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Studies of solvability for hyperbolic Volterra equations

Research Project

Project/Area Number 17540143
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionIbaraki University

Principal Investigator

OKA Hirokazu  Ibaraki University, College of Engineering, Associate Professor (90257254)

Co-Investigator(Kenkyū-buntansha) TANAKA Naoki  Shizuoka University, Faculty of Science, Professor (00207119)
NAKAMOTO Ritsuo  Ibaraki University, College of Engineering, Professor (80007799)
NISHIO Katsuyoshi  Ibaraki University, College of Engineering, Associate Professor (40001698)
HIRASAWA Go  Ibaraki University, College of Engineering, Associate Professor (10434002)
SAKAKIBARA Nobuhisa  Shibaura Institute of Technology, Faculty of Engineering, Associate Professor (30235139)
Project Period (FY) 2005 – 2007
Project Status Completed (Fiscal Year 2007)
Budget Amount *help
¥3,730,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥330,000)
Fiscal Year 2007: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2006: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2005: ¥1,200,000 (Direct Cost: ¥1,200,000)
Keywordsfinite difference approximations / stability condition / semilinear evolution equations / semierouos of locally Lipschitz operators / Cauchy problems / mild solutions / dissiiativity condition / nonlinear hyperbolic systems / 準線形双曲型方程式 / 半閉作用素 / Hyers-Ulam-Rassias安定性 / Grand Furuta不等式 / Bebiano-Lemos-Providencia不等式 / 作用素幾何平均 / 準線形発展方程式 / 退化準線形双曲型方程式 / 作用素不等式 / conelike半群 / Stieltjes完全性 / 非稠密な定義域をもつ作用素 / リプシッツ作用素半群 / Chernoffの積公式 / 古田不等式 / 作用素平均
Research Abstract

1. When we consider the initial-boundary problem for a partial differential equation in the space of continuous functions, the operator defined naturally from the equation has possibly non dense domain because of its boundary condition. This fact motivates us to study the initial-value problem for the equation of evolution governed by a family of closed linear operators whose common domain is not necessarily dense in the underlying Banach space. It is shown that an evolution operator is generated by such a family of operators, under the stability condition introduced from the viewpoint of finite difference approximations.
2. We discuss the global solvability fir a class of semilinear evolution equations which is the abstract version of the quasilinear wave equation with strong damping. The advantage of our formulation lies in the fact that it is possible to obtain a global solution by checking some energy inequalities concerning only low order derivatives.
3. We introduce the notion of semigroups of locally Lipschitz operators which provide us with mild solutions to the Cauchy problem for semilinear evolution equations, and characterize such semigroups of locally Lipschitz operators. This notion of the semigroups is derived from the well-posedness concept of the initial-boundary value problem for differential equations whose solution operators are not quasi-contractive even in a local sense but locally Lipschitz continuous with their initial data.
4. A new dissipativity condition is proposed in terms of a family of metric-like functionals, and a necessary and sufficient condition is given of the existence of semigroups of locally Lipschitz operators which provide us with mild solution of Cauchy problem for nonlinear evolution equations. The advantage of using a family of metric-like functionals instead of the metric induced by the original norm lies in the fact that the obtained result may be applied to some nonlinear hyperbolic systems.

Report

(4 results)
  • 2007 Annual Research Report   Final Research Report Summary
  • 2006 Annual Research Report
  • 2005 Annual Research Report
  • Research Products

    (49 results)

All 2008 2007 2006 2005 Other

All Journal Article (37 results) (of which Peer Reviewed: 6 results) Presentation (12 results)

  • [Journal Article] An application of semigroups of locally Lipschiz operators to Carrier equations with acoustic boundary conditions2008

    • Author(s)
      Yoshikazu, Kobayashi
    • Journal Title

      J. Math. Anal. Appl. 338

      Pages: 852-852

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] An application of semigroups of locally Lipschitz operators to Carrier equations with acoustic boundary conditions2008

    • Author(s)
      Yoshikazu Kobayashi
    • Journal Title

      J.Math.Anal.Appl. 338

      Pages: 852-872

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] A topology for semiclosed operators in a Hilbert space2007

    • Author(s)
      Go, Hirasawa
    • Journal Title

      Acta Sci. Math. (Szeged) 73

      Pages: 271-271

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Grand Furuta inequality and its variant2007

    • Author(s)
      Masatoshi, Fujii
    • Journal Title

      J. Math. Inequal. 1

      Pages: 437-437

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Superstability of multipliers and ring derivations on Banach algebras2007

    • Author(s)
      Takeshi, Miura
    • Journal Title

      Banach J. Math. Anal. 1

      Pages: 125-125

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Semigroups of locally Lipschitz operators associated with lution equations2007

