Theory of characterization and existence for entire solutions to reaction-diffusion equations in the multi-dimensional space.

• Project/Area Number: 21654025
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Single-year Grants
• Research Field: Global analysis
• Research Institution: Ryukoku University
• Principal Investigator: MORITA Yoshihisa 龍谷大学, 理工学部, 教授 (10192783)
• Co-Investigator(Kenkyū-buntansha): NINOMIYA Hirokazu 明治大学, 理工学部, 教授 (90251610)
• Project Period (FY): 2009 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥3,200,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥300,000); Fiscal Year 2011: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2010: ¥900,000 (Direct Cost: ¥900,000); Fiscal Year 2009: ¥1,000,000 (Direct Cost: ¥1,000,000)
• Keywords: 反応拡散方程式 / 無限次元力学系 / 進行波 / 全域解 / 分岐理論 / 安定性 / 非線形楕円型方程式 / ヘテロクリニック軌道 / 伝播速度 / FizHugh-Nagumo方程式 / 極限方程式 / 楕円型方程式 / 進行波解 / スポット進行波解 / Lotka-Volterra拡散競争系

Research Abstract

The reaction-diffusion equation is widely accepted as a model to describe pattern formations and propagations of some spatial pattern. In this research we study the equation in order to provide a new existence theory for entire solutions and a characterization of the spatial pattern of the solutions. In consequence we mathematically prove the existence and stability of new type of solutions to some class of reaction-diffusion equations.

