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Analysis of Nonlinear Waves and Nonlinear Diffusions

Research Project

Project/Area Number 09440066
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field 解析学
Research InstitutionTOKYO METOROPOLITAN UNIVERSITY

Principal Investigator

MOCHIZUKI Kiyoshi  Tokyo Metropolitan University, Faculty of Sci., Prof., 大学院・理学研究科, 教授 (80026773)

Co-Investigator(Kenkyū-buntansha) KURATA Kazuhiro  Tokyo Metropolitan University, Faculty of Sci., Associate Prof., 大学院・理学研究科, 助教授 (10186489)
SAKAI Makoto  Tokyo Metropolitan University, Faculty of Sci., Prof., 大学院・理学研究科, 教授 (70016129)
ISHII Hitoshi  Tokyo Metropolitan University, Faculty of Sci., Prof., 大学院・理学研究科, 教授 (70102887)
SUZUKI Ryuichi  Kokushikan U., Faculty of Eng., Asso. Prof., 工学部, 助教授 (00226573)
KAWANAGO Tadashi  Shizuoka U., Faculty of Eng., Associate Prof., 工学部, 助教授
肥田野 久二男  東京都立大学, 大学院・理学研究科, 助手 (00285090)
Project Period (FY) 1997 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥8,000,000 (Direct Cost: ¥8,000,000)
Fiscal Year 1999: ¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 1998: ¥2,400,000 (Direct Cost: ¥2,400,000)
Fiscal Year 1997: ¥2,800,000 (Direct Cost: ¥2,800,000)
Keywordslocalized dissipation near infinity / Degenerate parabolic equation / Kirchhoff equation / Sturm-Liouvill operator / Schrodinger operator / Hamilton-Jacobi equation / Hele-Shaw fluid / bifurcation phenomena / KPP方程式 / Sturm-Liouville作用素 / Hale-Shaw流 / Fefferman-Phong不等式 / 非線形散乱 / 解の爆発 / 反応拡散方程式 / Ginzburg-Landau方程式 / エネルギー減衰 / Life-span / 波動方程式 / KcrichhoH方程式 / 追化放物型方程式 / 爆発問題 / 漸近挙動
Research Abstract

Among various kind of problems on differential equations, in this project, we are mainly concerned with those related to the Applied Mathematics, Physics and Technology. Summarizing the results obtained by the investigators in the period 1997-99, we can say that the objective of this project is accomplished fruitfully.
The head investigator published 9 papers analyzing the nonlinear waves and nonlinear diffusions. The topics include the following:
(1) Decay and asymptotics of nonlinear waves: The existence of the scattering state is proved for acoustic wave equations with nonlinear dissipation. The existence of global small solution and its energy decay are established for the Kirchhoff equation (describing the vibration of elastic string) with dissipation localized near infinity.
(2) Semilinear or quasilinear degenerate parabolic equations: Reaction diffusion systems with nonlinear power source term are considered. The critical exponents which divide the blow-up and global existence of s … More olutions are shown to exist. Moreover, the critical blow-up, life span of blow-up solutions and the asymptotic behavior for time goes to infinity of global solutions are proved. Similar results are also obtained for the quasilinear equations describing fluids in porous media and combustion process in plasma.
(3) Spectral and scattering theory: A new formulation and results are obtained for the spectral inverse problem for the classical Sturm-Liouville operator. Scattering theory is established for the wave equation with small dissipation or hoarding. Moreover, by generalizing the former results, we expanded the applicability of the principle of limiting absorption for the Schrodinger operator oscillating long-range potential.
Each investigator developed many important nonlinear problems, among which are included e.g., homogenization of the Hamilton-Jacobi equation, free boundary problem of Hele-Shaw flow, variational problems, stability of Navier-Stokes equations and nonlinear scattering. Less

Report

(4 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • 1997 Annual Research Report
  • Research Products

    (38 results)

All Other

All Publications (38 results)

