Study on stationary patterns and dynamics of parabolic partial differential equations arising in chemistry and biology

• Project/Area Number: 21740116
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Global analysis
• Research Institution: Tokyo Institute of Technology
• Principal Investigator: MIYAMOTO Yasuhito 東京工業大学, 大学院・理工学研究科, 助教 (90374743)
• Project Period (FY): 2009 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000); Fiscal Year 2011: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2010: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2009: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 楕円型方程式 / 放物型方程式 / 非線形現象 / 大域的分岐構造 / 非線形楕円型方程式 / 対称性破壊分岐 / 放物型方程式系 / Chafee-Infante型問題 / Gierer-Meinhardt系 / ノイマン問題 / 第2固有値 / フェーズフィールドモデル / 大域的な分岐解の枝 / ホットスポット予想

Research Abstract

We studied the number and the Morse indices of the solutions of nonlinear elliptic partial differential equations arising in chemistry and biology, and clarified the solution structures. We also made a detailed analysis of a peak moving along the boundary in Gierer-Meinhardt system.

Research Products

• [Journal Article] Dynamics of boundary spike layer (2012)
  • Author(s): S.-I. Ei, K. Ikeda, and Y. Miyamoto
  • Journal Title: Communications on Pure and Applied Analysis Volume: 11 Pages: 115-145
  • Peer Reviewed
• [Journal Article] Asymptotic transversality and symmetry breaking bifurcation from concentrating solutions, Annales de l' Institute Henri Poincare (2012)
  • Author(s): Y. Miyamoto
  • Journal Title: Analyse Non Lin' eaire Volume: 29 Pages: 59-81
  • Peer Reviewed
• [Journal Article] Asymptotic transversality and symmetry breaking bifurcation from boundary concentrating solutions (2012)
  • Author(s): Yasuhito Miyamoto
  • Journal Title: Annales de l'Iinstitut Henri Poincare, Analyse Non Lineaire Volume: 29 Issue: 1 Pages: 59-81
  • DOI: 10.1016/j.anihpc.2011.09.003
  • Peer Reviewed
• [Journal Article] Dynamics of a boundary spike for the shadow Gierer-Meinhardt system (2012)
  • Author(s): S.-E.Ei, K.Ikeda, Y.Miyamoto
  • Journal Title: Communications on Pure and Applied Analysis Volume: 11 Issue: 1 Pages: 115-145
  • DOI: 10.3934/cpaa.2012.11.115
  • NAID: 120005324438
  • Peer Reviewed
• [Journal Article] Global branches of sign-changing solutions to a semilinear Dirichlet problem in a disk (2011)
  • Author(s): Y, Miyamoto
  • Journal Title: Advances in Differential Equations Volume: 16 Pages: 747-773
  • Peer Reviewed
• [Journal Article] Global bifurcation and stable two-phase separation for a phase field model in a disk (2011)
  • Author(s): Y. Miyamoto
  • Journal Title: Discrete and Continuous Dynamical Systems Volume: 30 Pages: 791-806
  • Peer Reviewed
• [Journal Article] No degeneracy of the second bifurcating branches for the Chafee-Infante problem on a planar symmetric domain (2011)
  • Author(s): Y. Miyamoto
  • Journal Title: Proceedings of the American Mathematical Society Volume: 139 Pages: 975-984
  • Peer Reviewed
• [Journal Article] Global bifurcation and stable two-phase separation for a phase field model in a disk (2011)
  • Author(s): Yasuhito Miyamoto
  • Journal Title: Discrete and continuous dynamical systems Volume: 30 Issue: 3 Pages: 791-806
  • DOI: 10.3934/dcds.2011.30.791
  • Peer Reviewed
• [Journal Article] Global branches from the second eigenvalue for a semilinear Neumann problem in a ball (2011)
  • Author(s): Yasuhito Miyamoto
  • Journal Title: Journal of Differential Equations Volume: 249 Pages: 1853-1870
  • Peer Reviewed
• [Journal Article] Nondegeneracy of the second bifurcating branches for the Chafee-Infante problem on a planar symmetric domain (2011)
  • Author(s): Yasuhito Miyamoto
  • Journal Title: Proceedings of the American Mathematical Society Volume: 139 Pages: 975-984
  • Peer Reviewed
• [Journal Article] Global branches from the second eigenvalue for a semilinear Neumann problem in a disk (2010)
  • Author(s): Y. Miyamoto
  • Journal Title: Journal of Differential Equations Volume: 249 Pages: 1853-1870
