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Interface motion and blow-up phenomena in nonlinear partial differential equations

Research Project

Project/Area Number 17340044
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionThe University of Tokyo

Principal Investigator

MATANO Hiroshi  The University of Tokyo, Graduate School of Mathematical Sciences the University of Tokyo, Professor (40126165)

Co-Investigator(Kenkyū-buntansha) FUNAKI Tadahisa  the University of Tokyo, Graduate School of mathematical Sciences, Professor (60112174)
WEISS Georg  the University of Tokyo, Graduate School of Mathematical Sciences, Associate professor (30282817)
TANIGUCHI Masaharu  Tokyo Institute of technology, Graduate School of Information Science and Technology, Associate professor (30260623)
MIZUMACHI Tetsu  Kyushu University, Graduate School of Mathematical Sciences, Associate professor (60315827)
NAKAMURA Ken-Ichi  University of Electro-Communications, Department of Computer Science, Faculty of electro-Communiations, Assistant (40293120)
Project Period (FY) 2005 – 2007
Project Status Completed (Fiscal Year 2007)
Budget Amount *help
¥15,620,000 (Direct Cost: ¥14,300,000、Indirect Cost: ¥1,320,000)
Fiscal Year 2007: ¥5,720,000 (Direct Cost: ¥4,400,000、Indirect Cost: ¥1,320,000)
Fiscal Year 2006: ¥4,400,000 (Direct Cost: ¥4,400,000)
Fiscal Year 2005: ¥5,500,000 (Direct Cost: ¥5,500,000)
Keywordsnonlinear analysis / partial differential equation / blow-up of solutions / interface motion / asymptotic method / singular limit / nonlinear diffusion equations / nonlinear heat equation / 進行波
Research Abstract

The aim of this research project is to make a theoretical study of various nonlinear problems related to interfacial motions and blow-up phenomena, by developing asymptotic methods based on the theory of infinite dimensional dynamical systems and stochastic methods, and also performing numerical simulations if necessary. We have obtained the following results.
(1) We have given an optimal estimate concerning the singular limit of Allen-Cahn type nonlinear diffusion equations, that has not been known previously. We also obtained similar results for FitzHugh-Nagumo systems and Lotka-Volterra competition systems.
(2) We have clarified the global dynamics of blow-up solutions in nonlinear diffusion equations. This was made possible by extending the existing theory on global attractors to the case where blow-up occurs.
(3) In nonlinear heat equations with power nonlinearity, the blow-up of solutions is classified into type I and type II, the latter being much more difficult to analyze than the … More former. By using a topological method based on the braid group theory, we have succeeded in determining all possible type II blow-up rates.
(4) We studied the stationary problem for the Allen-Cahn equation on the 2-dimensional space having lattice periodicity by using variational methods. We have shown that a necessary and sufficient condition for the existence of multi-layered stationary solution is that the set of single layered solutions has a gap somewhere ; in other words, this set should not be a continuum (foliation).
(5) We considered periodic traveling waves in a two-dimensional infinite strip whose boundaries are saw-tooth shaped. We determined the homogenization limit of such traveling waves as one lets the boundary undulation finer and finer.
(6) Various partial differential equations are derived from microscopic models via hydrodynamic limit. Those equations include the Stefan problem and some stochastic differential equations.
(7) Regularity of free boundaries arising in various elliptic equations such as the combustion model has bee established.
(8) Stability of V-shaped traveling wave in the equation of curvature-dependent motion has been established. Less

Report

(4 results)
  • 2007 Annual Research Report   Final Research Report Summary
  • 2006 Annual Research Report
  • 2005 Annual Research Report
  • Research Products

    (44 results)

All 2008 2007 2006 2005 Other

All Journal Article (29 results) (of which Peer Reviewed: 10 results) Presentation (15 results)

  • [Journal Article] The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system2008

    • Author(s)
      M.Alfaro, 他
    • Journal Title

      J.Differential Equations 245

      Pages: 505-565

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system2008

    • Author(s)
      M.Alfaro, D.Hilhorst, H.Matano
    • Journal Title

      J.Differential Equations 245

      Pages: 505-565

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] The singular limit of The Allen-Cahn equation and the FitzHugh-Nagumo system2008

    • Author(s)
      M.Alfaro, D.Hilhorst and H.Matano
    • Journal Title

      J.Differential Equations(掲載確定)

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Global dynamics of blow-up profiles in one-dimensional reaction diffusion equations2007

    • Author(s)
      B.Fiedler, 他
    • Journal Title

      J.Dynamics and Differential Equations 19

      Pages: 867-893

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Blow-up in nonlinear Heat equations with supercritical power nonlinearity2007

    • Author(s)
      H.Matano
    • Journal Title

      Contemporary Mathematics, Amer. Math. Soc. 446

      Pages: 385-412

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Singular limit of a spatially inhomogeneous Lotka-Volterra competition diffusion system2007

