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Singularity of solutions and stationary problems for nonlinear parabolic equations

Research Project

Project/Area Number 17K05333
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Mathematical analysis
Research InstitutionHiroshima University (2020)
Ehime University (2017-2019)

Principal Investigator

Naito Yuki  広島大学, 先進理工系科学研究科(理), 教授 (10231458)

Co-Investigator(Kenkyū-buntansha) 猪奥 倫左  東北大学, 理学研究科, 准教授 (50624607)
Project Period (FY) 2017-04-01 – 2021-03-31
Project Status Completed (Fiscal Year 2020)
Budget Amount *help
¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2019: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2018: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2017: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Keywords非線形解析 / 非線形楕円型方程式 / 非線形熱方程式 / 自己相似解 / 特異解 / 楕円型偏微分方程式 / 放物型偏微分方程式 / 定常問題
Outline of Final Research Achievements

We showed the existencde and the uniquness of the singular solutions to semilinear elliptic partial differential equations with Sobolev super-critical nonlinearity. We also showed the convergence of regular solutions to the singular solution.
We consider positive solutions of the semilinear heat equation with supercritical power nonlinearity, and construct peaking solutions by connecting a backward selfsimilar solution with a forward self-similar solution. In particular, we show the existence of incomplete blow-up solutions with blow-up profile above the singular steady state.

Academic Significance and Societal Importance of the Research Achievements

非線形楕円型偏微分方程式に対する境界値問題に対して、特異解から分岐構造を解析するという新たな手法の確立が期待できる。
特異定常解より大きな爆発形状をもつ不完全爆発解の構成により「爆発形状が特異定常解より大きければ完全爆発である」という直感を裏切る結果が得られた。これにより、非線形熱方程式における爆発問題において新たな進展が期待できる。

Report

(5 results)
  • 2020 Annual Research Report   Final Research Report ( PDF )
  • 2019 Research-status Report
  • 2018 Research-status Report
  • 2017 Research-status Report
  • Research Products

    (40 results)

All 2021 2020 2019 2018 2017 Other

All Int'l Joint Research (3 results) Journal Article (14 results) (of which Int'l Joint Research: 5 results,  Peer Reviewed: 13 results,  Open Access: 2 results) Presentation (23 results) (of which Int'l Joint Research: 9 results,  Invited: 12 results)

  • [Int'l Joint Research] University of Zagreb(クロアチア)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] Hanbat National University(韓国)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] Hanbat National University(韓国)

    • Related Report
      2017 Research-status Report
  • [Journal Article] Fundamental properties and asymptotic shapes of the singular and classical radial solutions for supercritical semilinear elliptic equations2020

    • Author(s)
      Miyamoto Yasuhito、Naito Yuki
    • Journal Title

      Nonlinear Differential Equations and Applications NoDEA

      Volume: 27 Issue: 6 Pages: 1-25

    • DOI

      10.1007/s00030-020-00658-4

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Asymptotically self-similar behaviour of global solutions for semilinear heat equations with algebraically decaying initial data2020

    • Author(s)
      Naito Yuki
    • Journal Title

      Proceedings of the Royal Society of Edinburgh: Section A Mathematics

      Volume: 150 Issue: 2 Pages: 789-811

    • DOI

      10.1017/prm.2018.97

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Existence of peaking solutions for semilinear heat equations with blow-up profile above the singular steady state2019

    • Author(s)
      Naito Yuki、Senba Takasi
    • Journal Title

      Nonlinear Analysis

      Volume: 181 Pages: 265-293

    • DOI

      10.1016/j.na.2018.12.001

    • Related Report
      2019 Research-status Report 2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Non-uniqueness for a critical heat equation in two dimensions with singular data2019

    • Author(s)
      Ioku Norisuke、Ruf Bernhard、Terraneo Elide
    • Journal Title

      Annales de l'Institut Henri Poincare C, Analyse non lineaire

      Volume: 36 Issue: 7 Pages: 2027-2051

    • DOI

      10.1016/j.anihpc.2019.07.004

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Attainability of the best Sobolev constant in a ball2019

