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Research on the Behavior of Solutions Evolution Equations in Time-Almost Periodic Noncylindrical Domains

Research Project

Project/Area Number 12640220
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionTOKAI UNIVERSITY

Principal Investigator

YAMAGUCHI Masaru  Tokai University, Department of Mathematics,, Professor, 理学部, 教授 (10056252)

Co-Investigator(Kenkyū-buntansha) MATSUYAMA Tokio  Tokai University, Department of Mathematics, Professor, 理学部, 教授 (70249712)
TANAKA Minoru  Tokai University, Department of Mathematics, Professor, 理学部, 教授 (10112773)
AKAMATSU Toyohiro  Tokai University, Department of Mathematics Professor, 理学部, 教授 (00112772)
Project Period (FY) 2000 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2002: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2001: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2000: ¥1,400,000 (Direct Cost: ¥1,400,000)
Keywordstime-periodic noncylindrical domain / nonlinear wave equation / periodic solution / time-quasiperiodic noncylindrical domain / linear wave equation / almost periodic solution / Diophantine inequality / 波動方程式 / 準周期解 / 吊り下げられた弦 / Diophantine条件 / Siegel条件 / 弱Poincare条件 / 境界値問題 / 時間周期解 / ディオファントス不等式 / ベッセル関数の零点 / 一次元力学系 / 回転指数 / 球対称ラプラシアン / 弦の振動
Research Abstract

In this project we dealt with linear and nonlinear wave equations in noncylindrical domain periodic or quasiperiodic in time. We studied the qualitative behavior of solutions of the initial boundary value problems (IBVP) and the boundary value problems (BVP). Our results are as follows.
(1) We considered BVP for 1-D nonlinear wave equations in time-periodic noncylindrical domains. If the nonlinear forcing term, the boundary functions and the boundary data are periodic in time with same period, BVP have time-periodic solutions. This problem had been regarded as one of the difficult problems.
(2) We considered IBVP for 1-D linear wave equations in time-quasiperiodic noncylindrical domains. The nonhomogeneous terms of the equations and the boundary data are also time quasiperiodic. As we showed in the previous Research Project, the solutions are generally almost periodic in time, hence bounded in time. We studied this phenomena more deeply, and found that there exist solutions which are the … More superpositions of time-unbounded waves.
(3) We considered IBVP for 3-D radially symmetric linear wave equations in time-quasiperiodic noncylindrical domains whose space-domains are surrounded by two balls. We showed that the solutions are generally almost periodic in time.
(4) We considered BVP for 3-D radially symmetric nonlinear wave equations in time-periodic noncylindrical domains whose space-domains are balls. Under the similar assumptions to those of (1) BVP have time-periodic solutions. The results seem to be interesting.
In order to solve the problems, we developed some useful method. This method consists of a transformation of BVP for wave equations to some functional equations and domain transformations that transform the noncylindrical domains to cylindrical domains. The former was established by M. Yamaguchi and the latter by M. Yamaguchi and H. Yoshida. This method is based on the Reduction Theorems by M. Herman and J. Yoccoz in periodic case and by M. Yamaguchi in quasiperiodic case. Less

Report

(4 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • 2000 Annual Research Report
  • Research Products

    (41 results)

All Other

All Publications (41 results)

