Qualitative properties of solutions to time evolution equations with damping terms

• Project/Area Number: 15K17581
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Fukuoka Institute of Technology
• Principal Investigator: Takeda Hiroshi 福岡工業大学, 工学部, 准教授 (10589237)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 消散項 / 漸近形 / 時間大域解 / 有限時間爆発 / 構造的消散項 / 平滑化効果 / 解の拡散現象 / 解の漸近形 / 無条件時間大域的適切性 / 強消散項 / 消散型波動方程式 / 高次漸近展開 / 解の大域挙動 / 時間大域適切性 / 正則性の損失 / 消散型梁方程式

Outline of Final Research Achievements

We studied the Cauchy problem of evolution equations with dissipative terms. Our main concern is to investigate the large time behavior of the solutions. As a result, we proved the asymptotic expansion formula of the solutions to various linear and nonlinear hyperbolic equations with damping terms.

Academic Significance and Societal Importance of the Research Achievements

非線形消散型波動方程式の高次漸近展開理論の確立は幅広い応用例を持つと考えられる。特に空間変数の次元に関しては3次元までと限定的ではあるが、高次元の場合もその漸近形は同じものと推測される。
弱い消散項を持つ非線形梁方程式の無条件時間大域適切性及びその解の漸近展開の証明に用いた線形主要部から導かれる平滑化効果を消散項の性質と併用する方法論も今後大いに進展していく余地がある。
弱消散項と強消散項を併せ持つ非線形波動方程式の時間大域解の存在・非存在に関する臨界指数の同定により、この方程式に対するより詳細な研究成果が今後得られるものと考えられる。

