2001 Fiscal Year Final Research Report Summary
Asymptotic analysis of ordinary differential equations, and its application to partial differential equations
| Project/Area Number |
12640179
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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 |
Basic analysis
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| Research Institution | HIROSHIMA UNIVERSITY |
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Principal Investigator |
USAMI Hiroyuki Hiroshima University, Faculty of Integrated Arts & Sciences, assistant professor, 総合科学部, 助教授 (90192509)
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| Co-Investigator(Kenkyū-buntansha) |
NAITO Manabu Ehime University, Faculty of Science, professor, 理学部, 教授 (00106791)
SHIBATA Tetsutaro Hiroshima University, Faculty of Integrated Arts & Sciences, assistant professor, 総合科学部, 助教授 (90216010)
YOSHIDA Kiyoshi Hiroshima University, Faculty of Integrated Arts & Sciences, professor, 総合科学部, 教授 (80033893)
MIZUTA Yohihiro Hiroshima University, Faculty of Integrated Arts & Sciences, professor, 総合科学部, 教授 (00093815)
NAITO Yuki Kobe University, Faculty of Engineering, assistant professor, 工学部, 助教授 (10231458)
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| Project Period (FY) |
2000 – 2001
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| Keywords | quasilinear equation / elliptic equation / positive solution / Sturm-Liouville problem / eigenvalue problem |
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| Research Abstract |
(1) Quasilinear ODEs of seond-order, which are generalizations of Emden's equation, are considered. Asymptotic representations of positive solutions are obtained explicitly. When nonlinear terms have singularities at the origin, uniqueness of decaying positive solutions is established.
(2) Two-term quasilinear ODEs of fourth-order are considered. Neessary and/or sufficient conditions are established for them to have no positive solutions existing near the infinity. Generalizations and applications of these results to 4-dimensional ordinary differential systems are also obtained.
(3) As an application of the results in (1), we obtain sufficient conditions for some types of quasilinear exterior elliptic BVPs to have positive solutions with specified asymptotic behavior near the infinity. As an application of the results in (2), we obtain sufficient conditions for some types of semilinear 2-dimensional exterior elliptic problems to have no positive solutions existing near the infinity.
(4) Eigenvalue problems for second-order semilinear ODEs on finite intervals are studied. We establish asymptotic properties of (variational) eigenvalues and eigenfunctions. Eigenvalue problems for n-th order linear ODEs are also studied on infinite intervals. We extend well-known Sturmian theory to these problems partially.
(5) We consider self-similar solutions of parabolic systems introduced by Keller and Segel to describe aggregation phenomena of molds due to chemotaxis. We find that such solutions must be radially symmetric, and then clarify the relation between parameters and various norms of solutions.
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