Investigations on structures of the solutions to nonlinear elliptic partial differential equations on the sphere.

• Project/Area Number: 19540224
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Global analysis
• Research Institution: Osaka Prefecture University
• Principal Investigator: KABEYA Yoshitsugu Osaka Prefecture University, 工学研究科, 教授 (70252757)
• Project Period (FY): 2007 – 2010
• Project Status: Completed (Fiscal Year 2010)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2010: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2009: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2008: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2007: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: 分岐理論 / 不完全分岐 / 球面上の微分方程式 / 楕円型偏微分方程式 / ホットスポット / 楕円型方程式 / 非線形楕円型方程式

Research Abstract

In this research period, I investigated the imperfect bifurcations on azimuthal solutions to a nonlinear elliptic equation when the domain covers almost the whole sphere and showed how the structure of bifurcations asymptotically looked like. As for the cases including non-azimuthal solutions, I obtained some results just before the end of the term. Also, as a related topics, I studies the behavior of solutions to a heat equation with a potential and showed that the time global behavior was governed by solutions to the stationary problem.

Research Products

• [Journal Article] Hot spots for the two dimensional heat equation with a rapidly decaying negative potential (2011)
  • Author(s): K. Ishige, Y. Kabeya
  • Journal Title: Discrete and Continuous Dynamical Systems Series S 4 Pages: 833-849
  • Peer Reviewed
• [Journal Article] Hot spots for the two dimensional heat equation with a rapidly decaying negative potential (2011)
  • Author(s): K.Ishige, Y.Kabeya
  • Journal Title: Discrete and Continuous Dynamical Systems Series S Volume: 4 Pages: 833-849
  • Peer Reviewed
• [Journal Article] Topics on the movement of hot spots for the heat equation with a potential (2010)
  • Author(s): K. Ishige, Y. Kabeya
  • Journal Title: 京都大学数理解析研究所講究録 1702 Pages: 71-87
• [Journal Article] Imperfect bifurcations in nonlinear elliptic equations on spherical caps (2010)
  • Author(s): C. Bandle, Y. Kabeya, H. Ninomiya
  • Journal Title: Communications on Pure and Applied Analysis 9 Pages: 1189-1208
  • Peer Reviewed
• [Journal Article] Imperfect bifurcations in nonlinear elliptic equations on spherical caps (2010)
  • Author(s): C.Bandle, Y.Kabeya, H.Ninomiya
  • Journal Title: Communications on Pure and Applied Analysis Volume: 9 Pages: 1189-1208
  • Peer Reviewed
• [Journal Article] Topics on the movement of hot spots for the heat equation with a potential (2010)
  • Author(s): K.Ishige, Y.Kabeya
  • Journal Title: 数理解析研究所考究録 Volume: 1702 Pages: 71-87
• [Journal Article] Hot spots for the heat equation with a rapidly decaying negative potential (2009)
  • Author(s): K. Ishige, Y. Kabeya
  • Journal Title: Advances in Differential Equations 14 Pages: 643-662
  • Peer Reviewed
• [Journal Article] Hot spots for the heat equation with a rapidly decaying negative potential (2009)
  • Author(s): K.Ishige, Y.Kabeya
  • Journal Title: Advances in Differential Equations 14 Pages: 643-662
  • Peer Reviewed
• [Journal Article] Large time behaviors of hot spots for the heat equation with a potential (2008)
  • Author(s): K. Ishige, Y. Kabeya
  • Journal Title: Journal of Differential Equations 244 Pages: 2934-2962
  • Peer Reviewed
• [Journal Article] Solutions having boundary layers to a nonlinear elliptic equation on a spherical cap (2008)
  • Author(s): C. Bandle, Y. Kabeya, H. Ninomiya
  • Journal Title: 京都大学数理解析研究所講究録 1588 Pages: 74-86
• [Journal Article] Large time behaviors of hot spots for the heat equation with a potential (2008)
  • Author(s): K. Ishige, Y. Kabeya
  • Journal Title: J.Differential Equations 244 Pages: 2934-2962
  • Peer Reviewed
• [Journal Article] Decay rates of the derivatives of the solutions of the heat equations in the exterior domain of a ball (2007)
  • Author(s): Ishige, Y. Kabeya
