Research on initial and boundary value problems, and space-time estimates for nonlinear partial differential equations

• Project/Area Number: 23684004
• Research Category: Grant-in-Aid for Young Scientists (A)
• Allocation Type: Single-year Grants
• Research Field: Global analysis
• Research Institution: Yamagata University (2013-2014); Tohoku University (2011-2012)
• Principal Investigator: NAKAMURA Makoto 山形大学, 理学部, 教授 (70312634)
• Project Period (FY): 2011-04-01 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥14,040,000 (Direct Cost: ¥10,800,000、Indirect Cost: ¥3,240,000); Fiscal Year 2014: ¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000); Fiscal Year 2013: ¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000); Fiscal Year 2012: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000); Fiscal Year 2011: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000)
• Keywords: 偏微分方程式 / 非線形 / 初期値問題 / 境界値問題 / 計量 / 波動方程式 / 分散型方程式 / 放物型方程式

Outline of Final Research Achievements

The space-time estimates by Lindblad and Rodnianski were generalized, and the initial value problem was considered for the system of wave equations with nonlinear terms which satisfy the null conditions in three spatial dimensions. The initial value problem was considered for heat equations with derivative nonlinear terms in Besov spaces and Triebel-Lizorkin spaces. The energy solutions of the initial value problem were constructed for nonlinear Klein-Gordon equations in the de Sitter space-time, and nonlinear Schroedinger equations derived by its nonrelativistic limit. The best constant of the Moser-Trudinger inequality was shown in Lebesgue spaces with a weight which has a singularity, and it is applied to the initial value problem for nonlinear Klein-Gordon equations with nonlinear terms of exponential type.