    • Author(s)
      Yoshikazu, Kobayashi
    • Journal Title

      J. Math. Anal. Appl. 330

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] A topology for semiclosed operators in a Hilbert space2007

    • Author(s)
      Go Hirasawa
    • Journal Title

      Acta Sci. Math.(Szeged) 73

      Pages: 271-282

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Grand Furuta inequality and its variant2007

    • Author(s)
      Masatoshi Fujii
    • Journal Title

      J. Math. Inequal. 1

      Pages: 437-441

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Superstability of multipliers and ring derivations on Banach algebras2007

    • Author(s)
      Takeshi Miura
    • Journal Title

      Banach J. Math. Anal. 1

      Pages: 125-130

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Semigroups of locally Lipschitz operators associated with semilinear evolution equations2007

    • Author(s)
      Yoshikazu Kobayashi
    • Journal Title

      J. Math. Anal. Appl. 330

      Pages: 1042-1067

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] On the continuity of positive definite functions on conelike semigroups2006

    • Author(s)
      T.M. Bisgaard
    • Journal Title

      京都大学数理解析研究所講究録 1478

      Pages: 127-127

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Norm inequalities for the geometric mean and its reverse.2006

    • Author(s)
      Ritsuo Nakamoto
    • Journal Title

      Sci. Math. Japonicae Online e-2006

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] On the continuity of positive definite functions on conelike semigroups2006

    • Author(s)
      T. M. Bisgaard
    • Journal Title

      Surikaisekikenkyujyo Kokyuroku No.1478

      Pages: 127-132

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Norm inequalities for the geometric mean and its reverse2006

    • Author(s)
      Ritsuo Nakamoto
    • Journal Title

      Sci. Math. Japonicae Online

      Pages: 1209-1214

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Norm inequalities for the geometric mean and its reverse2006

    • Author(s)
      Ritsuo Nakamoto
    • Journal Title

      Sci. Math. Japonicae(Online) e2006

      Pages: 1209-1214

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Concavity of the auxiliary function appearing in quantum reliability function2006

    • Author(s)
      Jun Ichi Fujii
    • Journal Title

      IEEE. Trans. Inform. Theory 52

      Pages: 3310-3313

    • Related Report
      2006 Annual Research Report
  • [Journal Article] On the continuity of positive definite function on conelike semi groups.2006

    • Author(s)
      Torben Maack Bisggard
    • Journal Title

      International Congress of Mathematicians Madrid 2006 1

      Pages: 69-69

    • Related Report
      2006 Annual Research Report
  • [Journal Article] On the continuity of positive definite functions on conelike semigroups.2006

    • Author(s)
      Torben Maack Bisgaard
    • Journal Title

      Proceedings of 2006 Hawaii International Conference on Statistics, Mathematics and Related Fields

      Pages: 209-209

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Stieltjes perfect semigroups are perfect2005

    • Author(s)
      T.M. Bisgaard
    • Journal Title

      Czech. Math. J. 55(130)

      Pages: 729-729

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Convergence and approximation of semigroups of Lipschitz operators2005

    • Author(s)
      Yoshikazu, Kobayashi
    • Journal Title

      Nonliear Anal. 61

      Pages: 781-781

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Evolution operators generated by non-densely defined operators2005

    • Author(s)
      Hirokazu, Oka
    • Journal Title

      Math. Nachr. 278

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Stieltjes perfect semigroups are perfect2005

    • Author(s)
      T. M. Bisgaard
    • Journal Title

      Czech. Math. J. 55(130)

      Pages: 729-753

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Convergence and approximation of semigroups of Lipschitz operators2005

    • Author(s)
      Yoshikazu Kobayashi
    • Journal Title

      Nonliear Anal. 61

      Pages: 781-821

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Evolution operators generated by non-densely defined operators2005

    • Author(s)
      Hirokazu Oka
    • Journal Title

      Math. Nachr. 278

      Pages: 1285-1296

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Evolution operators generated by non-densely defined operators.2005

    • Author(s)
      Hirokazu Oka
    • Journal Title

      Math.Nachr. 278・11

      Pages: 1285-1296

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Locally Lipschitz continuous integrated semigroups.2005

    • Author(s)
      Naoki Tanaka
    • Journal Title

      Studia Math. 167・1

      Pages: 1-16

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Covergence and approximation of semigroups of Lipschitz operators.2005

    • Author(s)
      Yoshikazu Kobayashi
    • Journal Title

      Nonlinear Anal. 61・5

      Pages: 781-821

    • Related Report
      2005 Annual Research Report
  • [Journal Article] An analysis on the internal structure of the celebrated Furuta inequality via operator mean.2005

    • Author(s)
      Masatoshi Fujii
    • Journal Title

      Sci. Math. Japonicae Online

      Pages: 425-431

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Upper bounds for the difference between symmetric operator means.2005

    • Author(s)
      Masatoshi Fujii
    • Journal Title

      Sci. Math. Japonicae Online

      Pages: 1-9

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Stieljes perfect semigroups are perfect.2005

    • Author(s)
      Torben Maack Bisgaard
    • Journal Title

      Czechoslovak Mathematical Journal 55・3

      Pages: 729-753

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Global solvability of abstract semilinear evolution equations

    • Author(s)
      Hirokazu, Oka
    • Journal Title

      Math. Nachr. (In press)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Global solvability of abstract semilinear evolution equations

    • Author(s)
      Hirokazu Oka
    • Journal Title

      Math.Nachr. (in press)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Semigroups of locally Lipschitz operators associated with semilinear evolution equations.

