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Analysis of linear and nonlinear wave phenomena and the inverse problem

Research Project

Project/Area Number 13640219
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionCHUO UNIVERSITY (2002-2003)
Tokyo Metropolitan University (2001)

Principal Investigator

MOCHIZUKI Kiyoshi  Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (80026773)

Co-Investigator(Kenkyū-buntansha) HIDANO Kunio  Chuo University, Faculty of Education, Assiociate, Professor, 教育学部, 助教授 (00285090)
OHARU Shinnnosuke  Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (40063721)
MURAMATSU Toshinobu  Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (60027365)
KADOWAKI Mitsuteru  Ehime University, Faculty of Engineering, Associate, Professor, 工学部, 助教授 (70300548)
SUZUKI Ryuichi  Kokushikan University, Faculty of Engineering, Associate, Professor, 工学部, 助教授 (00226573)
倉田 和浩  東京都立大学, 理学研究科, 助教授 (10186489)
酒井 良  東京都立大学, 理学研究科, 教授 (70016129)
Project Period (FY) 2001 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥3,200,000 (Direct Cost: ¥3,200,000)
Fiscal Year 2003: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2002: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2001: ¥1,300,000 (Direct Cost: ¥1,300,000)
Keywordsgeneralized KPP equation / Inverse spectral problem / semilinear wave equation / dissipative wave equation / scattering amplitude / large time asymptotics / Sturm-Liouville operator / Dirac operator / シュレディンガー方程式 / Sturm-Liouville作用素 / Dirac作用素 / Schr\"odinger作用素 / スペクトル表現 / 極限吸収の原理 / 散乱と逆散乱 / Inverse scattering problem / Inverse spectral problem / Non-conservative wave equation / Hele-Shaw fleid / Cahn-Hillard / Nonlinea scattering theory / Porous media equation / Low energy resolvent estimate
Research Abstract

In this project, we are mainly concerned with the analysis of wave phenomena governed by some basic partial differential equations in Applied Mathematics and Physics, and also by some model equations appearing in Chemistry and Biology. Summarizing the results obtained by the investigators in the period 200 1-2003, we can say the objective of this project is accomplished fruitfully.
The head investigator published 5 papers analyzing linear and nonlinear waves and nonlinear diffusions. The topics include the following:
(1)Nonlinear parabolic equations: Large time asymptotics of solutions for generalized KPP equation are studied. Precise formula obtained here is expected to have many applications in Biology.
(2)Inverse spectral problem: Inverse problem to determine the potential from some spectral data is studied. We obtained a uniqueness results for Sturm-Liouville operator and for Dirac operator on firiete interval. Our problem 'to determine the potential from interior spectral data' is a new formulation and is expected to be applied to many other operators.
(3)Semilinear wave equations: Asymptotic for wave equation with nonlinear dissipation is studied. We require that the dissipation is inhomogeneous in' space and time and sufficient conditions are obtained for solutions to be asymptoically free.
(4)Inverse scattering problem: First the direct scattering theory is established for wave equation with first time derivative term, and then the coefficient of the term is shown to be reconstructed from the scattering amplitude with a fixed energy.
Each investigator developed many interesting results on the related fields.

Report

(4 results)
  • 2003 Annual Research Report   Final Research Report Summary
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • Research Products

    (34 results)

All Other

All Publications (34 results)