  • Journal Title: Journal of Mathematical Society of Japan 59 Pages: 861-898
  • NAID: 10019540137
  • Peer Reviewed
• [Journal Article] Decay rates of the derivatives of the solutions of the heat equations in the exterior domain of a ball (2007)
  • Author(s): K. Ishige, Y. Kabeya
  • Journal Title: Journal of the Mathematical Society of Japan 59 Pages: 861-898
  • NAID: 10019540137
  • Peer Reviewed
• [Journal Article] Perturbation of structures of radial solutions to elliptic equations (2007)
  • Author(s): Y. Kabeya
  • Journal Title: Proceedings of the 14^<th> MSJ International Research Institute "Asymptotics Analysis and Singularity" Pages: 589-600
  • Peer Reviewed
• [Journal Article] Fundamental properties of solutions to a scalar-field type equation on the unit sphere (2007)
  • Author(s): Y. Kabeya, H. Ninomiya
  • Journal Title: Proceedings of the Conference held at Hsinchu, Taiwan Pages: 147-150
• [Journal Article] Decay rates of the derivative of the solutions to the heat equations and related topics (2007)
  • Author(s): K. Ishige, Y. Kabeya
  • Journal Title: 数理解析研究所講究録 1528 Pages: 12-35
• [Journal Article] Decay rates of the derivatives of the solutions of the heat equations in the exterior domain of a ball (2007)
  • Author(s): K. Ishige and Y. Kabeya
  • Journal Title: Journal of the Mathematical Society of Japan 59 Pages: 861-898
  • NAID: 10019540137
  • Peer Reviewed
• [Journal Article] Fundamental properties of solutions to a scalar-field type equation on the unit sphere (2007)
  • Author(s): Y. Kabeya and H. Ninomiya
  • Journal Title: Recent Advances in Nonlinear Analysis Pages: 147-160
• [Presentation] 逆二次で減衰する項を持つ線形熱方程式に対する解の減衰表と挙動 (2011)
  • Author(s): 壁谷喜継
  • Organizer: 別府解析セミナー
  • Place of Presentation: 別府市
  • Year and Date: 2011-01-17
• [Presentation] 逆二次で減衰する項を持つ線形熱方程式に対する解の減衰評価と挙動 (2011)
  • Author(s): 壁谷喜継
  • Organizer: 別府解析セミナー
  • Place of Presentation: 別府市
  • Year and Date: 2011-01-17
• [Presentation] Large time behavior of hot spots to a linear heat equation having a non-constant zeroth oder term (2010)
  • Author(s): 壁谷喜継
  • Organizer: 数理解析研究所研究集会
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2010-10-28
• [Presentation] Large time behavior of hot spots to a linear heat equation having a non-constant zeroth order term (2010)
  • Author(s): Y.Kabeya
  • Organizer: 数理解析研究所研究集会
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2010-10-28
• [Presentation] 逆二次で減衰する項を持つ線形熱方程式に対する解の最大値の挙動 (2010)
  • Author(s): 壁谷喜継
  • Organizer: 洞爺湖解析セミナー
  • Place of Presentation: 洞爺湖
  • Year and Date: 2010-09-27
• [Presentation] Large time behavior of hot spots to a linear heat equation having a non-constant zeroth oder term (2010)
  • Author(s): 壁谷喜継
  • Organizer: 第六回非線形現象の諸問題
  • Place of Presentation: 山口大学
  • Year and Date: 2010-09-18
• [Presentation] Large time behaviors of hot spots to a heat equation with a potential term (2010)
  • Author(s): 壁谷喜継
  • Organizer: 8^<th> AIMS International Conference on Partial Differential Equations
  • Place of Presentation: Dresden, Germany
  • Year and Date: 2010-05-22
• [Presentation] Large behavior of hot spots to a linear heat equation with a decaying potential (2009)
  • Author(s): 壁谷喜継
  • Organizer: 数理解析研究所研究集会
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2009-11-18
• [Presentation] Large behavior of hot spots to a linear heat equation with a decaying potential (2009)
  • Author(s): Y.Kabeya
  • Organizer: 京都大学数理解析研究所研究集会
  • Place of Presentation: 京都大学
  • Year and Date: 2009-11-18
• [Presentation] Nonlinear elliptic equations on a spherical domain and related topics (2009)
  • Author(s): 壁谷喜継
  • Organizer: 34^<th> Sapporo Symposium on Partial Differential Equations
  • Place of Presentation: 北海道大学
  • Year and Date: 2009-08-24
• [Presentation] Nonlinear elliptic equations on a spherical domain and related topics (2009)
  • Author(s): Y.Kabeya
  • Organizer: 34th Sapporo Symposium on Partial Differential Equations
  • Place of Presentation: 北海道大学
  • Year and Date: 2009-08-24
• [Presentation] The behavior of hot spots to a linear heat equation with a potential (2009)