  • [Publications] K. Mochizuki: "Global existence and energy decay of small solutions to the Kirchhoff equation with linear dissipation localized near inbinity"J. Natb. Kyoto Uniu. 39. 347-363 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Mukai, K.Mochizuki, Q. Huang: "Lerge time behaution and life span for a quasilinear pasabolic eguation with slowly decaying initial volues"Nonlenear Analysis. 39. 33-45 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Horie, H. Ishii: "Homogenization of Hamilton-Jacobi eguations on do mains with scale perisdic structure"Indiana Uniu. Math. J. 47. 1011-1058 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M. Sakai: "Shanp estimates of the distance from a fixed point to the frontier of a Hole-Chaw plow"Potential Analysis. 8. 277-302 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Kawanago: "Stability estimate for strong solutions of the Navier-Stokes system and its applications"Electronic J, Diffesential Equations 1998. 15. 1-23 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] R. Suzuki: "Asymptotic ******** of solutions of quasilinear parabolic equations with slowly decaying initial data"Adu. Math. Sci. Appl.. 9. 291-317 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] MOCHIZUKI,K.: "Global existence and energy decay of small solutions to the Kirchhoff equation with linear dissipation localized near infinity"J. Math. Kyoto Univ.. 39. 347-363 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] MUKAI,K., MOCHIZUKI,K., HUANG.Q.: "Large time behavior and life span for a quasilinear parabolic equation with slowly decaying initial values"Nonlinear Analysis. 39. 33-45 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] MOCHIZUKI,K., HUANG,Q.: "Existence and behavior of solutions for a weakly coupled system of reaction diffusion equations."Meth. Appl. Anual.. 5. 109-124 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] MOCHIZUKI,K., MOTAI,T.: "Asymptotics of energy for wave equations with a nonlinear dissipative term in RィイD1NィエD1, Non linear waves"GAKUTO International Series, Math. Sci. Appl.. 10. 349-352 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] MOCHIZUKI,K., SUZUKI,R.: "Critical exponent and critical blow-up for quasilinear parabolic equations"Israel J. Math.. 98(with SUZUKI,R.). 141-156 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] HORIE,K., ISHII,H.: "Homogenization of Hamilton-Jacobi equations on domains with scale periodic structure, Indiana Univ."Math. J.. 47. 1011-1058 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] "Sharp estimates of the distance from a fixed point to the frontier of a Hele-Shaw flow"Potential Analysis. 8. 277-302 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] AOLI,N., MORIYASU,K., OKA,M.: "Differentiable maps having hyperbolic sets"Topolgy Appl.. 82. 15-48 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] KURATA,K.: "Boundedness of integral operators on generalized Morrey spaces and its application to the Schrodinger operators"Proc. Amer. Math. Soc.. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] TAKAKUWA,S.: "A compactness theorem for harmonic maps."Differential and Integral Equations. 11. 169-178 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] HIDANO,K.: "Nonlinear small data scattering for the wave equation in RィイD14+1ィエD1"J. Math Soc.. 50. 253-292 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] KAWANAGO,T., SUZUKI,R.: "Stability estimate for strong solutions of the Navier-Stokes system and its applications, Asymptotic behavior of solutions of quasilinear parabolic equations with slowly decaying initial data"Adv. Math. Sci.. Appl., 9. 291-317 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] KADOWAKI,M.: "Low energy resolvent estimates for acoustic propagators in perturbed stratified media"J. Differential Equations. 141. 25-53 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K.Mukai, K.Mochizuki, Q.Hueng: "Large time behavior and life span for a quasilinear parabolic equation with slowly decaying initial values"Nonlinear Analysis. 30. 33-45 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] K.Mochizuki: "Global existence and energy decay of small solutions to the Kirchhoff equation with linear dissipation localyed near infinity"J. Math. Kyoto Univ.. 39. 347-363 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Ishii: "Homogenization of the Candy problem for Hainelton-Jacoki equations"Stochasticsnalysis control, optimization and applications, Birbhauser Boston MA. 305-324 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] M.Sakai: "Linear conbinations of harmonic measeeres and quadiatic domains of signet measures with small supports"Proc. Edinburgh Math. Soc.. 42. 433-444 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] K.Kurata: "Exiotence and semi-classical limit of the lest energy solutions to a nonlinear sohrodinger equation with electromagnetie fields"Nonlinear Analysis. (To appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] N.Aoki, K.Moriyasu, N.Sumi: "C^1-maps having hyperholic periodic points"Fund. Math.. (To appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] R.Suzuki: "Proc. Royal Soc. Edinburgh"Complete blow-up for qussilinear parabolic equatione (To appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] K.Mochizuki: "Global existence and energy decay of small solutione to the kirehhott equation with linear dissipation near infinity" J.Math.Kyoto Univ.to appear.

    • Related Report
      1998 Annual Research Report
  • [Publications] K.Mukai, K.Mochizuki and Q.Huang: "Large time behavior and life span for a qualilinear parabolic equation with slowly decaying initial values" Nonlinear Analysis TMA. to appear.

    • Related Report
      1998 Annual Research Report
  • [Publications] K.Mochizuki: "Blow-up, life span and large time behavior of solutions to a weakly coupled system of reaction-diffusion equations" Advances in Nonlinear Partial Differential Equations and Stochastics World Scientific. 48. 175-198 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Sakai: "Sharp estimates of the distance from a fiped point to the frontier of Hale-shaw flow" Potential Analysis. 8. 277-302 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] K.Kurata: "A Liouville type theorem for the Ginzburg-Landau equations with general potentials" Disrete and Continuous Dynamical Systems. 17-23 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] K.Hidano: "Nonlinear small data scattering for the wave equation in IR^<4+1>" J.Math.Soc.Japan. 50. 253-292 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] K.Mukai,K.Mochizuki and Q.Huang: "Large time behavion and life span for a quasilinear parabolic equation with slowly decaying initial values" Nonlinear Analysis TMA. (to appear).

    • Related Report
      1997 Annual Research Report
  • [Publications] K.Mochizuki and Q.Huang: "Existence and behavion of solntions for a weakly coupled system of reaction diffusion equations" Methods Appl.Analysis. (to appear).

    • Related Report
      1997 Annual Research Report
  • [Publications] K.Mochizuki and R.Suzuki: "Oritieal exponent and critical slow-up for guasilinear paralolic equations" Israel J.Math. 98. 141-156 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Q.Huang and K.Mochizuki: "A note on the global solutions of a degenerate paralolic system" Tokyo J.Math.20. 63-66 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] M.Arisawa,P.L.Lions and H.Ishii: "Un contre-example powr la th,Aiorie du contr,Atle ergodique en dimension infinie." C.R.Acad.Sci.Paris S,Air.I Math. 325. 37-41 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] K.Kurata: "Local boundedress and continuity for weak solutions of -(V-ib)^2μ+Vu=0" Math.Z.224. 641-653 (1997)

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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