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2014 Fiscal Year Annual Research Report

双曲型性と放物型性との間に横たわる階層構造の解明

Research Project

Project/Area Number 24654036
Research InstitutionWaseda University

Principal Investigator

大谷 光春  早稲田大学, 理工学術院, 教授 (30119656)

Co-Investigator(Kenkyū-buntansha) 西原 健二  早稲田大学, 政治経済学術院, 教授 (60141876)
小澤 徹  早稲田大学, 理工学術院, 教授 (70204196)
Project Period (FY) 2012-04-01 – 2015-03-31
Keywords放物型方程式 / 双曲型方程式 / 発展方程式 / 強い減衰項
Outline of Annual Research Achievements

A を実ヒルベルト空間 H 上の正値自己共役作用素とし H 上の強い減衰項 c A*α u' をもつ抽象方程式 (E) u" + A u + c A*α u' = 0, u(0) = a, u'(0) = b (A*α は A の α次の分数冪作用素)の放物型性に関して次の結果を得た.
(1) 任意のα>0 に対して,次の意味での時間正則性が得られた. S(t): (a,b) →u(t) は D(A*1/2)xH から D(A*1/2) への (0,∞) 上で無限回微分可能な写像となる.
(2)任意のα>0 に対して,次の意味での空間正則性が得られた.k(t): (a,b) → u*k(t) (u(t) の k 回導関数)は,任意の自然数 n に対して D(A*1/2)xH から D(A*n) への 有界な写像となる.
(3)特に, A*1/2 = L, c >2 のとき,方程式 (E) は (u' + αLu)'+ βL(u' + αLu) = 0 と分解され(α+ β=c, αβ=1) 作用素 L の放物型性(L が他のバナッハ空間 X において C*0 半群を生成する性質や比較定理が成り立つなど)が (E) にも引き継がれることを示した.

  • Research Products

    (8 results)

All 2015 2014

All Journal Article (3 results) (of which Peer Reviewed: 3 results,  Acknowledgement Compliant: 1 results) Presentation (5 results) (of which Invited: 3 results)

  • [Journal Article] Existence results for quasilinear elliptic equations with multivalued nonlinear terms2014

    • Author(s)
      M. Otani and V. Staicu
    • Journal Title

      Set-Valued and Variational Analysis

      Volume: 22 Pages: 859-877

    • DOI

      10.1007/s11228-014-0289-0

    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Analytic smoothing effect for nonlinear Schrodinger equation in two space dimensions2014

    • Author(s)
      G. Hoshino and T. Ozawa
    • Journal Title

      Osaka J. Math.

      Volume: 51 Pages: 609-618

    • Peer Reviewed
  • [Journal Article] Critical exponent for the Cauchy problem to the weakly coupled damped wave system2014

    • Author(s)
      K. Nishihara and Y. Wakasugi
    • Journal Title

      Nonlinear Analysis

      Volume: 108 Pages: 249-259

    • DOI

      10.1016/j.na.2014.06.001

    • Peer Reviewed
  • [Presentation] 一般領域における複素ギンツブルグ-ランダウ方程式の可解性2015

    • Author(s)
      清水 翔司・大谷 光春
    • Organizer
      実函数論分科会/日本数学会
    • Place of Presentation
      明治大学
    • Year and Date
      2015-03-21 – 2015-03-21
  • [Presentation] Quadratic Interactions in Dispersive Systems2014

    • Author(s)
      Tohru Ozawa
    • Organizer
      微分方程式の総合的研究
    • Place of Presentation
      京都大学
    • Year and Date
      2014-12-20 – 2014-12-20
    • Invited
  • [Presentation] The solvability of complex Ginzburg-Landau equation focusing on parabolicity2014

    • Author(s)
      清水 翔司・大谷 光春
    • Organizer
      実函数論分科会/日本数学会
    • Place of Presentation
      広島大学
    • Year and Date
      2014-09-28 – 2014-09-28
  • [Presentation] Large time behavior of solutions for Double-diffusive convection systems based on Brinkman-Forchheimer equation2014

    • Author(s)
      S. Uchida and M. Otani
    • Organizer
      The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications
    • Place of Presentation
      Madrid, Spain
    • Year and Date
      2014-07-09 – 2014-07-09
    • Invited
  • [Presentation] On the Cauchy problem for weakly coupled system of damped wave equations2014

    • Author(s)
      K. Nishihara
    • Organizer
      The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications
    • Place of Presentation
      Madrid, Spain
    • Year and Date
      2014-07-09 – 2014-07-09
    • Invited

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Published: 2016-06-01  

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