Initial and boundary value problems for nonlinear dispersive wave equations

• Project/Area Number: 19684002
• Research Category: Grant-in-Aid for Young Scientists (A)
• Allocation Type: Single-year Grants
• Research Field: Basic analysis
• Research Institution: Tohoku University
• Principal Investigator: NAKAMURA Makoto 東北大学, 大学院・理学研究科, 准教授 (70312634)
• Project Period (FY): 2007 – 2010
• Project Status: Completed (Fiscal Year 2010)
• Budget Amount: ¥10,400,000 (Direct Cost: ¥8,000,000、Indirect Cost: ¥2,400,000); Fiscal Year 2010: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000); Fiscal Year 2009: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000); Fiscal Year 2008: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000); Fiscal Year 2007: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000)
• Keywords: 関数方程式 / 偏微分方程式 / 非線形 / 初期値問題 / 外部問題 / 消散型波動方程式 / 長時間解 / 時間大域解 / 零条件 / 境界値問題 / 波動方程式 / 適切性 / 非線形シュレディンガー方程式 / ストリッカーツ評価 / 非線形波動方程式

Research Abstract

The well-posedness of the Cauchy problem for higher order dispersive wave equations was considered based on the generalization of the Keel-Smith-Sogge type estimate which is one of the weighted energy estimates for wave equations. The well-posedness of the Cauchy problem for higher order parabolic equations with power type nonlinear terms were constructed by the use of energy estimates for parabolic equations. Almost global solutions for localized dissipative wave equations with critical nonlinear terms were shown in exterior domains in three dimensional Euclidean spaces. And the global solutions were shown when the nonlinear terms satisfy the null conditions.

Research Products

• [Journal Article] Scattering theory for the Dirac Equation of Hartree type and the semirelativistic Hartree equation, Nonlinear Analysis (2012)
  • Author(s): M. Nakamura, K. Tsutaya
  • Journal Title: Series A: Theory, Methods and Applications Volume: 75 Pages: 3531-3542
  • Peer Reviewed
• [Journal Article] Global solutions for nonlinear wave equations with localized dissipations in exterior domains (2012)
  • Author(s): M. Nakamura
  • Journal Title: Journal of Differential Equations Volume: 252 Pages: 4742-4785
  • Peer Reviewed
• [Journal Article] Small global solutions for nonlinear complex Ginzburg-Landau equations and nonlinear dissipative wave equations in Sobolev spaces (2011)
  • Author(s): M. Nakamura
  • Journal Title: Reviews in Mathematical Physics Volume: 23 Pages: 903-931
  • Peer Reviewed
• [Journal Article] Remarks on Keel-Smith-Sogge estimates and some applications to nonlinear higher order wave equations (2011)
  • Author(s): M. Nakamura
  • Journal Title: Differential and Integral Equations Volume: 24 Pages: 519-540
  • Peer Reviewed
• [Journal Article] Global Existence and Uniqueness of Solutions to the Maxwell-Schroedinger Equations (2007)
  • Author(s): M. Nakamura, T. Wada
  • Journal Title: Communications in Mathematical Physics Volume: 276 Pages: 315-339
  • Peer Reviewed
• [Journal Article] General quasilinear wave equations with localized dissipation in exterior domains (2007)
  • Author(s): J. Metcalfe, M. Nakamura
  • Journal Title: Journal of Differential Equations Volume: 233 Pages: 313-344
  • Peer Reviewed
• [Journal Article] A generalization of the weighted Strichartz estimates for wave equations and an application to self-similar solutions (2007)
  • Author(s): J. Kato, M. Nakamura, T. Ozawa
  • Journal Title: Communications on Pure and Applied Mathematics Volume: 60 Pages: 164-186
  • NAID: 120000970574
  • Peer Reviewed
• [Journal Article] General quasilinear wave equations with localized dissipation in exterior domains (2007)
  • Author(s): Jason Metcalfe, M. Nakamura
  • Journal Title: Journal of Differential Equations 233 Pages: 313-344
  • Peer Reviewed
• [Journal Article] Global Existence and Uniqueness of Solutions to the Maxwell-Schrodinger Equations (2007)
  • Author(s): M. Nakamura, T. Wada
  • Journal Title: Communications in Mathematical Physics 276 Pages: 315-339
  • Peer Reviewed
• [Journal Article] Global existence and uniqueness of solutions to the Maxwell-Schroedinger equations (2007)
  • Author(s): Makoto Nakamura, Takeshi Wada
  • Journal Title: Communications in Mathematical Physics 276 Pages: 315-339
  • Peer Reviewed
• [Journal Article] General quasilinear wave equations with localized dissipation in exterior domains (2007)
  • Author(s): Jason Metcalfe, Makoto Nakamura
  • Journal Title: Journal of Differential Equations 233 Pages: 313-344
  • Peer Reviewed
• [Presentation] On global solutions for nonlinear wave equations with localized dissipations (2010)
  • Author(s): M. Nakamura
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 名古屋大学
  • Year and Date: 2010-09-25
• [Presentation] On Keel-Smith-Sogge estimates and some applications (2010)
  • Author(s): M. Nakamura
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 名古屋大学
  • Year and Date: 2010-09-25
• [Presentation] Global existence and asymptotic behavior of solutions for the Dirac equation of Hartree type (2010)
  • Author(s): M. Nakamura, K. Tsutaya
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 名古屋大学
  • Year and Date: 2010-09-24
• [Presentation] General quasilinear wave equations with localized dissipation in exterior domains (2008)
  • Author(s): M. Nakamura
  • Organizer: The 7th AIMS International Conference, University of Texas at Arlington
  • Place of Presentation: Arlington, USA
  • Year and Date: 2008-05-21
• [Presentation] Nonlinear wave equations with localized dissipations in exterior domains (2008)
  • Author(s): 中村誠
  • Organizer: The 7th AIMS International Conference
  • Place of Presentation: University of Texas at Arlington, Arhngton, 米国
  • Year and Date: 2008-05-21
• [Presentation] 非線形波動方程式の外部問題について (2007)
  • Author(s): M. Nakamura
  • Organizer: 日本数学会2007年度秋季総合分科会
  • Place of Presentation: 東北大学(特別講演)
  • Year and Date: 2007-09-23
• [Presentation] 非線形波動方程式の外部問題について (2007)
  • Author(s): 中村誠
  • Organizer: 日本数学会2007年度秋季総合分科会, 特別講演
  • Place of Presentation: 東北大学
  • Year and Date: 2007-09-23
• [Presentation] 非線形波動方程式の外部問題について (2007)
  • Author(s): 中村 誠
  • Organizer: 日本数学会
  • Place of Presentation: 東北大学
  • Year and Date: 2007-09-23
• [Book] Einstein方程式の初期値問題 (2009)
  • Author(s): 中村誠
  • Total Pages: 31
  • Publisher: 講義録
• [Book] 非線形波動方程式の外部問題について (2007)
  • Author(s): 中村誠
  • Publisher: 日本数学会, 2007年度秋季総合分科会, 函数方程式論分科会, 講演アブストラクト