    • Author(s)
      Yoshikazu Kobayashi
    • Journal Title

      J. Math. Anal. Appl. in press

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Abstract Cauchy problems for quasi-linear evolution equations with non-densely defined operators.

    • Author(s)
      Toshitaka Matsumoto
    • Journal Title

      Taiwanese J. Math. in press

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Approximation of abstract quasilinear evolution equations in the sense of Hadamard

    • Author(s)
      Naoki Tanaka
    • Journal Title

      Taiwanese J. Math. in press

    • Related Report
      2006 Annual Research Report
  • [Journal Article] On the continuity of positive definite functions on conelike semigroups.

    • Author(s)
      Torben Maack Bisgaard
    • Journal Title

      京都大学数理解析研究所講究録 (in press)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Abstract Cauchy problems for quasi-linear evolution equations with non-densely defined operators.

    • Author(s)
      Toshitaka Matsumoto
    • Journal Title

      Taiwanese J. Math. (in press)

    • Related Report
      2005 Annual Research Report
  • [Presentation] Local semigroup of locally Lipschitz operatorsの生成定理2008

    • Author(s)
      田中直樹
    • Organizer
      日本数学会年会
    • Place of Presentation
      近畿大学
    • Year and Date
      2008-03-25
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Bebiano-Lemos-Providencia不等式の逆とFuruta不等式について2008

    • Author(s)
      松本明美
    • Organizer
      日本数学会年会
    • Place of Presentation
      近畿大学
    • Year and Date
      2008-03-25
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Naoki Tanaka, A generation theorem of local semigroups of locally Lipschitz operators2008

    • Author(s)
      Naoki Tanaka
    • Organizer
      Spring Mathematical Society of Japan meeting
    • Place of Presentation
      Kinki University
    • Year and Date
      2008-03-25
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] A reverse of Bebiano-Lemos-Providencia inequality and Furuta inequality2008

    • Author(s)
      Akemi Matsumoto
    • Organizer
      Spring Mathematical Society of Japan meeting
    • Place of Presentation
      Kinki University
    • Year and Date
      2008-03-25
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] 有理数半群のある部分半群における切断型モー'メント問題2007

    • Author(s)
      榊原暢久
    • Organizer
      実解析学シンポジウム
    • Place of Presentation
      大阪教育大学
    • Year and Date
      2007-10-21
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Truncated moment problems on some subsemigroups of the semigroup of rational numbers2007

    • Author(s)
      Nobuhisa Sakakibara
    • Organizer
      Real Analysis Symposium
    • Place of Presentation
      Osaka Kyoiku University
    • Year and Date
      2007-10-21
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] 有理数半群のある部分集合上の切断型モーメント問題2007

    • Author(s)
      榊原暢久
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      東北大学
    • Year and Date
      2007-09-22
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Truncated moment problems on a subset of the semigroup of rational numbers2007

    • Author(s)
      Nobuhisa Sakakibara
    • Organizer
      Autumn Mathematical Society of Japan meeting
    • Place of Presentation
      Tohoku University
    • Year and Date
      2007-09-22
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] 局所リプシッツ作用素半群の生成定理2006

    • Author(s)
      田中直樹
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      大阪市立大学
    • Year and Date
      2006-09-22
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] A generation theorem of semigroups of locally Lipschitz operators2006

    • Author(s)
      Naoki Tanaka
    • Organizer
      Autumn Mathematical Society of Japan meeting
    • Place of Presentation
      Osaka City University
    • Year and Date
      2006-09-22
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On the continuity of positive definite functions on conelike semigroups2005

    • Author(s)
      Nobuhisa Sakakibara
    • Organizer
      Research Institute of Mathematical Sciences Workshop
    • Place of Presentation
      Kyoto University
    • Year and Date
      2005-11-02
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Semigroups of locally Lipschitz operators associated with semilinear evolution equations2005

    • Author(s)
      Toshitaka Matsumoto
    • Organizer
      Autumn Mathematical Society of Japan meeting
    • Place of Presentation
      Okayama University
    • Year and Date
      2005-09-22
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary

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Published: 2005-04-01   Modified: 2016-04-21  

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