Research Products

• [Journal Article] On the sharp constant of the weighted Trudinger-Moser type inequality and an application to nonlinear Klein-Gordon equations (2015)
  • Author(s): M. Ishiwata, M. Nakamura, H. Wadade
  • Journal Title: Discrete and Continuous Dynamical System - A Volume: 印刷中
  • Peer Reviewed
• [Journal Article] Remarks on global solutions of dissipative wave equations with exponential nonlinear terms (2015)
  • Author(s): M. Nakamura
  • Journal Title: Special issue in Communications on Pure and Applied Analysis Volume: 印刷中
  • Peer Reviewed
• [Journal Article] Energy solutions for nonlinear Klein-Gordon equations in de Sitter spacetime (2015)
  • Author(s): M. Nakamura
  • Journal Title: Current Trends in Analysis and Its Applications: Proceedings of the 9th ISAAC Congress, Krakow 2013, Volume: XVI Pages: 203-208
  • Peer Reviewed
• [Journal Article] Remarks on a dispersive equation in de Sitter spacetime (2015)
  • Author(s): M. Nakamura
  • Journal Title: Prodeedings. Analysis of hyperbolic PDEs. The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications in Madrid, Spain Volume: 印刷中
  • Peer Reviewed
• [Journal Article] On the sharp constant of the weighted Trudinger-Moser type inequality of the scaling invariant form (2014)
  • Author(s): M. Ishiwata, M. Nakamura, H. Wadade
  • Journal Title: Annales de l'Institut Henri Poincare - Analyse non lineaire Volume: 31 Pages: 297-314
  • Peer Reviewed
• [Journal Article] On the solutions for nonlinear wave equations with localized dissipations in exterior domains (2014)
  • Author(s): M. Nakamura
  • Journal Title: Nonlinear Dynamics in Partial Differential Equations, Advanced Studies in Pure Mathematics Volume: 64 Pages: 289-295
  • Peer Reviewed
• [Journal Article] The Cauchy problem of semilinear Schroedinger equations in curved background (2014)
  • Author(s): M. Nakamura
  • Journal Title: Proceedings. Wayamba International Conference, ``Sustainable Development Through Equitable Sharing," Volume: 無 Pages: 302-302
  • Peer Reviewed
• [Journal Article] Remarks on a weighted energy estimate and its application to nonlinear wave equations in one space dimension (2014)
  • Author(s): M. Nakamura
  • Journal Title: Journal of Differential Equations Volume: 256 Pages: 389-406
  • Peer Reviewed
• [Journal Article] The Cauchy problem for semi-linear Klein-Gordon equations in de Sitter spacetime (2014)
  • Author(s): M. Nakamura
  • Journal Title: Journal of Mathematical Analysis and Applications Volume: 410 Pages: 445-454
  • Peer Reviewed
• [Journal Article] On the sharp constant of the weighted Trudinger-Moser type inequality of the scaling invariant form (2014)
  • Author(s): M. Ishiwata, M. Nakamura, H. Wadade,
  • Journal Title: Annales de l'Institut Henri Poincare - Analyse non lineaire Volume: 印刷中
  • Peer Reviewed
• [Journal Article] Energy solutions for nonlinear Klein-Gordon equations in de Sitter spacetime (2014)
  • Author(s): M. Nakamura
  • Journal Title: Proceedings of the ISAAC Congress Volume: 印刷中
  • Peer Reviewed
• [Journal Article] On the solutions for nonlinear wave equations with localized dissipations in exterior domains (2014)
  • Author(s): M. Nakamura
  • Journal Title: Advanced Studies in Pure Mathematics Volume: 印刷中
  • Peer Reviewed
• [Journal Article] Global and almost global solutions for some nonlinear parabolic equations in Besov spaces and Triebel-Lizorkin spaces (2013)
  • Author(s): T. Iwabuchi, M. Nakamura
  • Journal Title: Adv. Differential Equations Volume: 18 Pages: 687-736
  • Peer Reviewed
• [Journal Article] Weighted energy esitmates and applications for nonlinear wave equations (2013)
  • Author(s): 中村誠
  • Journal Title: 「熊本大学応用解析セミナー」報告集 Volume: なし Pages: 11-16
• [Journal Article] On the Cauchy problem for nonlinear Klein-Gordon equations in de Sitter spacetime (2013)
  • Author(s): 中村誠
  • Journal Title: 「熊本大学応用解析セミナー」報告集 Volume: なし Pages: 87-90
• [Journal Article] Remarks on global solutions for nonlinear wave equations under the standard null conditions (2013)
  • Author(s): H. Lindblad, M. Nakamura, C. D. Sogge
  • Journal Title: Journal Differential Equations Volume: 254 Pages: 1396-1436
  • 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] Scattering theory for the Dirac Equation of Hartree type and the semirelativistic Hartree equation (2012)
  • Author(s): M. Nakamura, K. Tsutaya
  • Journal Title: Nonlinear Analysis Series A: Theory, Methods and Applications Volume: 75 Pages: 3531-3542
  • 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
• [Presentation] The Cauchy problem for nonlinear Schrodingier equation in fractional order Sobolev spaces (2015)
  • Author(s): M. Nakamura, T. Wada
  • Organizer: 日本数学会年会
  • Place of Presentation: 明治大学
  • Year and Date: 2015-03-21 – 2015-03-24
• [Presentation] On the effect of spatial expansion on nonlinear Schrodinger equations (2015)
  • Author(s): M. Nakamura
  • Organizer: 熊本大学セミナー
  • Place of Presentation: 熊本大学
  • Year and Date: 2015-03-14
  • Invited
• [Presentation] Remarks on Strichartz estimates and an application for nonlinear Schrodinger equations (2015)
  • Author(s): M. Nakamura
  • Organizer: 数理物理に現れる偏微分方程式研究会
  • Place of Presentation: 静岡県賀茂郡松崎町雲見
  • Year and Date: 2015-01-09 – 2015-01-11
  • Invited
• [Presentation] The Cauchy problem for nonlinear Schrodinger equations in de Sitter spacetime (2014)
  • Author(s): M. Nakamura
  • Organizer: International Conference on Recent Advances in Hyperbolic Partial Differential Equations
  • Place of Presentation: 広島国際会議場
  • Year and Date: 2014-12-04 – 2014-12-06
  • Invited
• [Presentation] On the Cauchy problem for nonlinear Schrodinger equations in de Sitter spacetime (2014)
  • Author(s): M. Nakamura
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 広島大学
  • Year and Date: 2014-09-25 – 2014-09-28
  • Invited
