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Autonomous Formation of Spatial Structures in Solutions of Parabolic Partial Differential Equations

Research Project

Project/Area Number 13440050
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionTohoku University

Principal Investigator

TAKAGI Izumi  Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (40154744)

Co-Investigator(Kenkyū-buntansha) NISHIURA Yasumasa  Hokkaido University, Institute of Electronic Sciences, Professor, 電子科学研究所, 教授 (00131277)
YANAGIDA Eiji  Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (80174548)
KOZONO Hideo  Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (00195728)
NAGASAWA Takeyuki  Saitama University, Faculty of Science, Professor, 理学部, 教授 (70202223)
FUJIIE Setsuro  Tohoku University, Graduate School of Science, Lecturer, 大学院・理学研究科, 講師 (00238536)
飯田 雅人  岩手大学, 人文社会科学部, 助教授 (00242264)
Project Period (FY) 2001 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥14,500,000 (Direct Cost: ¥14,500,000)
Fiscal Year 2004: ¥2,700,000 (Direct Cost: ¥2,700,000)
Fiscal Year 2003: ¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 2002: ¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 2001: ¥5,700,000 (Direct Cost: ¥5,700,000)
Keywordsreaction-diffusion systems / pattern formation / collapse of patterns / blow-up of solutions / activator-inhibitor system / scattering of spot-like solutions / spike-layered solutions / 活性因子・抑制因子 / 非常に薄い領域 / 反応拡散系 / 集中現象 / 特異摂動 / 軌道の極限集合 / 半線型楕円型偏微分方程式 / 領域の境界の曲率 / 安定性 / 界面 / 空間的構造
Research Abstract

This objective of this project is to pursue the behavior of solutions of nonlinear partial differential equations of parabolic type.
In collaboration with Wei-Ming Ni (University of Minnesota) and Kanako Suzuki (Tohoku University), Takagi studied the behavior of solutions of a reaction-diffusion system of activator-inhibitor type proposed by Gierer and Meinhardt and obtained the following results : (i)In the case where the initial data are constant functions, there exist solutions blowing up in finite time if the activator activates the its production stronger than that of the inhibitor. There are two types of blow-up solutions. Either (a) only the activator blows up or (b) both the activator and the inhibitor blow up. In the former case, we can choose the initial value so that the inhibitor converges to any specified positive number. (ii)In the case where the equation for the activator does not contain the source term, no solution blows up in finite time if the activator produces the inhibitor more than itself. Moreover, in this case the collapse of patterns can occur. Here, by the collapse of patterns we mean that the solution converges to the origin as the time variable tends to infinity.
Nishiura studied scattering phenomena of pulse solutions and spot solutions. He found that various input/output relationships can be formed depending on the local dynamics in the neighborhood of unstable steady-states or periodic solutions and on the location of solution orbits.
Yanagida considered a certain quasilinear parabolic equation and showed that the solution is either (a) globally increasing, (b) a traveling wave, or (c) extinct in finite time, depending on the initial data. The asymptotic behavior of the solution is also investigated.
Kozono proved that in three dimensional exterior domains one can construct weak solutions of the Navier-Stokes equations which satisfy the strong energy inequality for all square-integrable initial data.

Report

(5 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • Research Products

    (29 results)

All 2005 2004 2003 Other

All Journal Article (11 results) Publications (18 results)

  • [Journal Article] Stability of stationary interfaces of binary-tree type2005

    • Author(s)
      R.Ikota, E.Yanagida
    • Journal Title

      Calculus of Variations and Partial Differential Equations 22

      Pages: 375-389

    • NAID

      120000981472

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Stability of stationary interfaces of binary-tree type2005

    • Author(s)
      R.Ikota, B.Yanagida
    • Journal Title

      Calculus of Variations and Partial Differential Equations 22

      Pages: 375-389

    • NAID

      120000981472

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Destabilization of fronts in a class of bistable systems2004

    • Author(s)
      A.Doelman, D.Iron, Y.Nishiura
    • Journal Title

      SIAM Journal of Mathematical Analysis 35

      Pages: 1420-1450

    • NAID

      120001041012

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Grow-up rate of solutions for a supercritical semilinear diffusion equation2004

    • Author(s)
      M.Fila, M.Winkler, E.Yanagida
    • Journal Title

      Journal of Differential Equations 205

      Pages: 365-389

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Destabilization of fronts in a class of bistable systems2004

    • Author(s)
      A.Doelman, D.Iron, Y.Nishiura
    • Journal Title

      SIAM J.Amath.Anal. 35(6)