  • [Publications] K.Mochizuki, I.A.Shishmarev: "Large time asymptotics of small solutions to generalized Kolmogorov-Petrovskii-Piskunov equations"Funkcialai Ekvacioj. 44. 99-117 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Mochizuki, I.Trooshin: "Inverse problem for interior spectral data of the Sturm-Liouville operator"J.Inverse III-posed Problems. 9. 425-433 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Mochizuki, I.Trooshin: "Inverse problem for interior spectral data of the Dirac operator on a finite interval"Publ.Res.Inst.Math.Sci.. 38. 387-395 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 望月 清: "Strichartz評価と半線形波動方程式"Seminar Notes on Mathematical Sciences. 6. 117-130 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Mochizuki: "Inverse scattering for a small nonselfadjoint perturbation of the wave equation"Analysi and Applications-ISSAC 2001(H.G.W.Berger, R.P.Gilbert, M.W.Wong eds.). 303-316 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Mochizuki, M.Nakao: "Total energy decay for the wave equation with a dissipation near infinity in exterior domains"Preprint 2003.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Mochizuki, I.A.Shishmarev: "Large time asymptotics of small solutions to generalized Kohnogorov-Petrovskii-Piskunov equation"Fukcialai Ekvacioj. 44. 99-117 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Mochizuki, I.Trooshin: "Inverse problem for interior spectral data of the Sturn-Liouville opearor"J. Inverse III-posed Problems. 9. 425-433 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Mochizuki, I.Trooshin: "Inverse problem for interior spectral data of the Dirac operator on a finite interval"Publ. Res. Inst. Math. Sci.. 38. 387-395 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Mochizuki: "Strichartz estimates and semilinear wave equations"Seminar Notes of Mathematical Sciences 6, Ibaraki Univ.. February. 117-130 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Mochizuki: "Inverse scattering for a small nonselfadjoint perturbation of the wave equation"Analysis and Applications-ISSAK 2001 (H.G.W.Begehr, R.Gilbert, M.W.Wong (eds.))(Kluwer Academic Publishers). 303-316 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] B.Gustaffson, M.Sakai: "Sharp estimates of the curvature of some free boundaries in two dimensions"Ann. Acad. Sci. Fenn. Math.. 13. 133-142 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Kurata, M.Shibata, K.Tada: "Existence of positive solutions for some nonlinear elliptic equations on unbounded domains with cylindrical ends"Nonlinear Anal.. 55. 401-422 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Matsuura, S.Oharu: "Mathematical models of bone remodeling phenomena and numerical simulations"J. Math. Soc. Japan. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Muramatu, S.Taoka: "The initial value problem for the 1-D semilinear Schro~'dinger equation in Besov spaces"J. Math. Soc. Japan. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Hidano: "Scattering and self-similar solutions for the nonlinear wave equation"Differential Integral Eqs.. 15. 405-462 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] R.Suzuki: "Asymptotic behavior of solutions of quasilinear parabolic equations with supercritical nonlinearity"J Differential Eqs.. 190. 150-181 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Kadowaki: "Low energy resolvent estimates for acoustic piopagators in perturbed stratified media"J Differential Eqs.. 179. 246-277 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Mochizuki: "Decay, asymptotics and scattering for the SchrO" dinger equation with time dependent complex potentials"Preprint 2004.

    • Related Report
      2003 Annual Research Report
  • [Publications] 望月 清: "Strichartz評価と半線形波動方程式"Seminar Notes of Mathematical Sciences. 6. 117-130 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Mochizuki, M.Nakao: "Total energy decay for the wave equation with a dissipation near infinity in exterior domain"Preprint 2003.

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Muramatu, S.Taoka: "The initial value problem for the 1-D semilinear SchrO" dinger equation in Besov spaces"J.Math.Soc.Japan. (To appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] Y.Matsuura, S.Oharu: "Mathematical models ob bone remodeling phenomena and numerical simulations"Adv.Math.Sci.Appl.. 13. 401-422 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] R.Suzuki: "Asymptotic behavior of solutions of quasilinear parabolic equations with supercritical nonlinearity"J.Differential Eqs.. 190. 150-181 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Mochizuki, I.Trooshinn: "Inverse problem for interior spectral data of the Dirac operator on a finite domain"Research Institute for Mathematica Sciences. 38/2. 387-395 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Mochizuki, I.Trooshinn: "Inverse problem for interior spectral data of the Sturm-Liouville operator"Journal of Inverse and Ill-Posed Problems. 9/4. 425-433 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Mochizuki: "An improvement of a result of J\"ager and Reito about oscillating long-range potentials"Seminar Notes of Mathematical Sciences. 5. 49-63 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Mochizuki: "Inverse scattering for a small nonselfadjoint perturbation of the wave equation"Proceeding 3rd International IZAAK Congress 2001, Kluwer. (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Mochizuki, I.A.Shishmarev: "Large time asymptotics of small solutions to generalized kolmogorov-Petrovski-Piskunov equation"funkcialaj ekuacioj. 44・1. 99-117 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Mochizuki, I.Trooshin: "Inverse problem for interior spectral data of the Sturm-Liouville operator"J. Inverse and Ill-Posed Problems. 9・4. 425-433 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Kurata, M.Shibata: "On a one-dimensional variational problem related to Cahn-Hilliard energy in a bent strip-like domain"Nonlinear Analysis. 47. 1059-1068 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Hidano: "Scattering problem for the nonlinear wave equation in the finite energy and conformal charge space"J. Functional Analysis. 187・2. 274-307 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Hidano: "Scattering and self-similar solutions for the nonlinear wave equation"Differential Integral Equations. 15・4. 405-462 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Kadowaki: "Low and High energy resolvent estimates for wave propagation in stratified media and their applications"Differential Integral Equations.

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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