2005 Fiscal Year Final Research Report Summary
The global behavior of solutions of evolution equations in noncylindrical domain with time-moving boundaries
| Project/Area Number |
15540213
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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 | Tokai University |
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Principal Investigator |
YAMAGUCHI Masaru Tokai University, School of Science, Professor, 理学部, 教授 (10056252)
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| Co-Investigator(Kenkyū-buntansha) |
AKAMATSU Toyohiro Tokai University, School of Science, Professor, 理学部, 教授 (00112772)
ITOH Tatsuo Tokai University, School of Science, Professor, 理学部, 教授 (20151516)
TANAKA Minoru Tokai University, School of Science, Professor, 理学部, 教授 (10112773)
MATSUYAMA Tokio Tokai University, School of Science, Professor, 理学部, 教授 (70249712)
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| Project Period (FY) |
2003 – 2005
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| Keywords | nonlinear wave equation / Klein-Gordon equation / Suspended string equation / periodic solution / almost periodic solution / Diophantine approximation / cusp |
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| Research Abstract |
1) Let the space dimension be 1 to 5. Consider nonlinear sphere-symmetric autonomous wave equations in ball. We showed that the equations have infinitely many time-periodic solutions. In the proof we used the Diophantine inequalities on the eigenvalues of Laplacian and the periods. To this end we studied in detail numerical properties of the zero points of the Bessel functions, using the asymptotic expansions of the zero points.
2) We considered IBVP for linear equations of heavy suspended string. Assume that time-quasiperiodic force works to the string. We assume the general Diophantine conditions on the eigenvalues of the string operator and the quasi perios Then we completely made clear the strunture of almost periodic structure of the solutions of IBVP. To show this statement, we defined well-matched function spaces, solved the eigenfunction problem in these function spaces and constructed the spectral theory.
3) We considered BVP for nonlinear sphere-symmetric wave equations in ball with periodically moving boundaries. Then we showed the existence of periodic solutions of the BVP.
4) The following theorem on the geodesics on manifolds is proved by M.Tanaka.
Theorem : Let M be a real analytic Riemaniann manifold homeomorphic to a 2-sphere. If the Gaussian curvature of M is positive, then the conjugate locus of each point consists of a single point or has at least 4 cusps.
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