  • Peer Reviewed
• [Journal Article] The "hot spots" conjecture for a certain class of planar convex domains (2009)
  • Author(s): Y. Miyamoto
  • Journal Title: Journal of Mathematical Physics Volume: 50
  • Peer Reviewed
• [Journal Article] Non-existence of a secondary bifurcation point for a semilinear elliptic problem in the presence of symmetry (2009)
  • Author(s): Y. Miyamoto
  • Journal Title: Journal of Mathematical Analysis and Applications Volume: 357 Pages: 89-97
  • Peer Reviewed
• [Journal Article] Global branches of non-radially symmetric solutions to a semilinear Neumann problem in a disk (2009)
  • Author(s): Y. Miyamoto
  • Journal Title: Journal of Functional Analysis Volume: 256 Pages: 747-776
  • Peer Reviewed
• [Journal Article] Global branches of non-radially symmetric solutions to a semilinear Neumann problem in a disk (2009)
  • Author(s): Yasuhito Miyamoto
  • Journal Title: Journal of Functional Analysis 256 Pages: 747-776
  • Peer Reviewed
• [Journal Article] Non-existence of a secondary bifurcation point for a semilinear elliptic problem in the presence of symmetry (2009)
  • Author(s): Yasuhito Miyamoto
  • Journal Title: Journal of Mathematical Analysis and Applications 357 Pages: 89-97
  • Peer Reviewed
• [Journal Article] The "hot spots" conjecture for a certain class of planar convex domains (2009)
  • Author(s): Yasuhito Miyamoto
  • Journal Title: Journal of Mathematical Physics 50
  • Peer Reviewed
• [Presentation] 境界に凝集する解からの対称性破壊分岐について (2012)
  • Author(s): 宮本安人
  • Organizer: 日本数学会2012年度年会
  • Place of Presentation: 東京理科大学
  • Year and Date: 2012-03-27
• [Presentation] 境界に凝集する解からの対称性破壊分岐について (2012)
  • Author(s): 宮本安人
  • Organizer: 日本数学会2012年度年会(函数方程式分科会)
  • Place of Presentation: 東京理科大学
  • Year and Date: 2012-03-27
• [Presentation] Stable patterns and solutions with Morse index one (2011)
  • Author(s): Y. Miyamoto
  • Organizer: 日仏合同会議Modeling and Analysis in the Life Sciences
  • Place of Presentation: 東京大学
  • Year and Date: 2011-11-29
• [Presentation] 円板領域上のフェーズフィールドモデルの安定解の形状と大域的分岐構造について (2011)
  • Author(s): 宮本安人
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 信州大学
  • Year and Date: 2011-09-28
• [Presentation] 円板領域上のフェーズフィールドモデルの安定解の形状と大域的分岐構造について (2011)
  • Author(s): 宮本安人
  • Organizer: 日本数学会秋期総合分科会(函数方程式分科会)
  • Place of Presentation: 信州大学
  • Year and Date: 2011-09-28
• [Presentation] Stable patterns and solutions with Morse index one (2011)
  • Author(s): Y. Miyamoto
  • Organizer: 4^<th> MSJ-SI
  • Place of Presentation: 九州大学
  • Year and Date: 2011-09-12
• [Presentation] Stable patterns and solutions with Morse index one (2011)
  • Author(s): Yasuhito Miyamoto
  • Organizer: 4^<th> MSJ-SI, Nonlinear dynamics in Partial Differential Equations
  • Place of Presentation: 九州大学医学部百年講堂
  • Year and Date: 2011-09-12
• [Presentation] Nonradial maximizers for a Henon type problem and symmetry breaking bifurcations for a Liouville-Gelfand equation with a vanishing coefficient (2011)
  • Author(s): Y. Miyamoto
  • Organizer: 2^<nd> Italian-Japanese Workshop Geometric Properties for Parabolic and Elliptic PDE's
  • Place of Presentation: コルトナ(イタリア)
  • Year and Date: 2011-06-23
• [Presentation] Nonradial maximizers for a H'enon type problem and symmetry breaking bifurcations for a Liouvulle-Gel'fand equation with a vanishing coefficient (2011)
  • Author(s): Yasuhito Miyamoto
  • Organizer: 2nd Italian-Japanese Workshop, Geometric properties for parabolic and elliptic PDE's
  • Place of Presentation: コルトナ(イタリア)
  • Year and Date: 2011-06-23
• [Presentation] 球領域におけるNeumann問題の第2固有値からの分岐について (2011)
  • Author(s): 宮本安人
  • Organizer: 日本数学会(函数方程式分科会)
  • Place of Presentation: 早稲田大学
  • Year and Date: 2011-03-21
• [Presentation] 円板領域上のフェーズフィールドモデルの安定解の形状と大域的分岐構造について (2011)
  • Author(s): 宮本安人
  • Organizer: 日本数学会(函数方程式分科会)
  • Place of Presentation: 早稲田大学
  • Year and Date: 2011-03-21
• [Presentation] The "hot spots" conjecture for a certain class of planar convex domains (2010)
  • Author(s): Y. Miyamoto
