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Geometric properties and asymptotic behavior of solutions of diffusion equations

Research Project

Project/Area Number 23340035
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionTohoku University

Principal Investigator

Ishige Kazuhiro  東北大学, 理学(系)研究科(研究院), 教授 (90272020)

Co-Investigator(Kenkyū-buntansha) KOZONO Hideo  早稲田大学, 理工学術院, 教授 (00195728)
OGAWA Takayoshi  東北大学, 大学院理学研究科, 教授 (20224107)
KOIKE Shigeaki  東北大学, 大学院理学研究科, 教授 (90205295)
YAMADA Sumio  学習院大学, 理学部, 教授 (90396416)
Co-Investigator(Renkei-kenkyūsha) YANAGIDA Eiji  東京工業大学, 理工学研究科, 教授 (80174548)
KABEYA Yoshitsugu  大阪府立大学, 大学院工学研究科, 教授 (70252757)
Project Period (FY) 2011-04-01 – 2016-03-31
Project Status Completed (Fiscal Year 2015)
Budget Amount *help
¥17,680,000 (Direct Cost: ¥13,600,000、Indirect Cost: ¥4,080,000)
Fiscal Year 2014: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2013: ¥5,200,000 (Direct Cost: ¥4,000,000、Indirect Cost: ¥1,200,000)
Fiscal Year 2012: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2011: ¥5,460,000 (Direct Cost: ¥4,200,000、Indirect Cost: ¥1,260,000)
Keywords放物型方程式 / 形状解析 / 漸近解析 / 爆発集合 / 放物型凸 / 最大点挙動 / 動的境界条件 / 放物型凸性 / 非線形放物型方程式系 / 非線形楕円型方程式 / O.D.E.型解 / 最適減衰評価 / 爆発問題 / 熱方程式 / 解の形状 / 最大点集合 / 高次漸近展開 / 等高面 / 凸性 / 初期値問題
Outline of Final Research Achievements

We developed the method for studying geometric properties and asymptotic behavior of solutions of parabolic equations, and obtained the asymptotic behavior of hot spots and the optimal decay rates of the Lebesgue norms for the heat equation with a potential. Furthermore, we established a method of obtaining the higher order asymptotic expansions of the solutions behaving like the heat kernel. In addition, we study the location of the blow-up set for a semilinear heat equation by the profile of the solution just before the blow-up time. In particular, we gave a sufficient condition for no boundary blow-up.

Report

(5 results)
  • 2015 Final Research Report ( PDF )
  • 2014 Annual Research Report
  • 2013 Annual Research Report
  • 2012 Annual Research Report
  • 2011 Annual Research Report
  • Research Products

    (27 results)

All 2015 2014 2013 2012 2011

All Journal Article (23 results) (of which Peer Reviewed: 23 results,  Acknowledgement Compliant: 3 results,  Open Access: 1 results) Presentation (3 results) (of which Int'l Joint Research: 2 results,  Invited: 3 results) Funded Workshop (1 results)

  • [Journal Article] Parabolic power concavity and parabolic boundary value problems2014

    • Author(s)
      Kazuhiro Ishige and Paolo Salani
    • Journal Title

      Mathematische Annalen

      Volume: 358 Issue: 3-4 Pages: 1091-1117

    • DOI

      10.1007/s00208-013-0991-5

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Global solutions for a nonlinear integral equation with a generalized heat kernel2014

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

      Discrete and Continuous Dynamical Systems - Series S

      Volume: 7 Issue: 4 Pages: 767-783

    • DOI

      10.3934/dcdss.2014.7.767

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Blow-up set for type I blowing up solutions for a semilinear heat equation2014

    • Author(s)
      Y. Fujishima and K. Ishige
    • Journal Title

      Ann. Inst. H. Poincare Anal. Non Lineaire

      Volume: 31 Issue: 2 Pages: 231-247

    • DOI

      10.1016/j.anihpc.2013.03.001

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A note on parabolic power concavity2014

    • Author(s)
      Kazuhiro Ishige and Paolo Salani
    • Journal Title

      Kodai Mathematical Journal

      Volume: 37 Issue: 3 Pages: 668-679

    • DOI

      10.2996/kmj/1414674615

    • NAID

      130004705918

    • ISSN
      0386-5991, 1881-5472
    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Asymptotics for a nonlinear integral equation with a generalized heat kernel2014

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

      Journal of Evolution Equations

      Volume: 14 Issue: 4-5 Pages: 749-777

    • DOI

      10.1007/s00028-014-0237-3

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Large time behavior of solutions of a semilinear elliptic equation with a dynamical boundary condition2013

    • Author(s)
      Marek Fila, Kazuhiro Ishige, and Tatsuki Kawakami
    • Journal Title

      Adv. Differential Equations

      Volume: 18 Pages: 69-100

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Convection-diffusion equation with absorption and non-decaying initial data2013

    • Author(s)
      Kazuhiro Ishige and Kanako Kobayashi
    • Journal Title

      J. Differential Equations

      Volume: 254 Issue: 3 Pages: 1247-1268

    • DOI

      10.1016/j.jde.2012.10.014

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Decay rate of L^q norms of critical Schroedinger heat semigroups2013

    • Author(s)
      Kazuhiro Ishige and Yoshitsugu Kabeya
    • Journal Title

      Springer INdAM Seriries

      Volume: 2 Pages: 165-178

    • DOI

      10.1007/978-88-470-2841-8_11

    • ISBN
      9788847028401, 9788847028418
    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Local quasi-concavity of the solutions of the heat equation with a nonnegative potential2013

    • Author(s)
      D. Andreucci and Kazuhiro Ishige
    • Journal Title

      Ann. Mat. Pura Appl.