Research Products

• [Journal Article] Spectrum comparison for a conserved reaction-diffusion system with a variational property (2012)
  • Author(s): Y. Morita
  • Journal Title: Journal of Applied. Analysis and Computation Volume: Vol.2 Pages: 57-71
  • Peer Reviewed
• [Journal Article] Spectrtum comparison for a conserved reaction-diffusion system with a variational property (2012)
  • Author(s): Y.Morita
  • Journal Title: Journal of Applied. Analysis aad Computation Volume: 2 Pages: 57-71
  • Peer Reviewed
• [Journal Article] Stack ed Fronts for Cooperative Systems wi th Equal Diffusion Coefficients (2011)
  • Author(s): M. Iida, R. Lui, H. Ninomiya
  • Journal Title: SIAM J. Math. Anal Volume: Vol.43 Pages: 1369-1389
  • Peer Reviewed
• [Journal Article] Fast reactio n limit of a three-component reactio n-diffusion system (2011)
  • Author(s): H. Murakawa, H. Ninomiya
  • Journal Title: Journal of Mathe matical Analysis and Applications Volume: Vol.379 Pages: 150-170
  • Peer Reviewed
• [Journal Article] Fast reaction limit of a three-component reaction-difiusion system (2011)
  • Author(s): H.Murakawa
  • Journal Title: Journal of Mathematical Analysis and Aplication Volume: 379 Pages: 150-170
  • Peer Reviewed
• [Journal Article] Stacked Fronts for Cooperative Systems with Equal Diffusion Coefficien (2011)
  • Author(s): M.Iida
  • Journal Title: SIAM J.Math.Anal. Volume: 43 Pages: 1369-1389
  • Peer Reviewed
• [Journal Article] Fast reaction limit of a three-component reaction-diffusion system (2011)
  • Author(s): H.Murakawa, H.Ninomiya
  • Journal Title: J.Mathematical Analysis and Applications Volume: Vol.379 Pages: 150-170
  • Peer Reviewed
• [Journal Article] Imperfect bifurcations in nonlinear elliptic equations on spherical cap s (2010)
  • Author(s): C. Bandle, Y. Kabeya and H. Ninomiya
  • Journal Title: Communications on Pure and Applie d Analysis Volume: Vol.9No.5 Pages: 1189-1208
  • Peer Reviewed
• [Journal Article] Existence and uniqueness of stabili zed propagating wave segments in wav e front interaction model (2010)
  • Author(s): J. S. Guo, H. Ninomiya, J. C. Tsai
  • Journal Title: Physica D : Nonlinear Phenomena Volume: Vol.239No.3-4 Pages: 230-239
  • Peer Reviewed
• [Journal Article] Traveling wa ve solutions and entire solutions to reaction-diffusion equations (2010)
  • Author(s): Y. Morita, H. Ninomiya
  • Journal Title: SUGAKU Expositions Volume: Vol.23No.2 Pages: 213-233
  • NAID: 130004558838
• [Journal Article] Stability and b ifurcation of nonconstant solutions to a reaction-diffusion system with conservation of a mass (2010)
  • Author(s): Y. Morita, T. Ogawa
  • Journal Title: Nonlinearity Volume: Vol.23 Pages: 1387-1411
  • Peer Reviewed
• [Journal Article] Traveling wave solutions and entire solutions to reaction-diffusion equations (2010)
  • Author(s): Y.Morita, H.Ninomiya
  • Journal Title: Sugaku Expositions Volume: Vol.23 Pages: 213-233
  • NAID: 130004558838
• [Journal Article] Stability and bifurcation of nonconstant solutions to a reaction-diffusion system with conservation of a mass (2010)
  • Author(s): Y.Morita, T.Ogawa
  • Journal Title: Nonlinearity Volume: Vol.23 Pages: 1387-1411
  • 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: Vol.9 Pages: 1189-1208
  • Peer Reviewed
• [Journal Article] Existence and uniqueness of stabilized propagating wave segments in wave front interaction model (2010)
  • Author(s): J.-S.Guo, H.Ninomiya, J.-C.Tsai
  • Journal Title: Physica D : Nonlinear Phenomena 239 Pages: 230-239
  • Peer Reviewed
• [Journal Article] An enti re solution to the Lotka-Volterra co mpetition-diffusion equations (2009)
  • Author(s): Y. Morita and K. Tachibana
  • Journal Title: SIAM J. Math. Anal Volume: Vol.40 Pages: 567-584
  • Peer Reviewed
• [Journal Article] An entire solution to the Lotka-Volterra competition-diffusion equations (2009)
  • Author(s): Y.Morita, K.Tachibana
  • Journal Title: SIAM J.Math.Anal. 40 Pages: 567-584
  • Peer Reviewed
• [Presentation] Traveling spots and rotating spirals of the wave front interaction model (2012)
  • Author(s): H. Ninomiya
  • Organizer: 20th Annual Workshop of Differential Equations
  • Place of Presentation: Tamkang University, Taipei
• [Presentation] Localized patterns in a reaction-diffusion system with conservation of a mass (2011)
  • Author(s): Y.Morita
  • Organizer: International Workshop on Far-From-Equilibrium Dynamics
  • Place of Presentation: RIMS, Kyoto University(Kyoto)
  • Year and Date: 2011-01-07
• [Presentation] The traveling spots and rotating waves of the wave front interaction model (2011)
  • Author(s): H. Ninomiya
