2005 Fiscal Year Final Research Report Summary
Boundary value problems for higher-order nonlinear ordinary differential equations
| Project/Area Number |
15340048
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 | EHIME UNIVERSITY |
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Principal Investigator |
NAITO Manabu Ehime University, Faculty of Science, Professor, 理学部, 教授 (00106791)
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| Co-Investigator(Kenkyū-buntansha) |
SAKAGUCHI Shigeru Ehime University, Faculty of Science, Professor, 理学部, 教授 (50215620)
HASHIMOTO Takahiro Ehime University, Faculty of Science, Assistant, 理学部, 助手 (60291499)
USAMI Hiroyuki Hiroshima University, Faculty of Integrated Arts and Sciences, Associate Professor, 総合科学部, 助教授 (90192509)
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| Project Period (FY) |
2003 – 2005
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| Keywords | quasilinear ODE / positive solution / degenerate elliptic equation / heat equation / boundary value problems / eigenvalue problems / poly-harmonic |
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| Research Abstract |
The aim of this research is to study the existence, the uniqueness, the number of zeros and the distribution of zeros of solutions of boundary value problems for higher-order ordinary differential equations, and to obtain the detailed information for the set of solutions of higher-order elliptic differential equations on the base of the results for ordinary differential equations.
The new results are as follows :
1.For a fourth-order or even-order quasilinear ordinary differential equation, necessary and sufficient conditions for the existence of a positive solution are obtained. For a 2-system of the second-order ordinary differential equations and a 2-system of the second-order elliptic equations, the existence of a positive solution is discussed.
2.A degenerate elliptic equation with arbitrary nonlinearity is considered on exterior domain, and necessary and sufficient conditions for all solutions to be oscillatory are established. A degenerate elliptic equation is considered on strip-like domain, and the nonexistence of a positive solution is discussed.
3.For an initial value problem for the heat equation, stationary isothermic surfaces and uniformly dense domains are discussed, and an interaction between degenerate diffusion and shape of domain is discussed.
4.For a singular eigenvalue problem to a higher-order ordinary differential equation on an infinite interval, it is shown that there is a countable sequence of eigenfunctions having exactly n zeros.
5.For a fourth-order nonlinear elliptic differential equation including the poly-harmonic operator, the existence results and nonexistence results are obtained.
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