      Volume: 192 Issue: 3 Pages: 329-348

    • DOI

      10.1007/s10231-011-0226-x

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Sharp decay estimates of L^q norms of nonnegative Schroedinger heat semigroups2013

    • Author(s)
      Norisuke Ioku, Kazuhiro Ishige and Eiji Yanagida
    • Journal Title

      J. Funct. Anal.

      Volume: 264 Issue: 12 Pages: 2764-2783

    • DOI

      10.1016/j.jfa.2013.03.009

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Asymptotic expansions of solutions of the Cauchy problem for nonlinear parabolic equations2013

    • Author(s)
      Kazuhiro Ishige and Tatsuki Kawakami
    • Journal Title

      J. Anal. Math.

      Volume: 121 Issue: 1 Pages: 317-351

    • DOI

      10.1007/s11854-013-0038-6

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Large time behavior of the solution for a two dimensional semilinear elliptic equation with a dynamical boundary condition2013

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

      Asymptot. Anal.

      Volume: 85 Issue: 1-2 Pages: 107-123

    • DOI

      10.3233/asy-131183

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Global solutions for a semilinear heat equation in the exterior domain of a compact set2012

    • Author(s)
      K. Ishige and M. Ishiwata
    • Journal Title

      Discrete Contin. Dyn. Syst.

      Volume: 32 Issue: 3 Pages: 847-866

    • DOI

      10.3934/dcds.2012.32.847

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Blow-up for a semilinear parabolic equation with large diffusion on R^N. II2012

    • Author(s)
      Y. Fujishima and K. Ishige
    • Journal Title

      J. Differential Equations

      Volume: 252 Issue: 2 Pages: 1835-1861

    • DOI

      10.1016/j.jde.2011.08.040

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition2012

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

      Commun. Pure Appl. Anal.

      Volume: 11 Issue: 3 Pages: 1285-1301

    • DOI

      10.3934/cpaa.2012.11.1285

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] L^p norms of nonnegative Schroedinger heat semigroup and the large time behavior of hot spots2012

    • Author(s)
      K. Ishige and Y. Kabeya
    • Journal Title

      J. Funct. Anal.

      Volume: 262 Issue: 6 Pages: 2695-2733

    • DOI

      10.1016/j.jfa.2011.12.024

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Refined asymptotic profiles for a semilinear heat equation2012

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

      Math. Ann.

      Volume: 353 Issue: 1 Pages: 161-192

    • DOI

      10.1007/s00208-011-0677-9

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Heat equation with a singular potential on the boundary and the Kato inequality2012

    • Author(s)
      K. Ishige and M. Ishiwata
    • Journal Title

      J. Anal. Math.

      Volume: 118 Issue: 1 Pages: 161-176

    • DOI

      10.1007/s11854-012-0032-4

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Blow-up set for a semilinear heat equation and pointedness of the initial data2012

    • Author(s)
      Y. Fujishima and K. Ishige
    • Journal Title

      Indiana Univ. Math. J.

      Volume: 61 Issue: 2 Pages: 627-663

    • DOI

      10.1512/iumj.2012.61.4596

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Hot spots for the two dimensional heat equation with a rapidly decaying negative potentia2011

    • Author(s)
      K. Ishige and Y. Kabeya
    • Journal Title

      Discrete Contin. Dyn. Syst. Ser. S

      Volume: 4

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On a new kind of convexity for solutions of parabolic problems2011

    • Author(s)
      K. Ishige and P. Salani
    • Journal Title

      Discrete Contin. Dyn. Syst. Ser. S

      Volume: 4

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Blow-up for a semilinear parabolic equation with large diffusion on R^N2011

    • Author(s)
      Y. Fujishima and K. Ishige
    • Journal Title

      J. Differential Equations

      Volume: 250

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Initial trace for a doubly nonlinear parabolic equation2011

    • Author(s)
      K. Ishige and J. Kinnunen
    • Journal Title

      J. Evol. Equ.

      Volume: 11

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Presentation] 放物型方程式の解の凸性について2015

    • Author(s)
      石毛和弘
    • Organizer
      日本数学会函数方程式分科会
    • Place of Presentation
      明治大学
    • Year and Date
      2015-03-23
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Heat equation with a nonlinear boundary condition and uniformly local $L^r$ spaces2014

    • Author(s)
      石毛和弘
    • Organizer
      6th Euro-Japanese Workshop on Blow-up
    • Place of Presentation
      東京工業大学
    • Year and Date
      2014-09-04
    • Related Report
      2014 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Heat equation with a nonlinear boundary condition and uniformly local L^r spaces2014

    • Author(s)
      石毛和弘
    • Organizer
      第39回偏微分方程式論札幌シンポジウム
    • Place of Presentation
      北海道大学
    • Year and Date
      2014-08-27
    • Related Report
      2014 Annual Research Report
    • Int'l Joint Research / Invited
  • [Funded Workshop] 6th Euro-Japanese Workshop on Blow-up2014

    • Place of Presentation
      東京工業大学
    • Year and Date
      2014-09-01
    • Related Report
      2014 Annual Research Report

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Published: 2011-04-06   Modified: 2019-07-29  

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