1999 Fiscal Year Final Research Report Summary
Behavior of solutions of IBVP witrh periodically moving boundary conditions of evolution equations
| Project/Area Number |
09640223
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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 |
解析学
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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) |
SUGITA Kimio Tokai University, School of Science, Professor, 理学部, 教授 (60056083)
ITOH Tatuo Tokai University, School of Science, Professor, 理学部, 教授 (20151516)
AKAMATSU Toyohiro Tokai University, School of Science, Professor, 理学部, 教授 (00112772)
楢崎 隆 東海大学, 理学部, 助教授 (70119692)
TANAKA Minoru Tokai University, School of Science, Professor, 理学部, 教授 (10112773)
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| Project Period (FY) |
1997 – 1999
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| Keywords | Vibration of string / wave equation / noncylindrical domain / periodic boundary condition / Diophantine inequality / quasiperiodic solution / reduction theorem / rotation number |
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| Research Abstract |
We are concerned with IBVP for one or two dimensional wave equations defined in domains with periodically or quasiperiodically moving boundaries and boundary functions. The purpose of our research is the behavior of solutions of the IBVP, Main results of our research are as follows.
1. A simple composed function A defined by two boundary functions plays an essential role in the behavior of the solutions. An important mapping by the reflective characteristics is defined by the composed function. By the reflective characteristics the geometric structure of problem was clarified.
2. Consider the case where the above A defines periodic dynamical system (PDS). If the rotation number of the PDS satisfies the Diophantine inequality from number theory, then every solution is quasiperiodic both in space and time. This result is epoch-making in this subject. That is, the problem in this case is almost completely solved. From this result the interesting result by J. Cooper is the exceptional case.
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Also the necessary condition was considered and some results were obtained, by using some results from number theory.
3. We found the new interesting domain mapping which tr4nsforms the periodic noncylin-drical domain onto a cylindrical domain. The important fact is that the wave operator is preserved by this mapping, different from other results. By this we can show the existence of periodic solutions of BVP for nonlinear wave equations which seemed difficult to show Also we made clear the behavior of the solutions of IBVP in case of nonhomogeneous wave equations.
4. Generally the case where the above A defines quasiperiodic dynamical systems is difficult. To obtain the corresponding results of the periodic case, we defined the new concept the upper (lower) rotation number. Using this, we showed the important Reduction Theorem, and applying this Theorem, we had the results. Different from the PDS, in this case the boundary functions are of the perturbed form. In this sense the results are not global, while the periodic case is of global character.
In the membrane oscillation we considered only the case where the boundaries are circles and the solutions should be spherically symmetric. The general case is extremely difficult. Less
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