    • Author(s)
      N. Ioku
    • Journal Title

      Mathematische Annalen

      Volume: in press Issue: 1-2 Pages: 1-16

    • DOI

      10.1007/s00208-018-1776-7

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Critical dissipative estimate for a heat semigroup with a quadratic singular potential and critical exponent for nonlinear heat equations2019

    • Author(s)
      Norisuke Ioku and Takayoshi Ogawa
    • Journal Title

      J. Differential Equations

      Volume: 266 Issue: 4 Pages: 2274-2293

    • DOI

      10.1016/j.jde.2018.08.030

    • Related Report
      2019 Research-status Report 2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Rectifiable and nonrectifiable solution curves of half-linear differential systems2018

    • Author(s)
      Yuki Naito, Mervan Pasic, Satoshi Tanaka
    • Journal Title

      Math. Slovaca

      Volume: 68 Issue: 3 Pages: 575-590

    • DOI

      10.1515/ms-2017-0126

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Separation structure of radial solutions for semilinear elliptic equations with exponential nonlinearity2018

    • Author(s)
      Soohyun Bae and Yuki Naito
    • Journal Title

      Discrete Contin. Dyn. Syst.

      Volume: 38 Issue: 9 Pages: 4537-4554

    • DOI

      10.3934/dcds.2018198

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Singular extremal soluitons for supercritical elliptic equations in a ball2018

    • Author(s)
      Yasuhito Miyamoto and Yuki Naito
    • Journal Title

      J. Differential Equations

      Volume: 265 Issue: 7 Pages: 2842-2885

    • DOI

      10.1016/j.jde.2018.04.055

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Existence and nonexistence of solutions for the heat equation with a superlinear source term2018

    • Author(s)
      Yohei Fujishima and Norisuke Ioku
    • Journal Title

      J. Math. Pures Appl. (9)

      Volume: 118 Pages: 128-158

    • DOI

      10.1016/j.matpur.2018.08.001

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Asymptotically self-similar behaviour of global solutions for semilinear heat equations with algebraically decaying initial data2018

    • Author(s)
      Yuki Naito
    • Journal Title

      Proc. Roy. Soc. Edinburgh Sect. A

      Volume: 印刷中

    • Related Report
      2017 Research-status Report
  • [Journal Article] A priori bounds for superlinear elliptic equations with semidefinite nonlinearity2017

    • Author(s)
      Yuki Naito, Takashi Suzuki, Yohei Toyota
    • Journal Title

      Nonlinear Anal.

      Volume: 151 Pages: 18-40

    • DOI

      10.1016/j.na.2016.11.016

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Canceling effects in higher-order Hardy-Sobolev inequalities2017

    • Author(s)
      N. Ioku and A. Cianchi
    • Journal Title

      Calculus of Variations and Partial Differential Equations

      Volume: 56 Issue: 2

    • DOI

      10.1007/s00526-017-1112-1

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / 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] Blow-up criteria for the classical Keller-Segel model of chemotaxis in higher dimensions2021

    • Author(s)
      内藤 雄基
    • Organizer
      九州関数方程式セミナー
    • Related Report
      2020 Annual Research Report
    • Invited
  • [Presentation] Existence and uniqueness of singular solutions for supercritical semilinear elliptic equations2021

    • Author(s)
      内藤雄基
    • Organizer
      日本数学会 函数方程式分科会一般講演
    • Related Report
      2020 Annual Research Report
  • [Presentation] Incomplete blow-up of solutions for semilinear heat equations with supercritical nonlinearity2019

    • Author(s)
      内藤 雄基
    • Organizer
      信州大学偏微分方程式研究集会
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] Threshold solutions for semilinear heat equations with polynomial decay initial data2019

    • Author(s)
      Yuki Naito
    • Organizer
      Equadiff 2019
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] 特異定常解より大きい爆発形状を持つ不完全爆発解の存在について2019

    • Author(s)
      内藤 雄基、仙葉 隆
    • Organizer
      日本数学会 函数方程式分科会一般講演
    • Related Report
      2019 Research-status Report
  • [Presentation] Threshold solutions for semilinear heat equations with slowly decaying initial data2019

