2007 Fiscal Year Final Research Report Summary
Study of asymptotic behaviors to wave equations
| Project/Area Number |
16540205
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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 |
NARAZAKI Takashi Tokai University, School of Science, Professor (70119692)
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| Co-Investigator(Kenkyū-buntansha) |
YAMAGUCHI Masaru Tokai University, School of Science, Professor (10056252)
TANAKA Minoru Tokai University, School of Science, Professor (10112773)
AKAMATSU Toyohiro Tokai University, School of Science, Professor (00112772)
ITO Tatsuo Tokai University, School of Science, Professor (20151516)
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| Project Period (FY) |
2004 – 2007
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| Keywords | damped wave equation / odd data problem / asymptotic behavior / oscillating boundary / suspended string / cut locus / Tononogov's theorem / hypoellipticity |
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| Research Abstract |
Under certain restriction on dissipation term, Matsuyama and Tanaka proved the existence of scattering states and no decay of energy in time to an initial-boundary-value problem for the wave equation in the exterior domain outside a compact obstacle.
Narazaki studied the Cauchy problem to nonlinear wave equation with linear dissipation. Using rapid decay estimates of the solutions to the corresponding linear equations with odd initial data, he proved existence of global solutions in time to nonlinear damped wave equations, when order of nonlinear is over new critical exponent proposed by him. Moreover, he studied the case where initial data belong to L^<P> (1<p), and they proved that time global solutions exist when order of nonlinear term is over 1+2/np , where n is a space dimension that is not larger than 3.
Yamaguchi studied an initial boundary value problem for a one-dimensional wave equation in a domain with time-quasiperiodically oscillating boundary, he indicated general conditions under which every solution of the stated problem is quasiperiodic. Moreover, he studied nonlinear wave equation that is model for suspended string. He proved the existence periodic solutions, bounded time global solutions and regularity of solutions.
Tanaka showed the structure of cut locus of any points on revolutionary torus surfaces in Euclidean space, and he extend the conjecture on conjugate locus "Jacobi's last geometric statement" to wider class of Liouville surfaces. Moreover, he determined the structure of the cut locus of a class of two-spheres of revolution, which includes all ellipsoids of revolution. Furthermore, he showed that the subclass of this class gives a new model surface for Topogonov's comparison theorem.
Akamatsu proved the hypoellipticity of second order partial differential operator with principal symbol that changes sign.
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[Presentation] A generalized Toponogov comparison theorem2006
Author(s)
M, Tanaka
Organizer
Proceedings of KMITL International Conference on Science and Applied Science
Place of Presentation
King Mongkut's Institute of Technology Ladkrabang(KMITL), Thailand
Year and Date
2006-03-10
Description
「研究成果報告書概要(欧文)」より
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[Presentation] 比較定理と最小跡2005
Author(s)
田中 實
Organizer
佐賀大学微分幾何学研究集会
Place of Presentation
佐賀大学
Year and Date
2005-12-17
Description
「研究成果報告書概要(和文)」より
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