  • Author(s): Y.Kabeya
  • Organizer: International Workshop on Nonlinear PDE and Applications at Pusan National University
  • Place of Presentation: 韓国釜山大学
  • Year and Date: 2009-06-29
• [Presentation] Bifurcation diagrams on a nonlinear elliptic problem on a spherical cap (2009)
  • Author(s): 壁谷喜継
  • Organizer: 6^<th> International Conference on Elliptic and Parabolic Partial Differential Equations
  • Place of Presentation: Gaeta, Italy
  • Year and Date: 2009-05-26
• [Presentation] Bifurcation diagrams on a nonlinear elliptic problem on a spherical cap (2009)
  • Author(s): Y.Kabeya
  • Organizer: International Conference on Elliptic and Parabolic Partial Differential Equations
  • Place of Presentation: Gaeta, Italy
  • Year and Date: 2009-05-26
• [Presentation] Bifurcation diagrams for a nonlinear elliptic problem on a spherical cap (2009)
  • Author(s): 壁谷喜継
  • Organizer: Workshop on Viscosity Solutions and Related Topics
  • Place of Presentation: 埼玉大学サテライトオフィス, 東京都千代田区
  • Year and Date: 2009-01-30
• [Presentation] Bifurcation diagrams for a nonlinear elliptic problem on a spherical cap (2009)
  • Author(s): 壁谷 喜継
  • Organizer: Workshop on Viscosity Solutions and Related Topics
  • Place of Presentation: 埼玉大学東京駅サテライトキャンパス
  • Year and Date: 2009-01-30
• [Presentation] Large behavior of hot spots to a linear heat equation with a decaying potential (2009)
  • Author(s): Y. Kabeya, K. Ishige
  • Organizer: 京都大学数理解析研究所研究集会
  • Place of Presentation: Kyoto, Japan
• [Presentation] Nonlinear elliptic equations on a spherical domain and related topics (2009)
  • Author(s): Y. Kabeya, C. Bandle, H. Ninomiya
  • Organizer: At 34th Sapporo Symposium on Partial Differential Equations
  • Place of Presentation: Sapporo, Japan
• [Presentation] The behavior of hot spots to a linear heat equation with a potential (2009)
  • Author(s): Y. Kabeya, K. Ishige
  • Organizer: At International Workshop on Nonlinear PDE and Applications at Pusan National University
  • Place of Presentation: Busan, Korea
• [Presentation] Bifurcation diagrams on a nonlinear elliptic problem on a spherical cap (2009)
  • Author(s): Y. Kabeya, C. Bandle, H. Ninomiya
  • Organizer: At 6^<th> International Conference on Elliptic and Parabolic Partial Differential Equations
  • Place of Presentation: Gaeta, Italy
• [Presentation] Imperfect bifurcations arising in a nonlinear elliptic problem on the unit sphere (2008)
  • Author(s): 壁谷喜継
  • Organizer: Workshop on the stationary problems in the nonlinear partial differential equations
  • Place of Presentation: 神戸大学工学部
  • Year and Date: 2008-12-10
• [Presentation] Imperfect bifurcations arising in a nonlinear elliptic problem on the unitsphere (2008)
  • Author(s): 壁谷 喜継
  • Organizer: ワークショップ「非線形偏微分方程式における定常問題」
  • Place of Presentation: 神戸大学
  • Year and Date: 2008-12-09
• [Presentation] Imperfect bifurcations arising in a nonlinear elliptic problem on the unit sphere (2008)
  • Author(s): Y. Kabeya
  • Organizer: At Workshop on the stationary problems in the nonlinear partial differential equations
  • Place of Presentation: Kobe
• [Presentation] Imperfect bifurcations arising in a nonlinear elliptic equation on a spherical cap (2007)
  • Author(s): Y. Kabeya
  • Organizer: Recent Advances on Nonlinear Parabolic and Elliptic Differential Equations
  • Place of Presentation: 龍谷大学瀬田キャンパス
  • Year and Date: 2007-12-05
• [Presentation] Solutions having boundary layers to a nonlinear elliptic equation on a spherical cap (2007)
  • Author(s): Y. Kabeya
  • Organizer: 数理解析研究所研究集会「非線形発展方程式と現象の数理」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2007-10-22
• [Presentation] Imperfect bifurcations arising in a nonlinear elliptic equation on a spherical cap (2007)
  • Author(s): Y. Kabeya
  • Organizer: Recent Advances on Nonlinear Parabolic and Elliptic Differential Equations
  • Place of Presentation: Ryukoku University, Otsu, Japan
• [Book] フーリエ解析と偏微分方程式入門 (2010)
  • Author(s): 壁谷喜継
  • Total Pages: 135
  • Publisher: 共立出版
• [Remarks] ホームページ等
  • URL: http://www.ms.osakafu-u.ac.jp/~kabeya/index.html