEs 20232023

    • Related Report
      2023 Annual Research Report

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Published: 2020-04-28   Modified: 2025-01-30  

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