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Problems on Partial Differential Equations

Research Project

Project/Area Number 61460003
Research Category

Grant-in-Aid for General Scientific Research (B)

Allocation TypeSingle-year Grants
Research Field 解析学
Research InstitutionOsaka University

Principal Investigator

TANABE Hiraki  Professor,Faculty of Science,Osaka University, 理学部, 教授 (70028083)

Co-Investigator(Kenkyū-buntansha) NAKAO Shintaro  Associate Professor,Faculty of Science,Osaka University, 理学部, 助教授 (90030783)
KOMATSU Gen  Associate Professor,Faculty of Science,Osaka University, 理学部, 助教授 (60108446)
IKAWA Mitsuru  Professor,Faculty of Science,Osaka University, 理学部, 教授 (80028191)
WATANABE Takeshi  Professor,Faculty of Science,Osaka University, 理学部, 教授 (50028081)
IKEDA Nobuyuki  Professor,Faculty of Science,Osaka University, 理学部, 教授 (00028078)
Project Period (FY) 1986 – 1987
Project Status Completed (Fiscal Year 1987)
Budget Amount *help
¥4,400,000 (Direct Cost: ¥4,400,000)
Fiscal Year 1987: ¥1,800,000 (Direct Cost: ¥1,800,000)
Fiscal Year 1986: ¥2,600,000 (Direct Cost: ¥2,600,000)
Keywordstime delay / parabolic equation / evolution operator / controllability / observability semigroup / decay of wave motions / poles of scattering matrix / 散乱行列の極 / ハミルトン作用素 / ヴォルテラ型積分微分方程式 / 初期境界値問題 / 比重付楕円型評価 / 漸近行動 / 漸近展開 / 解の減衰
Research Abstract

The evolution operator for parabolic evolution equations with time delay was successfully constructed.As for the solvability of that kind of equations various results are know under mild smsoothness hypothesis on the coefficient of the delay term.If the coefficient is Holder continuous,it was found that the evolution operator can be constructed.A remarkable difference from the case without time delay is that a singularity of the time derivative of the evolution operator appear at each integral multiple of the delay interval.With the aid of this evolution operator a considerable part of the theoty of control for equations with only bounded operators in delay terms which has been developped by S.Nakagiri of Kobe University can be extended to the case where operators in delay terms are unbouned.For that purpose the basic space is enlarged so that the solution is expressed by a semigroup S(t).However,the adjoint operatorS*(t)appears and the results of Nakagiri cannot be extended directly.H … More ence,assuming that the basic space is a Hilbert space and the main operator is associated with a srongly elliptic sesquilinear form and considering the equation also in the space of negative norm as occasion demands, it has been found that a fairly large part of Nakagiri's resutls can be extended. Namely,the first structural operator F which connects the semigroup S_T(t)associated with the equation adjoint to the original one and S(t) is defined and through the second structural operator defined by means of the above mentioned evolution operator the relationsFS(t)=S*_t(t)F,S*(t)F*=F*S_t(t)were established.In this manner the equivalence of the controllability of the original equation and the observability /f the adjoint equation was proved.However,in order to develop this type of theoty there are problems on whether the spectra of the infinitesimal generator of the semigroup S(t)are discrete or not and whether the generalized eigenfunctions are complete or not.In some special case there is an affirmative answer to this problem; however,in the general case it will be a future subject.
In addition some new results were established by other investigators on the decay of wave motions and the poles of scattering matrix,some approxiamation theory for stochastic differential equations,and essential self-adjointness of quantum Hamiltonians. Less

Report

(2 results)
  • 1987 Final Research Report Summary
  • 1986 Annual Research Report
  • Research Products

    (17 results)

All Other

All Publications (17 results)

  • [Publications] 田辺広域(Hiroki Tanabe): Annali della Scuols Normais Superiore-Classe di Scienze.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1987 Final Research Report Summary
  • [Publications] 田辺広域(Hiroki Tanabe): Journal of Differential equations.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1987 Final Research Report Summary
  • [Publications] 井川満(Mitsuru Ikawa): Annales de 1'Institut Fourier.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1987 Final Research Report Summary
  • [Publications] 井川満(Mitsuru Ikawa): Prospect of Algibraic Analysie.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1987 Final Research Report Summary
  • [Publications] 中尾慎太郎(Shintaro Nakao): Seminaire de Probabilitites XXII, Lecture Notes in Mathimatics(Springer).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1987 Final Research Report Summary
  • [Publications] 楳田登美男: Scientific Reports, College of General Education, Osaka Unibersity. 36. 1-6 (1987)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1987 Final Research Report Summary
  • [Publications] Hiroki Tanabe: "On the asymptotic behavior of linear parabolic equations in L^1 space (with D.G.Park)" Annali della Scuola Normale Superore-Classe di Scienze.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1987 Final Research Report Summary
  • [Publications] Hiroki Tanabe: "On fundemental solutions of linear parabolic equations of higher order in time and associated Volterra equations" Journal of Differential Equations.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1987 Final Research Report Summary
  • [Publications] Mitsuru Ikawa: "Decay of solutions of the wave equation in the exxterior of several convex conves bodies" Annales de 1'Institute Fourier.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1987 Final Research Report Summary
  • [Publications] Mitsuru Ikawa: "On the poles of the scattering matrix for several convex bodies" Prospect of Algebraic Analysis.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1987 Final Research Report Summary
  • [Publications] Shintaro Nakao: "A note on approximation for stochastic differential equations(withS. S.Kaneko)" Seminaire de Probabilitites XXII,Lecture Notes in Mathematics.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1987 Final Research Report Summary
  • [Publications] 田辺広城: Journal of the Faculty of Science,University of Tokyo. 34. (1987)

    • Related Report
      1986 Annual Research Report
  • [Publications] 井川満: Annales de l'Institut Fourier.

    • Related Report
      1986 Annual Research Report
  • [Publications] 池田信行: Lecture Notes in Control and Information Sciences. 78. 195-205 (1986)

    • Related Report
      1986 Annual Research Report
  • [Publications] 中尾愼太郎,H.Kaneko: Lecture Notes in Mathematics(Springer)S【e!´】minaire de Probabilit【e!´】s.

    • Related Report
      1986 Annual Research Report
  • [Publications] 八木厚志: Mathematische Annalen.

    • Related Report
      1986 Annual Research Report
  • [Publications] 八木厚志: Proceedings of the Centre of Mathematical Analysis Australian National University.

    • Related Report
      1986 Annual Research Report

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Published: 1987-03-31   Modified: 2016-04-21  

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