2002 Fiscal Year Final Research Report Summary
Study of the Mathematical structure of singularities
| Project/Area Number |
11214202
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| Research Category |
Grant-in-Aid for Scientific Research on Priority Areas (B)
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| Allocation Type | Single-year Grants |
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| Review Section |
Science and Engineering
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| Research Institution | The University of Tokyo |
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Principal Investigator |
MATANO Hiroshi University of Tokyo, Graduate school of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (40126165)
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| Co-Investigator(Kenkyū-buntansha) |
MASUDA Kyuya Meiji University, Faculty of Science and Technology, Professor, 理工学部, 教授 (10090523)
WEISS Georg Graduate school of Mathematical Sciences, Associate Professor, 大学院・数理科学研究科, 助教授 (30282817)
FUNAKI Tadahisa Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (60112174)
SHISHIKURA Mitsuhiro Kyoto University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (70192606)
YANAGIDA Eiji Tohoku University, Graduate School Sciences, Professor, 大学院・理学研究科, 教授 (80174548)
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| Project Period (FY) |
1999 – 2002
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| Keywords | singularity / singular perturbation method / singular limit / blow-up of solutions / nonlinear problems / dynamical systems / bifurcation theory / interface |
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| Research Abstract |
(1) Singular limit of diffusion equations In certain kinds of diffusion equations, transition layers appear as the diffusion coefficients tends to zero. Matano has studied the singular limit for diffusion equations with spatially inhomogeneous coefficients. Yanagida has used this singular limit technique to proved the existence of stable periodic solutions. Funaki has investigated the behavior of interfaces that appear in equations with random deviation.
(2) Blow-up in nonlinear heat equations Some blow-up solutions of nonlinear heat equations can be continued beyond the blow-up time in a certain sense. Matano has studied the dynamics of such blow-up solutions. Yanagida has obtained new formula for estimating the blow-up time.
(3) Singularities in free boundary problems Weiss has applies his new method obtain the Hausdorff dimension of the singularity set of a 2-phase obstacle problem.
(4) Singularities and bifurcation in comlex dynamical system Shishikura has obtained a new systematic method to study bifurcation problems and rigidity of Julia sets.
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