• [Presentation] Einstein の重力場方程式の初期値問題－１ (2014)
  • Author(s): M. Nakamura
  • Organizer: Encounter with Mathematics, 第62回 波動方程式 --古典物理から相対論まで--
  • Place of Presentation: 中央大学
  • Year and Date: 2014-09-15 – 2014-09-16
  • Invited
• [Presentation] Einstein の重力場方程式の初期値問題－２ (2014)
  • Author(s): M. Nakamura
  • Organizer: Encounter with Mathematics, 第62回 波動方程式 --古典物理から相対論まで--
  • Place of Presentation: 中央大学
  • Year and Date: 2014-09-15 – 2014-09-16
  • Invited
• [Presentation] Dissipative property of nonlinear Schrodinger equations in de Sitter spacetime (2014)
  • Author(s): M. Nakamura
  • Organizer: Workshop in Mathematical Sciences
  • Place of Presentation: Wayamba University of Sri Lanka, Sri Lanka
  • Year and Date: 2014-08-31
  • Invited
• [Presentation] The Cauchy problem of semilinear Schrodinger equations in curved backgroun (2014)
  • Author(s): M. Nakamura
  • Organizer: Wayamba International Conference
  • Place of Presentation: Wayamba University of Sri Lanka, Sri Lanka
  • Year and Date: 2014-08-29 – 2014-08-30
  • Invited
• [Presentation] Remarks on nonlinear Schrodinger equations in curved spacetime (2014)
  • Author(s): M. Nakamura
  • Organizer: ICM 2014 Satellite Conference in Harmonic Analysis
  • Place of Presentation: Chosun University, Korea
  • Year and Date: 2014-08-04 – 2014-08-08
• [Presentation] Global solutions for nonlinear Schrodinger equations in spatial expansion (2014)
  • Author(s): M. Nakamura
  • Organizer: 2nd East Asian Conference in Harmonic Analysis and Applications
  • Place of Presentation: Mudanjiang Normal University, China
  • Year and Date: 2014-07-10 – 2014-07-16
  • Invited
• [Presentation] Remarks on a dispersive equation in de Sitter spacetime (2014)
  • Author(s): M. Nakamura
  • Organizer: Analysis of hyperbolic PDEs, 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications in Madrid
  • Place of Presentation: Madrid, Spain
  • Year and Date: 2014-07-07 – 2014-07-11
  • Invited
• [Presentation] 波動の数学解析 (2014)
  • Author(s): M. Nakamura
  • Organizer: 第18回 理工学研究科セミナー
  • Place of Presentation: 山形大学
  • Year and Date: 2014-05-27
  • Invited
• [Presentation] Nonlinear Schrodinger equations in curved spacetime (2014)
  • Author(s): M. Nakamura
  • Organizer: Hakodate workshop on nonlinear wave equations
  • Place of Presentation: 公立はこだて未来大学
  • Year and Date: 2014-05-24
  • Invited
• [Presentation] Dissipative structure of nonlinear Klein-Gordon equations in expanding spacetime (2014)
  • Author(s): M. Nakamura
  • Organizer: Recent topics in Partial Differential Equations
  • Place of Presentation: 山形大学
  • Year and Date: 2014-05-18
• [Presentation] An application of the Trudinger-Moser inequality to nonlinear Klein-Gordon equations with singular weights (2014)
  • Author(s): M. Nakamura
  • Organizer: Recent topics in Partial Differential Equations
  • Place of Presentation: 山形大学
  • Year and Date: 2014-05-17
• [Presentation] Remarks on global solutions for nonlinear wave equations under the standard null conditions (2012)
  • Author(s): 中村誠
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 九州大学
• [Presentation] The Cauchy problem for dissipative wave equations with weighted nonlinear terms (2012)
  • Author(s): 中村誠, 和田出秀光
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 九州大学
• [Presentation] 重み付きMoser-Trudinger型不等式の最良定数について (2011)
  • Author(s): 石渡通徳、中村誠、和田出秀光
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 信州大学
  • Year and Date: 2011-09-29
• [Presentation] Global and almost global solutions for some nonlinear parabolic equations in Besov spaces and Triebel-Lizorkin spaces (2011)
  • Author(s): 岩渕司、中村誠
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 信州大学
  • Year and Date: 2011-09-29
• [Presentation] Global solutions for nonlinear Klein-Gordon equation in de Sitter spacetime
  • Author(s): M. Nakamura
  • Organizer: 1st East Asian Conference in Harmonic analysis and Applications
  • Place of Presentation: Seoul National University, Korea
  • Invited
• [Presentation] Energy solutions for nonlinear Klein-Gordon equations in de Sitter spacetime
  • Author(s): M. Nakamura
  • Organizer: 9th International ISAAC Congress
  • Place of Presentation: Pedagogical University, Krakow, Poland
  • Invited
• [Presentation] On small solutions for nonlinear wave equations in exterior domains
  • Author(s): M. Nakamura
  • Organizer: Lectures, School of Mathematical Sciences
  • Place of Presentation: 北京大学, 中国.
  • Invited
• [Presentation] Global solutions for nonlinear Klein-Gordon equations in de Sitter spacetime
  • Author(s): M. Nakamura
  • Organizer: 第13回さいたま数理解析セミナー
  • Place of Presentation: 埼玉大学サテライトキャンパス
  • Invited
• [Presentation] Remarks on Klein-Gordon equations with exponential nonlinearity
  • Author(s): M. Nakamura
  • Organizer: Recent Progress in Wave Models
  • Place of Presentation: 中央大学
  • Invited
• [Presentation] Local and global solutions for nonlinear dissipative equations in energy spaces
  • Author(s): M. Nakamura
  • Organizer: Recent topics related to partial differential equations
  • Place of Presentation: 山形大学
• [Presentation] Remarks on a weighted energy estimate and its application to nonlinear wave equations in one space dimension
  • Author(s): M. Nakamura
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 愛媛大学
• [Presentation] The Cauchy problem for nonlinear Klein-Gordon equations in de Sitter spacetime
  • Author(s): M. Nakamura
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 愛媛大学
• [Presentation] Nonlinear Klein-Gordon equation in de Sitter spacetime and its related topics
  • Author(s): M. Nakamura
  • Organizer: 京都大学NLPDEセミナー
  • Place of Presentation: 京都大学.
  • Invited
• [Presentation] Klein-Gordon equations with singular nonlinear terms
  • Author(s): M. Nakamura
  • Organizer: Seminar for recent topics in PDE
  • Place of Presentation: 北海道工業大学