      Pages: 1420-1450

    • NAID

      120001041012

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Some analytical results on the Gray Scott model2004

    • Author(s)
      D.Fathi, L.Elisabeth, Y.Nishiura
    • Journal Title

      Asymptotic Analysis 39(3-4)

      Pages: 225-261

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Nonstabilizing solutions and grow-up set for a supercritical semilinear diffusion equations2004

    • Author(s)
      P.Polacik, B.Yanagida
    • Journal Title

      Differential and Integral Equations 17(5-6)

      Pages: 535-548

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Grow-up rate of solutions for a supercritical semilinear diffusion equations2004

    • Author(s)
      M.Fila, M.Winkler, E.Yanagida
    • Journal Title

      J.Differential Equations 205

      Pages: 365-389

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Bifurcating critical points of bending energy under constraints related to the shape of red blood cells2003

    • Author(s)
      T.Nagasawa, I.Takagi
    • Journal Title

      Calculus of Variations and Partial Differential Equations 16

      Pages: 63-111

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Dynamic transitions through scatters in dissipative systems2003

    • Author(s)
      Y.Nishiura, T.Teramoto, K.-I.Ueda
    • Journal Title

      Chaos 13

      Pages: 962-972

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The Eckhaus and zigzag instability criteria in gradient/skew-gradient dissipative systems2003

    • Author(s)
      M.Kuwamura, E.Yanagida
    • Journal Title

      Physica D 175

      Pages: 185-195

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Publications] M.Kuwamura, E.Yanagida: "The Eckhaus and zigzag instability criteria in gradient/skew-gradient dissipative systems"Physica D. 175. 186-195 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] R.Ikota, B.Yanagida: "A stability criterion for stationary curves to the curvature-driven motion with a triple junction"Differential and Integral Equations. 16. 707-726 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Yagisita, E.Yanagida: "A remark on stable subharmonic solutions of time-periodic reaction-diffusion equations"J.Math.Anal.Appl.. 286. 795-803 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] P.Polacik, E.Yanagida: "On bounded and unbounded global solutions of a supercitical semilinear heat equations"Math.Ann.. 327. 795-803 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Yasumasa Nishiura, Takashi Terao, Kei-Ichi Ueda: "Scattering and separators in dissipative systems"Physical Review E. 67. 056210-1-056210-7 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Yasumasa Nishiura, Takashi Terao, Kei-Ichi Ueda: "Dynamic transitions through scatters in dissipative systems"Chaos. 13・3. 962-972 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Nagasawa, I.Takagi: "Bifurcating critical points of bending energy under constraints related to the shane of red blood cells"Calculus of Variations. 16. 63-111 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] P.Polacik, E.Yanagida: "Stable subharmonic solutions of reaction-diffusion equations on an arbitrary domain"Disc.Cont.Dyn.Systems.. 8. 209-218 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] E.Yanagida: "Standing pulse solutions in reaction-diffusion systems with skew-gradient structure"J.Dyn.Diff.Eqs.. 4. 189-205 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Teramoto, Y.Nishiura: "Stable gyroid morphology in a gradient system with nonlocal effects"J.Phys.Soc.Japan. 71(7). 1611-1614 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.J.Choe, H.Kozono: "Stokes problem for Lipshitz domain"Indiana Univ.Math.J.. 51. 1235-1260 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Fumihiko Nakano: "Absence of transport in Anderson localization"Reviews in Mathematical Physics. 14(4). 375-407 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Wei-Ming Ni, Eiji Yanagida, Izumi Takagi: "Stability of least energy patterns of the shodow system for an activator-inhibitor model"Japan Journal of Industrial and Applied Mathematics. 18. 259-272 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Chang-Shou Lin, Izumi Takagi: "Method of rotating planes applied to a singularly perturbed Neumann problem"Calculus of Variations. 13. 519-536 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Morishita, E.Ynagida, S.Yotsutani: "Structural change of solutions for a scalar curvature equation"Differential and Integral Equations. 169. 255-280 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] W.M.Ni, P.Polacik, E.Yanagida: "Monotonicity of stable solutions in shadow systems"Transactions of the American Mathematical Society. 353. 5057-5069 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Y.Nishiura, D.Ueyama: "Spatio-temporal chaos for the Gray-Scott model"Physica D. 150. 137-162 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Kozono: "Rapid time-Decay and net force to the obstacles by the Stokes flow in exterior domains"Math. Ann.. 320. 709-730 (2001)

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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