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Blowup problems for nonlinear heat equations

Research Project

Project/Area Number 26800065
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeMulti-year Fund
Research Field Mathematical analysis
Research InstitutionAkita University

Principal Investigator

Harada Junichi  秋田大学, 教育文化学部, 准教授 (00580169)

Project Period (FY) 2014-04-01 – 2018-03-31
Project Status Completed (Fiscal Year 2017)
Budget Amount *help
¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2015: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2014: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Keywords爆発問題 / 非線形熱方程式 / 同時・非同時爆発 / 日自己相似爆発 / 解の爆発問題 / 複素数値熱方程式 / 同次・非同時爆発 / 爆発プロファイル / 偏微分方程式 / 複素熱方程式
Outline of Final Research Achievements

We studied the mechanism of singularities for nonlinear diffusion equations. Since singularities include much information about the character of each equation, we can understand new phenomenons hidden in the equation. In this study, we focused on singularities appearing in diffusion equations for a higher dimension. We extended conventional theories and treated wide class of nonlinear problems. Consequently, we found a new type of singularity.

Report

(5 results)
  • 2017 Annual Research Report   Final Research Report ( PDF )
  • 2016 Research-status Report
  • 2015 Research-status Report
  • 2014 Research-status Report
  • Research Products

    (17 results)

All 2018 2017 2016 2015 2014

All Journal Article (7 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 7 results,  Acknowledgement Compliant: 2 results) Presentation (10 results) (of which Int'l Joint Research: 2 results,  Invited: 4 results)

  • [Journal Article] Construction of type II blow-up solutions for a semilinear parabolic system with higher dimension2018

    • Author(s)
      Junichi Harada
    • Journal Title

      Calculus of Variations and Partial Differential Equations

      Volume: 56 Issue: 4

    • DOI

      10.1007/s00526-017-1213-x

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Nonsimultaneous blowup for a complex valued semilinear heat equation2017

    • Author(s)
      Junichi Harada
    • Journal Title

      Journal of Differential Equations

      Volume: 263 Issue: 8 Pages: 4503-4516

    • DOI

      10.1016/j.jde.2017.05.024

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Blow-up sets for a complex-valued semilinear heat equation2017

    • Author(s)
      原田潤一
    • Journal Title

      Journal of Evolution Equations

      Volume: 印刷中 Issue: 2 Pages: 849-867

    • DOI

      10.1007/s00028-016-0341-7

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Blowup profile for a complex valued semilinear heat equation2016

    • Author(s)
      原田潤一
    • Journal Title

      Journal of Functional Analysis

      Volume: 270 Issue: 11 Pages: 4213-4255

    • DOI

      10.1016/j.jfa.2016.03.015

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Blow-up behavior of solutions to the heat equation with nonlinear boundary conditions2015

    • Author(s)
      原田潤一
    • Journal Title

      Advances in Differential Equations

      Volume: 20 Pages: 23-76

    • Related Report
      2014 Research-status Report
    • Peer Reviewed
  • [Journal Article] Positive solutions to the Laplace equation with nonlinear boundary conditions on the half space2014

    • Author(s)
      原田潤一
    • Journal Title

      Calculus of Variations and Partial Differential Equations

      Volume: 50 Issue: 1-2 Pages: 399-435

    • DOI

      10.1007/s00526-013-0640-6

    • Related Report
      2014 Research-status Report
    • Peer Reviewed
  • [Journal Article] Non self-similar blow-up solutions to the heat equation with nonlinear boundary conditions2014

    • Author(s)
      原田潤一
    • Journal Title

      Nonlinear Analysis

      Volume: 102 Pages: 36-83

    • DOI

      10.1016/j.na.2014.01.028

    • Related Report
      2014 Research-status Report
    • Peer Reviewed
  • [Presentation] Boundary behavior for solutions of the heat equation with a nonlinear boundary condition2017

    • Author(s)
      原田潤一
    • Organizer
      日本数学会2017年度会
    • Place of Presentation
      首都大学東京
    • Year and Date
      2017-03-26
    • Related Report
      2016 Research-status Report
  • [Presentation] 非線形境界条件熱方程式の爆発解の境界付近での漸近挙動について2017

    • Author(s)
      原田潤一
    • Organizer
      福島における偏微分方程式論 集中ワークショップ
    • Place of Presentation
      コラッセふくしま
    • Year and Date
      2017-02-12
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Type II blowup solutions for a semilinear parabolic system2017

    • Author(s)
      原田潤一
    • Organizer
      九州における偏微分方程式研究集会
    • Place of Presentation
      九州大学
    • Year and Date
      2017-02-01
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Exact blow-up profile for a heat equation with a nonlinear boundary condition2017

    • Author(s)
      原田潤一
    • Organizer
      RIMS 偏微分方程式の解の形状解析
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Simultaneous and nonsimultaneous blow-up for a semilinear parabolic system in higher dimension2015

    • Author(s)
      原田潤一
    • Organizer
      3rd Chile-Japan Workshop on Nonlinear PDE
    • Place of Presentation
      大阪大学
    • Year and Date
      2015-12-08
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 複素数値半線形熱方程式における非同時爆発の可能性について2015

    • Author(s)
      原田潤一
    • Organizer
      第三回北海道-東北コンソーシアムセミナー
    • Place of Presentation
      小樽商科大学
    • Year and Date
      2015-07-18
    • Related Report
      2015 Research-status Report
  • [Presentation] 複素数値半線形熱方程式における非同時爆発の可能性について2015

    • Author(s)
      原田潤一
    • Organizer
      2015年度 日本数学会 年会
    • Place of Presentation
      明治大学(東京都・千代田区)
    • Year and Date
      2015-03-22
    • Related Report
      2014 Research-status Report
  • [Presentation] Blow-up points for a semilinear complex-valued heat equation2014

    • Author(s)
      原田潤一
    • Organizer
      第23回なかもず解析セミナー
    • Place of Presentation
      大阪府立大学(大阪府・羽曳野市)
    • Year and Date
      2014-11-28
    • Related Report
      2014 Research-status Report
  • [Presentation] 複素数に値をとる半線形熱方程式の爆発点について2014

    • Author(s)
      原田潤一
    • Organizer
      第5回移流と拡散の数理
    • Place of Presentation
      九州大学西新プラザ(福岡県・福岡市)
    • Year and Date
      2014-11-22
    • Related Report
      2014 Research-status Report
  • [Presentation] ある非線形熱方程式系の爆発点について2014

    • Author(s)
      原田潤一
    • Organizer
      2014年度 日本数学会 秋季総合文科会
    • Place of Presentation
      広島大学(広島県・東広島市)
    • Year and Date
      2014-09-28
    • Related Report
      2014 Research-status Report

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Published: 2014-04-04   Modified: 2019-03-29  

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