2005 Fiscal Year Final Research Report Summary
Global solution structure and the stability of nonlocal nonlinear second order boundary value problems with definite integrals
| Project/Area Number |
15540220
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Ryukoku University |
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Principal Investigator |
YOTSUTANI Shoji Ryukoku Univ., Faculty of Sci. and Tech., Professor, 理工学部, 教授 (60128361)
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| Co-Investigator(Kenkyū-buntansha) |
MORITA Yoshihisa Ryukoku Univ., Faculty of Sci. and Tech., Professor, 理工学部, 教授 (10192783)
MATSUMOTO Waichiro Ryukoku Univ., Faculty of Sci. and Tech., Professor, 理工学部, 教授 (40093314)
OKA Hiroe (國府 宏枝) Ryukoku Univ., Faculty of Sci. and Tech., Professor, 理工学部, 教授 (20215221)
NINOMIYA Hirokazu Ryukoku Univ., Faculty of Sci. and Tech., Ass.Professor, 理工学部, 助教授 (90251610)
YANAGIDA Eiji Tohoku Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (80174548)
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| Project Period (FY) |
2003 – 2005
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| Keywords | cross-diffusion / nonlinear elliptic equation / nonlocal / nonlinear boundary value problem / periodic boundary condition / Ginzburg-Landau equation / positive radial solution / singular solution |
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| Research Abstract |
First, Lou-Ni-Yotsutani [DCDS 2004] investigated a limiting equation to a cross-diffusion equation that appears in mathematical biology. We showed that it has different kinds of singular solutions and revealed the structure of all solutions.
This problem is a nonlocal nonlinear elliptic boundary problem, for which no method was known to solve it. We discovered a new method, which are the combination of the modern method of PDE and classical analysis and algebra.
There are a lot of interesting problems for which our method is applicable.
A problem of the Oseen's spiral flow is one of them, for which we obtained the complete bifurcation diagram in Ikeda-Kondo-Okamoto-Yotsutani [CPAA 2003].
Matsumoto-Murai-Yotsutani [Pisa, 2005] gave the complete answer for a problem to determine curves with the least energy under the given length.
Second, Kosugi-Morita-Yotsutani [CPAA 2005, J.Math.Phy. 2005] have revealed the complete Global bifurcation branches one dimensional Ginzburg-Landau equations with periodic boundary conditions.
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