    • Author(s)
      H.Hilhorst, 他
    • Journal Title

      Comm. Partial Diff. Equations 32

      Pages: 879-933

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Global dynamics of blow-up profiles in one-dimensional reaction diffusion equations2007

    • Author(s)
      B.Fiedler, H.Matano
    • Journal Title

      J.Dynamics and Differential Equations 19

      Pages: 867-893

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Blow-up in nonlinear heat equations with supercritical power nonlinearity2007

    • Author(s)
      H.Matano
    • Journal Title

      Contemporary Mathematics(Amer.Math.Soc.) 446

      Pages: 385-412

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Singular limit of a spatially inhomogeneous Lotka-Volterra competition diffusion system2007

    • Author(s)
      D.Hilhorst, G.Karali, H.Matano, K.Nakashima
    • Journal Title

      Comm.Partial Diff.Equations 32

      Pages: 879-933

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Global dynamics of blow-up profiels in one-dimensional reaction diffusion equations2007

    • Author(s)
      B.Fiedler and H.Matano
    • Journal Title

      J, Dynamics Differential Equations 19

      Pages: 867-893

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Singular limit of a spatially inhomogeneous Lotka-volterra competition diffusion system2007

    • Author(s)
      D.Hilhorst, G.Karali, H.Matano and K.Nakashima
    • Journal Title

      Comm.Partial Diff.Equqations 32

      Pages: 879-933

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Instability of vortex solitions for 2D focusing NLS2007

    • Author(s)
      Mizumachi, T.
    • Journal Title

      Adv. Differential Equations 12

      Pages: 261-264

    • Related Report
      2006 Annual Research Report
  • [Journal Article] On the necessity of gaps2006

    • Author(s)
      H.Matano, 他
    • Journal Title

      J.Euro. Math. Soc. 8

      Pages: 355-373

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Periodic taveling waves in an undulating band domain and their homogenization limit2006

    • Author(s)
      H.Matano, 他
    • Journal Title

      Networks and Heterogeneous Media 1

      Pages: 537-568

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] On the necessity of gaps2006

    • Author(s)
      H.Matano, P.Rabinowitz
    • Journal Title

      J.Euro.Math.Soc. 8

      Pages: 355-373

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit2006

    • Author(s)
      B.Lou, H.Matano, K.-I.Nakamura
    • Journal Title

      Networks and Heterogeneous Media 1

      Pages: 537-568

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] On the necessity of gaps2006

    • Author(s)
      Matano, H (Rabinowitz, P.)
    • Journal Title

      J. Euro. Math. Soc. {bf 8} 8

      Pages: 355-373

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Periodic taveling waves in an undulating band domain and their homogenization limit2006

    • Author(s)
      Matano, H., Nakamura, K.-I.(Lou, B)
    • Journal Title

      Networks and Heterogeneous Media 1

      Pages: 537-568

    • Related Report
      2006 Annual Research Report
  • [Journal Article] The Two-Phase Membrane Problem-an Intersection-Comparison Approach to the Regularity at Branch Points2006

    • Author(s)
      Weiss, G.S.(Shahgolian, H.)
    • Journal Title

      Adv. Math. 205

      Pages: 487-503

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Convergence to V-shaped fronts in curvature flows for spatially non-decaying initial perturbations2006

    • Author(s)
      Taniguchi, M. (Nara, M.)
    • Journal Title

      Discrete and Continuous Dynamical Systems 16

      Pages: 137-156

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Bifurcations of steady states for the Eguchi-Oki-Matsumura model of phase separation.2006

    • Author(s)
      Nakamura, K.-I.(Ishimura, N, Nakamura, M.)
    • Journal Title

      Applicable Analysis 85

      Pages: 831-843

    • NAID

      120006933453

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Global stability of traveling curved fronts in the Allen-Cahn equations2006

    • Author(s)
      Taniguchi, M.(Ninomiya, H.)
    • Journal Title

      Discrete and Continuous Dynamical Systems 15

      Pages: 819-832

    • NAID

      120006581802

    • Related Report
      2005 Annual Research Report
  • [Journal Article] A remark on linearly unstable standing wave solutions to NLS2006

    • Author(s)
      Mizumachi, T.
    • Journal Title

      Nonlinear Analysis : Theory, Methods and Applications 64

      Pages: 657-676

    • Related Report
      2005 Annual Research Report
  • [Journal Article] On the steadily rotating spirals2006

    • Author(s)
      Nakamura K.-I.(Guo, J.-S., Ogiwara, T., Tsai, J.-C.)
    • Journal Title

      Japan J.Industrial and Appl.Math. 23(印刷中)

    • NAID

      10018138632

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Immediate regularization after blow-up2005

    • Author(s)
      H.Fila, 他
    • Journal Title

      SIAM J.Math.Anal. 37

      Pages: 752-776

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Immediate regularization after blow-up2005

    • Author(s)
      M.Fila, H.Matano, P.Polacik
    • Journal Title

      SIAM J.Math.Anal. 37

      Pages: 752-776

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Immediate regularization after blow-up2005