Research Products

• [Journal Article] Large time behavior of global solutions to nonlinear wave equations with frictional and viscoelastic damping terms (2019)
  • Author(s): Ryo Ikehata and Hiroshi Takeda
  • Journal Title: Osaka Journal of Mathematics Volume: 印刷中
  • NAID: 120006768703
  • Peer Reviewed
• [Journal Article] Asymptotic profiles of solutions for structural damped wave equations (2019)
  • Author(s): Ryo Ikehata and Hiroshi Takeda
  • Journal Title: J. Dynamics and Differential equations Volume: 31 Issue: 1 Pages: 537-571
  • DOI: 10.1007/s10884-019-09731-8
  • Peer Reviewed
• [Journal Article] Unconditional small data global well-posedness for nonlinear beam equations with weak damping (2018)
  • Author(s): Hiroshi Takeda
  • Journal Title: Nonlinear Analysis. Real World Applications Volume: 41 Pages: 1-33
  • DOI: 10.1016/j.nonrwa.2017.10.007
  • Peer Reviewed
• [Journal Article] Uniform Energy Decay for Wave Equations with Unbounded Damping Coefficients (2018)
  • Author(s): Ryo Ikehata and Hiroshi Takeda
  • Journal Title: Funkcialaj Ekvacioj Volume: 印刷中
  • NAID: 130007858187
  • Peer Reviewed
• [Journal Article] Critical exponent for nonlinear wave equations with frictional and viscoelastic damping terms (2017)
  • Author(s): Ryo IKEHATA and Hiroshi TAKEDA
  • Journal Title: Nonlinear Analysis Volume: 148 Pages: 228-253
  • DOI: 10.1016/j.na.2016.10.008
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Higher order asymptotic expansions to the solutions for a nonlinear damped wave equation (2016)
  • Author(s): Tatsuki Kawakami and Hiroshi Takeda
  • Journal Title: Nonlinear Differ. Equ. Appl. Volume: 23 Issue: 5 Pages: 1-30
  • DOI: 10.1007/s00030-016-0408-8
  • Peer Reviewed
• [Journal Article] Large time behavior of solutions for a nonlinear damped wave equation (2016)
  • Author(s): Hiroshi takeda
  • Journal Title: Commun. Pure Appl. Anal. Volume: 15 Issue: 1 Pages: 41-55
  • DOI: 10.3934/cpaa.2016.15.41
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Asymptotic Profile of Solutions to a Hyperbolic Cahn-Hilliard Equation (2015)
  • Author(s): Hiroshi Takeda, Yasunori Maekawa and Shuichi Kawashima
  • Journal Title: Bulletin of the Institute of Mathematics Academia Sinica New Series Volume: 10 Pages: 479-539
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] Asymptotic expansion of solutions for wave equations with structural damping and its application (2019)
  • Author(s): Hiroshi Takeda
  • Organizer: Waseda Workshop on Partial Differential Equations 2019
  • Int'l Joint Research / Invited
• [Presentation] 構造的消散項を持つ波動方程式の解の漸近展開とその応用について (2018)
  • Author(s): 竹田寛志
  • Organizer: 熊本応用解析セミナー
  • Invited
• [Presentation] Asymptotic expansion of solutions to a nonlinear strong damped wave equation (2018)
  • Author(s): Hiroshi Takeda
  • Organizer: Critical exponent and nonlinear evolution equations 2018
  • Int'l Joint Research / Invited
• [Presentation] 構造的消散項をもつ波動方程式の解の大域挙動とその応用について (2017)
  • Author(s): 竹田寛志
  • Organizer: 第2回大分解析セミナー
  • Invited
• [Presentation] 非線形強消散型波動方程式の時間大域解の挙動について (2017)
  • Author(s): 竹田寛志
  • Organizer: 九州関数方程式セミナー
  • Invited
• [Presentation] 強消散項を持つ非線形波動方程式の解の漸近形について (2017)
  • Author(s): 竹田寛志
  • Organizer: 東北大学応用数学セミナー
  • Invited
• [Presentation] Asymptotic expansion of the solution to a nonlinear wave equation with strong damping (2017)
  • Author(s): 竹田寛志
  • Organizer: 解析セミナー(神戸大学)
  • Invited
• [Presentation] Large time behavior of solutions to strong damped wave equations (2017)
  • Author(s): Hiroshi Takeda
  • Organizer: The 11th ISAAC congress
  • Int'l Joint Research / Invited
• [Presentation] Asymptotic profile of solutions to a nonlinear wave equation with strong damping (2017)
  • Author(s): Hiroshi Takda
  • Organizer: Ito Workshop on Partial Differential Equations
  • Int'l Joint Research / Invited
• [Presentation] Large time behavior of solutions to strong damped wave equations (2016)
  • Author(s): Hiroshi Takeda
  • Organizer: 2nd Workshop Analysis for Nonlinear problems in UCSB
  • Place of Presentation: カリフォルニア大学サンタバーバラ校
  • Year and Date: 2016-08-17
  • Int'l Joint Research
• [Presentation] Unconditional global well-posedness for a nonlinear damped plated equation and decay property of the global solution (2016)
  • Author(s): Hiroshi Takeda
  • Organizer: PDE/Applied Math Seminar
  • Place of Presentation: カリフォルニア大学サンタバーバラ校
  • Year and Date: 2016-04-22
  • Int'l Joint Research / Invited
• [Presentation] 弱い消散項を持つ非線形梁方程式の解の大域挙動 (2016)
  • Author(s): 竹田寛志
  • Organizer: 第4回室蘭連続講演会
  • Place of Presentation: 室蘭工業大学
  • Year and Date: 2016-01-07
  • Invited
• [Presentation] Unconditional global well-posedness for a nonlinear damped beam equation and the decay property of the solution (2015)
  • Author(s): H. Takeda
  • Organizer: Korea-Japan Workshop on Nonlinear PDEs
  • Place of Presentation: Ulsan National Institute of Science and Technology
  • Year and Date: 2015-11-30
  • Int'l Joint Research / Invited
• [Presentation] 奇関数を初期値とする非線形消散型梁方程式の解の挙動 (2015)
  • Author(s): 竹田寛志
  • Organizer: 第5回佐賀FIT微分方程式勉強会
  • Place of Presentation: 佐賀大学
  • Year and Date: 2015-09-08