  • [Publications] M.Yamaguchi: "Periodic solutions of nonlinear 3D wave equations in sphere-symmetric domain with periodically oscillating boundaries"Proceedings of the ISSAC (World Scientific co.). (To appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Yamaguchi: "Periodic solutions of nonlinear equations of string with periodically Ocillating boundaries"Funkcialaj Ekvacioj. 45. 397-416 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Yamaguchi: "3D wave equations in sphere-symmetric domain with periodically oscillating boundaries"Discrete and Continuous Dynamical Systems. 7. 385-396 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Yamaguchi: "Nonhomogeneous string problem with periodically moving boundaries (with Hiroshi Yashida)"Fields Institute Communications. 25. 565-574 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Akamatsu: "Remarks on the rank of a Lie algebra and necessary conditions for hypoellipticity of a degenerate parabolic operator"Proc.School of Sci.Tokai Univ.. 38. 21-31 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Matsuyama: "Asymptotic behavior of solutions for wave equation with an effective dissipation around the boundary"J.Math.Anal.Appl.. 271. 467-492 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Matsuyama: "Asymptotics for the nonlinear dissipative wave equation"Trans.AMS. 355. 865-899 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Matsuyama: "L^2-behavior of solutions to the linear heat and wave equations in exterior domains (with R.Ikehata)"Scientiae Mathematicae Japonicae. 55. 33-42 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Matsuyama: "Remarks on the L^2 estimates of the density for the compressible Navier Stokes flow in R^3 (with R.Ikehata and T.Kobayashi)"Nonlinear Anal. 47. 2519-2526 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Tanaka: "Characterization of a differentiable points of the distance function to the cut locus"J.Math.Soc.Japan. 55. 231-241 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Tanaka: "Compactification and maximal diameter theorem for noncompact manifolds with radial curvature bounded below"Math.Z.. 241. 341-351 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Tanaka: "A Sard theorem for the distance functions (with J.Itoh)"Math.Ann.. 320. 1-10 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M. Yamaguchi: "Periodic solutions of nonlinear 3D wave equations in sphere-symmetric domain with periodically oscillating boundaries"Proceedings of the ISSAC (World Scientific co.). to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M. Yamaguchi: "Periodic solutions of nonlinear equations of string with periodically Oscillating boundaries"Funkcialaj Ekvacioj. Vol.45, No.3. 397-416 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M. Yamaguchi: "3D wave equations in spheresymmetric domain with periodically oscillating boundaries"Discrete and Continuous Dynamical Systems. Vol.7, No.2. 385-396 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M. Yamaguchi, (with Hiroshi Yoshida): "Nonhomogeneous string problem with periodically moving boundaries"Fields Institute Communications. 25. 565-574 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T. Akamatsu: "Remarks on the rank of a Lie algebra and necessary conditions for hypoellipticity of a degenerate parabolic operator"Proc. School of Sci. Tokai Univ.. 38. 21-31 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T. Matsuyama: "Asymptotic behavior of solutions for wave equation with an effective dissipation around the boundary"J. Math. Anal. Appl.. 271. 467-492 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T. Matsuyama: "Asymptotics for nonlinear dissipative wave equation"Trans. AMS. 355. 865-899 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T. Matsuyama, (with R. Ikehata): "L^2-behavior of solutions to the linear heat and wave equations in exterior domains"Scientiae Mathematicae Japonicae. Vol.55, No.1. 33-42 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T. Matsuyama, (with R. Ikehata and T. Kobayashi): "Remarks on the L^2 estimates of the density for the compressible Navier Stokes flow R^3"Nonlinear Anal.. 47. 2519-2526 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M. Tanaka: "Characterization of a differentiable points of the distance function to the cut locus"J. Math. Soc. Japan. Vol.55, No.1. 231-241 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M. Tanaka: "Compactification and maximal diameter theorem for noncompact manifolds with radial curvature bounded below"Math. Z.. 241. 341-351 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M. Tanaka, (with J. Itoh): "A Sard theorem for the distance functions"Math. Ann.. 320. 1-10 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Yamaguchi: "Periodic solutions of nonlinear equations of string with periodically oscillating boundaries"Funkcialaj Ekvacioj (日本数学会関数方程式分科会誌). Vol.45 No.3. 397-416 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Yamaguchi: "3D wave equations in sphere symmetric domain with periodically oscillating boundaries"Discrete and Continuous Dynamical Systems. Vol.7 No.2. 385-396 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Yamaguchi: "Nonhomogeneous string problem with periodically moving boundaries"Fields Institute Communications. Vol.25. 565-574 (2000)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Tanaka, K.Shiohama: "Compactification and maximal diameter theorem for noncompact manifolds With radial curvature bounded below"Math. Zeit.. Vol.241 No.2. 341-351 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Tanaka: "Characterization of a differentiable point of the distance function to the cut locus"J. Math. Soc. Japan. Vol.55 No.1. 231-241 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Yamaguchi: "Periodic solutions of nonlinear wave equations in spheresymmetric domain in R^3 with periodically osillating boundaries"Proceedings of ISAAC. (発表予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Yamaguchi: "3D wave equations in sphere-symmetric domain with periodically oscillating boundaries"Discrete and Continuous Dynamical Systems. Vol.7.No.3. 385-396 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Yamaguchi, H.Yoshida: "Nonhomogeneous string problem with periodically moving boundaries"Fields Institute Communications. 25. 565-574 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Yamaguchi: "Nonhomogeneous wave equations in a domain with periodically oscillating boundaries"Proceedings of the 4-th Workshop on Differential Equations. 210-212 (1999)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Tanaka, K Shiohama: "Compactification and maximal diametertheorem for noncompaet manifolds with rodial curvature bounded below"Math.2eit. (発表予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Tanaka, J.Itoh: "A Sard theorem for the distance functions"Math.Anallen. Vol.320,No.1. 1-10 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Yamaguchi: "3D wave equation in shere-symmetric domain with periodically oscillating boundaries"Discrete and Continuous Dynamical Systems. Vol.7,No.2. 385-391 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Yamaguchi and H.Yoshida: "Nonhomogeneous string problem with periodically moving boundaries"Fields Institute Communications. Vol.28. 565-574 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] R.Ikehata,T.Matsuyama and T.Kobayashi,: "Remark on the L2 estimates of the density for the compressible Navier-Stokes flow in R3"To appear in Proceedings of the IIIrd World Congress of Nonlinear Analysts.

    • Related Report
      2000 Annual Research Report
  • [Publications] R.Ikehata and T.Matsuyama: "Remarks on the behaviour of solutions to the linear wave equations in unbounded domains"Proceedings of the School of Science of Tokai University. Vol.31. 1-13 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Tanaka: "A Sard Theorem for the distance function"To appear in Math.Annalen.

    • Related Report
      2000 Annual Research Report
  • [Publications] J.Itoh and M.Tanaka: "The Lipshitz continuity of the distance function to the cut locus"Trans.Amer.Math.Soc.. Vol.353. 21-40 (2000)

    • Related Report
      2000 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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