  • Organizer: Reaction-Diffusion Systems in Mathematics and the Life Sciences : A conference in honor of Jacques Demongeot and Masayasu Mimura
  • Place of Presentation: University of Montpellier II, France
• [Presentation] Reaction-diffusion approximation and related topics (2011)
  • Author(s): H. Ninomiya
  • Organizer: One Forum, Two Cities : Aspect of Nonlinear PDE
  • Place of Presentation: National Taiwan University, Taipei
• [Presentation] Monostable-type traveling waves of a bistable reaction-diffusion equation in the multi-dimensional space (2011)
  • Author(s): 森田善久
  • Organizer: RIMS研究集会「非線形現象に現れる界面運動の数理解析・数値解析」
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] Non-planar traveling waves and entire solutions of Allen-Cahn equations (2011)
  • Author(s): H. Ninomiya
  • Organizer: Asymptotic Dynamics Driven by Solitons and Traveling Fronts in Nonlinear PDE
  • Place of Presentation: Universidad de Chile, Santiago
• [Presentation] Non-planar traveling waves of reaction-diffusion equations (2011)
  • Author(s): H. Ninomiya
  • Organizer: 19th Workshop on Differential Equations
  • Place of Presentation: National Cheng Kung University, Taiwan
• [Presentation] Localized patterns in a reaction-diffusion system with conservation of a mass (2011)
  • Author(s): Y. Morita
  • Organizer: Far-from-Equilibrium Dynamics
  • Place of Presentation: RIMS and Shiran-Kaikan Kyoto
• [Presentation] Non-planar traveling waves of reaction-diffusion equations (2010)
  • Author(s): H. Ninomiya
  • Organizer: Workshop on Pattern formation in Chemical and Biological Systems
  • Place of Presentation: Eotvos University, Budapest, Hungry
• [Presentation] A reaction-diffusion system with conservation of a mass (2010)
  • Author(s): Y. Morita
  • Organizer: The Third China-Japan Colloquium of Mathematical Biology
  • Place of Presentation: Sea North Green Garden-Beijing, Beijing, China
• [Presentation] Traveling waves in the various shapes (2010)
  • Author(s): H. Ninomiya
  • Organizer: The 8th AIMS Conference on Dynamical Systems, Differential Equations and Applications
  • Place of Presentation: Dresden, Germany
• [Presentation] Stability and bifurcation of solutions to a mass-conserved reaction-diffusion system (2010)
  • Author(s): Y. Morita
  • Organizer: The 8th AIMS Conference on Dynamical Systems, Differential Equations and Applications
  • Place of Presentation: Dresden University of Technology Dresden Germany
• [Presentation] Stability and bifurcation of solutions to a reaction-diffusion system with conservation of a mass (2010)
  • Author(s): Y. Morita
  • Organizer: The 10th International Workshop on Differential Equations in Memory of the late Professor Jeong Seon Baek
  • Place of Presentation: Chonnam National University, Gwangju, Korea
• [Presentation] On the diffusionpreventing blowup of reactiondiffusion equations (2010)
  • Author(s): H. Ninomiya
  • Organizer: Spatio-Temporal Patterns from Mathematics to Biomedical Applications
  • Place of Presentation: Japanese French Meeting ReaDiLab, ARCHAMPS, FRANCE
• [Presentation] Traveling segments and rotating waves of wave front interaction model (2010)
  • Author(s): 二宮弘和
  • Organizer: 第27回九州における偏微分方程式研究集会
  • Place of Presentation: 九州大学
• [Presentation] Some entire solutions of reaction-diffusion equations (2009)
  • Author(s): H. Ninomiya
  • Organizer: The Second Chile-Japan Workshop on Nonlinear Elliptic and Parabolic PDEs
  • Place of Presentation: Meiji University, Japan
• [Presentation] Propagating waves in wave front interaction model (2009)
  • Author(s): 二宮弘和
  • Organizer: RIMS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
• [Presentation] Traveling waves of a reaction-diffusion equation in the heigher-dimensional space (2009)
  • Author(s): Y. Morita
  • Organizer: Equadiff
  • Place of Presentation: Masaryk University, Brno Czech Republic
• [Presentation] A mass-conserved reaction-diffusion system (2009)
  • Author(s): Y. Morita
  • Organizer: The 2nd International Conference on Reaction-Diffusion Systems and Viscosity Solutions
  • Place of Presentation: Providence University, Taichung, Taiwan
• [Presentation] Blowup in reaction-diffusion equations (2009)
  • Author(s): H. Ninomiya
  • Organizer: 6th European Conference on Elliptic and Parabolic Problems
  • Place of Presentation: Gaeta, Italy
• [Book] Handbook of Differential Equations, Hungary : North-Holland (2009)
  • Author(s): D. Hilhorst, M. Mimura, H. Ninomiya
• [Book] Evolutionary Equations, Vol 5, Handbook of Differential Equations (2009)
  • Author(s): D.Hilhorst, M.Mimura, H.Ninomiya
  • Total Pages: 64
  • Publisher: North-Holland