  • Organizer: International Workshop on PDE : Concentration and Related Topics in Nonlinear Problems
  • Place of Presentation: 東北大学
  • Year and Date: 2010-11-22
• [Presentation] The "hot spots" conjecture for a certain class of planar convex domains (2010)
  • Author(s): 宮本安人
  • Organizer: International Workshop on PDE Conference and Related Topics in Nonlinear Problems
  • Place of Presentation: 東北大学
  • Year and Date: 2010-11-22
• [Presentation] 2次元対象領域におけるChafee-Infante問題の分岐解の非退化性について (2010)
  • Author(s): 宮本安人
  • 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] Nondegeneracy of the second bifurcating branches for the Chafee-Infante problem on a planar symmetric domain (2010)
  • Author(s): 宮本安人
  • Organizer: 日本数学会(函数方程式分科会)
  • Place of Presentation: 名古屋大学
  • Year and Date: 2010-09-24
• [Presentation] Stable patterns and solutions with Morse index one (2010)
  • Author(s): Y. Miyamoto
  • Organizer: 35^<th>偏微分方程式札幌シンポジウム
  • Place of Presentation: 北海道大学
  • Year and Date: 2010-08-23
• [Presentation] Stable patterns and solutions with Morse index one (2010)
  • Author(s): 宮本安人
  • Organizer: The 35^<th> Sapporo symposium on Partial Differential Equations
  • Place of Presentation: 北海道大学
  • Year and Date: 2010-08-23
• [Presentation] 対称性を持たない2次元凸領域におけるホットスポット予想について (2010)
  • Author(s): 宮本安人
  • Organizer: 日本数学会2011年度年会
  • Place of Presentation: 慶應義塾大学
  • Year and Date: 2010-03-25
• [Presentation] 対称性を持たない2次元凸領域におけるホットスポット予想について (2010)
  • Author(s): 宮本安人
  • Organizer: 日本数学会(函数方程式分科会)
  • Place of Presentation: 慶應義塾大学
  • Year and Date: 2010-03-25
• [Presentation] 安定パターンの形状と非線形ホットスポット予想 (2009)
  • Author(s): 宮本安人
  • Organizer: 現象数理学:冬の学校「パターンダイナミクス1-2-3」
  • Place of Presentation: 明治大学紫紺館
  • Year and Date: 2009-12-10
• [Presentation] Stable patterns for shadow systems and a nonlinear hot spots conjecture (2009)
  • Author(s): Y. Miyamoto
  • Organizer: 2^<nd> Chile-Japan Workshop on Nonlinear Elliptic and Parabolic PDEs
  • Place of Presentation: 明治大学
  • Year and Date: 2009-12-03
• [Presentation] 対称性を持たない2次元凸領域におけるホットスポット予想について (2009)
  • Author(s): 宮本安人
  • Organizer: 杜の学校2009・盛岡応用数学小研究集会
  • Place of Presentation: 岩手大学
  • Year and Date: 2009-11-15
• [Presentation] 外力項と減衰項のあるシュレディンガー方程式の孤立波の存在について (2009)
  • Author(s): 宮本安人
  • Organizer: 最近の解析学の話題とその周辺から
  • Place of Presentation: 広島大学
  • Year and Date: 2009-10-10
• [Presentation] Neumann問題の2次分岐点の非存在について (2009)
  • Author(s): 宮本安人
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 大阪大学
  • Year and Date: 2009-09-25
• [Presentation] Neumann 問題の2次分岐点の非存在について (2009)
  • Author(s): 宮本安人
  • Organizer: 日本数学会(函数方程式分科会)
  • Place of Presentation: 大阪大学
  • Year and Date: 2009-09-25
• [Presentation] Asymptotic behavior of solutions for the shadow reaction-diffusion system with the nonlinearity of the FitzHugh-Nagumo type (2009)
  • Author(s): 宮本安人
  • Organizer: 第31回発展方程式若手セミナー
  • Place of Presentation: 国立女性教育会館
  • Year and Date: 2009-09-01
• [Presentation] Stable patterns for shadow systems and a nonlinear "hot spots" conjecture (2009)
  • Author(s): Y. Miyamoto
  • Organizer: 1^<st> Italian-Japanese workshop Geometric Properties for Parabolic and Elliptic PDE's
  • Place of Presentation: 東北大学
  • Year and Date: 2009-06-16
• [Presentation] Stable patterns for shadow systems and a nonlinear "hot spots" conjecture (2009)
  • Author(s): 宮本安人
  • Organizer: 1st Italian-Japanese workshop on geometric properties for parabolic and elliptic PDE's
  • Place of Presentation: 東北大学
  • Year and Date: 2009-06-16
• [Presentation] $\Delta u+\lambda f(u)=0$の Neumann 問題の分岐について (2009)
  • Author(s): 宮本安人
  • Organizer: 京都駅前セミナー
  • Place of Presentation: キャンパスプラザ京都
  • Year and Date: 2009-05-22
• [Presentation] $\Delta u+\lambda f(u)=0$の Neumann 問題の分岐について (2009)
  • Author(s): 宮本安人
  • Organizer: 解析ゼミ
  • Place of Presentation: 埼玉大学
  • Year and Date: 2009-05-19
• [Presentation] Neumann 問題の分岐とその周辺1&2 (2009)
  • Author(s): 宮本安人
  • Organizer: 札幌NSCセミナー
  • Place of Presentation: 北海道大学電子科学研究所