    • Author(s)
      Matano, H.(Fila, M., Polacik, P.)
    • Journal Title

      SIAM J.Math.Anal. 37

      Pages: 752-776

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Hydrodynamic limit and nonlinear PDEs with singularities

    • Author(s)
      Funaki, T.
    • Journal Title

      "Asymptotic Analysis and Singularity", Advanced Studies in Pure Mathematics (掲載予定)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Cross-shaped and degenerate singularities in an unstable elliptic free boundary problem

    • Author(s)
      Weiss, G.S.(Andersson, J.)
    • Journal Title

      J.Differential Equations (掲載予定)

    • Related Report
      2005 Annual Research Report
  • [Presentation] Front propagation in the presence of obstacles2007

    • Author(s)
      Hiroshi Matano
    • Organizer
      A confenrence in honor of Avner Friedman's 75th birthday:Differential Equations and Math Biology
    • Place of Presentation
      オハイオ州立大学
    • Year and Date
      2007-11-16
    • Related Report
      2007 Annual Research Report
  • [Presentation] Front propagation in the presence of obstacles2007

    • Author(s)
      H.Matano
    • Organizer
      A confenrence in honor of Avner Friedman's 75th birthday : Differential Equations and Math Biology
    • Place of Presentation
      コロンブス(米国)
    • Year and Date
      2007-11-15
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Front propagation in the presence of obstacles2007

    • Author(s)
      H.Matano
    • Organizer
      Confenrence in honor of Avner Friedman's 75th birthday : Differential Equations and Math Biology
    • Place of Presentation
      Columbus, USA
    • Year and Date
      2007-11-15
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Propagation of fronts in spatially heterogeneous media2007

    • Author(s)
      H.Matano
    • Organizer
      ReaDilab Conference on Mathematical Modeling and Analysis in Biological and Chemical Systems
    • Place of Presentation
      オルセー(フランス)
    • Year and Date
      2007-09-05
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Propagation of fronts in spatially heterogeneous media2007

    • Author(s)
      H.Matano
    • Organizer
      ReaDilab Conference on Mathematical Modeling and Analysis in Biological and Chemical Systems
    • Place of Presentation
      Orsay, France
    • Year and Date
      2007-09-05
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Traveling waves for an integro-difference equation2006

    • Author(s)
      H.Matano
    • Organizer
      International Conference on Difference Equations and Applications
    • Place of Presentation
      京都大学
    • Year and Date
      2006-07-28
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Traveling waves for an integro-difference equation2006

    • Author(s)
      H.Matano
    • Organizer
      International Conference on Difference Equations and Applications
    • Place of Presentation
      Kyoto
    • Year and Date
      2006-07-28
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Traveling waves in a saw-toothed domain and their homogenization limit2006

    • Author(s)
      H.Matano
    • Organizer
      Launching Meeting of Networks and Heterogeneous Media
    • Place of Presentation
      マイオリ(イタリア)
    • Year and Date
      2006-06-22
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Traveling waves in a saw-toothed domain and their homogenization limit2006

    • Author(s)
      H.Matano
    • Organizer
      Launching Meeting of Networks and Heterogeneous Media
    • Place of Presentation
      Maiori, Italy
    • Year and Date
      2006-06-22
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] A variational approach for quasi-periodic fronts in Allen-Cahn model equations2005

    • Author(s)
      H.Matano
    • Organizer
      Frontiers of Applied Analysis
    • Place of Presentation
      ピッツバーグ(米国)
    • Year and Date
      2005-09-08
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] A variational approach for quasi-periodic fronts in Allen-Cahn model equations2005

    • Author(s)
      H.Matano
    • Organizer
      Frontiers of Applied Analysis
    • Place of Presentation
      Pittsburgh, USA
    • Year and Date
      2005-09-08
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Complete and incomplete blow-up in a nonlinear heat equation2005

    • Author(s)
      H.Matano
    • Organizer
      EQUADIFF 11
    • Place of Presentation
      Bratislava, Slovakia
    • Year and Date
      2005-07-25
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Quasi-periodic fronts for an Allen-Cahn equation in spatially periodic media2005

    • Author(s)
      H.Matano
    • Organizer
      CIRM Conference on Infinite Dimensional Dynamical Systems
    • Place of Presentation
      リュミニ(フランス)
    • Year and Date
      2005-07-06
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Quasi-periodic fronts for an Allen-Cahn equation in spatially periodic media2005

    • Author(s)
      H.Matano
    • Organizer
      CIRM Conference on Infinite Dimensional Dynamical Systems
    • Place of Presentation
      Luminy, Fance
    • Year and Date
      2005-07-06
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Complete and incomplete blow-up in a nonlinear heat equation2005

    • Author(s)
      H.Matano
    • Organizer
      EQUADIFF 11
    • Place of Presentation
      ブラティスラヴァ(スロヴァキア)
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary

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Published: 2005-04-01   Modified: 2016-04-21  

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