Concentration phenomena in spatially inhomogeneous nonlinear reaction-diffusion systems

• Project/Area Number: 20740090
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Global analysis
• Research Institution: Tokyo University of Marine Science and Technology
• Principal Investigator: NAKASHIMA Kimie 東京海洋大学, 海洋科学技術研究科, 准教授 (10318800)
• Project Period (FY): 2008 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥4,521,106 (Direct Cost: ¥3,477,774、Indirect Cost: ¥1,043,332); Fiscal Year 2011: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2010: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2009: ¥1,271,106 (Direct Cost: ¥977,774、Indirect Cost: ¥293,332); Fiscal Year 2008: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 非線形現象 / 反応拡散方程式系 / 特異摂動問題 / 遷移層 / 非線形反応拡散系 / スパイク / 空間非一様性 / 微分方程式 / 非線形反応拡散方程式系 / 特異極限 / 界面 / 非線型反応拡散方程式系 / モース指数 / 変分法 / 写像度 / 非線形拡散系 / Allen-Cahn方程式 / 関数方程式論 / 非線型拡散系 / シャドウシステム

Research Abstract

In reaction-diffusion systems, when diffusion coefficients are very small, solutions sometimes form transition layers. In this research, we consider a model in population genetics which is described by a reaction diffusion equation. We will rigorously prove, under certain conditions, that this equation has a unique nontrivial steady-state, it is linearly stable and has transition layers.

Research Products

• [Journal Article] Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domainsDiscrete and Continuous Dynamical (2012)
  • Author(s): Fang Li and Kimie Nakashima
  • Journal Title: Systems- Series A (DCDS-A) Volume: Vol.32 Issue: 4 Pages: 1391-1420
  • DOI: 10.3934/dcds.2012.32.1391
  • Peer Reviewed
• [Journal Article] Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains (2011)
  • Author(s): Fang Li, Kimie Nakashima
  • Journal Title: Discrete and Continuous Dynamical Systems Volume: 32 Pages: 1391-1420
  • Peer Reviewed
• [Journal Article] An Indefinite Nonlinear Diffusion Problem in Population Genetics I (2010)
  • Author(s): Linlin Su, Kimie Nakashima, Wei-MingNi
  • Journal Title: Existence and Limiting Profiles,Discrete and Continuous Dynamical Systems - Series A Volume: 27 Issue: 2 Pages: 617-641
  • DOI: 10.3934/dcds.2010.27.617
  • Peer Reviewed
• [Journal Article] Wei-Ming Ni An Indefinite Nonlinear Diffusion Problem in Population Genetics, I:Existence and Limiting Profiles (2010)
  • Author(s): Linlin Su, Kimie Nakashima
  • Journal Title: Discrete and Continuous Dynamical Systems-Series A 27 Pages: 617-641
  • Peer Reviewed
• [Journal Article] An Indefinite Nonlinear Diffusion Problem in Population Genetics, I : Existence and Limiting Profiles (2010)
  • Author(s): Linlin Su, Kimie Nakashima, Wei-Ming Ni
  • Journal Title: Discrete and Continuous Dynamical Systems-Series A 27(2010) Volume: 27 Pages: 617-641
  • Peer Reviewed
• [Journal Article] Stability from the point of view of diffusion, relaxation and spatial Inhomogeneity (2008)
  • Author(s): Fang Li, Kimie Nakashima andWei-Ming Ni
  • Journal Title: Discrete and Continuous Dynamical Systems - Series A Volume: 20 Pages: 259-27
  • Peer Reviewed
• [Journal Article] Ni Stability from the point of view of diffusion, relaxation andspatial inhomogeneity (2008)
  • Author(s): Fang Li, Kimie Nakashima, Wei-Ming.
  • Journal Title: Discrete and Continuous Dynamical Systems-Series A 20 Pages: 259-274
  • Peer Reviewed
• [Journal Article] Stability from the point of view of diffusion, relaxation and spatial inhomogeneity (2008)
  • Author(s): F. Li, K. Nakashima, and W-M. Ni
  • Journal Title: Discrete and Continuous Dynamical Systems-Series A 20 Pages: 259-274
  • Peer Reviewed
• [Journal Article] Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains
  • Author(s): Fang Li, Kimie Nakashima
  • Journal Title: Discrete and Continuous Dynamical Systems-Series A 印刷中
  • Peer Reviewed
• [Journal Article] Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains
  • Author(s): Fang Li, Kimie Nakashima
  • Journal Title: Discrete and Continuous Dynamical Systems-Series A Volume: (印刷中)
  • Peer Reviewed
• [Journal Article] An Indefinite Nonlinear Diffusion Problem in Population Genetics, I : Existence and Limiting Profiles
  • Author(s): Linlin Su, Kimie Nakashima, Wei-Ming Ni
  • Journal Title: Discrete and Continuous Dynamical Systems-Series A (印刷中)
  • Peer Reviewed
• [Presentation] 非線形解析と可積分系の数理 (2010)
  • Author(s): Kimie Nakashima
  • Organizer: Location of layers for a spatially inhomogeneous balanced bistable equation
  • Place of Presentation: 龍谷大学
  • Year and Date: 2010-11-18
• [Presentation] Location of layers for a spatially inhomogeneous balanced bistable equation (2010)
  • Author(s): Kimie Nakashima
  • Organizer: 非線形解析と可積分系の数理
  • Place of Presentation: 龍谷大学
  • Year and Date: 2010-11-18
• [Presentation] Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains (2010)
  • Author(s): Kimie Nakashima
  • Organizer: Nonlinear Evolutionary PDEs and theirEquilibrium States
  • Place of Presentation: 早稲田大学
  • Year and Date: 2010-06-11
• [Presentation] Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains (2010)
  • Author(s): Kimie Nakashima
  • Organizer: Nonlinear Evolutionary PDEs and their Equilibrium States
  • Place of Presentation: 早稲田大学
  • Year and Date: 2010-06-11
• [Presentation] Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains (2010)
  • Author(s): Kimie Nakashima
  • Organizer: Nonlinear Evolutionary PDEs and their Equilibrium States
  • Place of Presentation: 早稲大学
  • Year and Date: 2010-06-11