    • Author(s)
      Yuki Naito
    • Organizer
      2019 International Workshop on Nonlinear PDEs and Its Application
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Blow-up criteria for the simplest Keller-Segel model of chemotaxis in higher dimensions2019

    • Author(s)
      Yuki Naito
    • Organizer
      Chemotaxis and Nonlinear Parabolic Equations
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Blow-up criteria for the parabolic-elliptic Keller-Segel system in higher dimensions2019

    • Author(s)
      内藤 雄基
    • Organizer
      数学と現象:Mathematics and Phenomena in Miyazaki 2019
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] Existence and uniqueness of singular solutions for supercritical semilinear elliptic equations2019

    • Author(s)
      内藤 雄基
    • Organizer
      RIMS 共同研究(グループ型) 「常微分方程式の手法による非線形問題の探究」
    • Related Report
      2018 Research-status Report
  • [Presentation] Asymptotic behavior of global solutions for semilinear heat equations with slowly decaying initial data2018

    • Author(s)
      内藤 雄基
    • Organizer
      RIMS共同研究(公開型) 「保存則をもつ偏微分方程式の解の正則性,特異性および漸近挙動の研究」
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] 非線形熱方程式の不完全爆発解の存在について2018

    • Author(s)
      内藤 雄基
    • Organizer
      九州関数方程式セミナー
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] An ODE approach to backward self-similar solutions for semilinear heat equations2018

    • Author(s)
      Yuki Naito
    • Organizer
      NTHU Department of Mathematics Visiting Scholar Colloquium
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Positive singular solutions for semilinear elliptic equationswith supercritical nonlinearity2018

    • Author(s)
      Yuki Naito
    • Organizer
      The 12Th AIMS conference, SS101 "Structure of solutions for nonlinear elliptic equations"
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research
  • [Presentation] Incomplete blow-up solutions for semilinear heat equations with supercritical nonlinearity2018

    • Author(s)
      Yuki Naito
    • Organizer
      The 12Th AIMS conference, SS102 "Asymptotics for Nonlinear Diffusion Equations and Related Topics"
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Singular sextremal solutions for supercritical elliptic equations in a ball2018

    • Author(s)
      宮本安人、内藤雄基
    • Organizer
      日本数学会秋期総合分科会 関数方程式分科会
    • Related Report
      2018 Research-status Report
  • [Presentation] Incomplete blow-up of solutions for semilinear heat equations with supercritical nonlinearity2018

    • Author(s)
      内藤雄基
    • Organizer
      名古屋微分方程式セミナー
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Singular extremal solutions for supercritical elliptic equations in a ball2018

    • Author(s)
      内藤雄基
    • Organizer
      HMAセミナー・冬の研究会 2018
    • Related Report
      2017 Research-status Report
  • [Presentation] A shooting approach to backward self-similar solutions for semilinear heat equations2018

    • Author(s)
      内藤雄基
    • Organizer
      RIMS共同研究「非線形問題への常微分方程式の手法によるアプローチ」
    • Related Report
      2017 Research-status Report
  • [Presentation] Asymptotic behavior of global solutions for semilinear heat equations with slowly decaying initial data2018

    • Author(s)
      内藤雄基
    • Organizer
      日本数学会2018年度年会 関数方程式分科会
    • Related Report
      2017 Research-status Report
  • [Presentation] Singular sextremal solutions for supercritical elliptic equations in a ball2017

    • Author(s)
      Yuki Naito
    • Organizer
      Equadiff2017
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research
  • [Presentation] Asymptotic behavior of global solutions for semilinear heat equations with slowly decaying initial data2017

    • Author(s)
      内藤 雄基
    • Organizer
      愛媛大学における微分方程式セミナー
    • Related Report
      2017 Research-status Report
  • [Presentation] Asymptotically self-similar behavior of global solutions to semilinear heat equations2017

    • Author(s)
      Yuki Naito
    • Organizer
      Qualitative Theory on Nonlinear Partial Differential Equations
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Asymptotically self-similar behavior of global solutions to semilinear heat equations2017

    • Author(s)
      Yuki Naito
    • Organizer
      Nonlinear Analysis, PDEs, and Applications
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited

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Published: 2017-04-28   Modified: 2022-02-22  

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