Project/Area Number  09640223 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
解析学

Research Institution  Tokai University 
Principal Investigator 
YAMAGUCHI Masaru Tokai University, School of Science, Professor, 理学部, 教授 (10056252)

CoInvestigator(Kenkyūbuntansha) 
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)
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)

Project Fiscal Year 
1997 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥3,000,000 (Direct Cost : ¥3,000,000)
Fiscal Year 1999 : ¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 1998 : ¥900,000 (Direct Cost : ¥900,000)
Fiscal Year 1997 : ¥1,300,000 (Direct Cost : ¥1,300,000)

Keywords  Vibration of string / wave equation / noncylindrical domain / periodic boundary condition / Diophantine inequality / quasiperiodic solution / reduction theorem / rotation number / 弦の振動 / 波動方程式 / 非柱状領域 / 周期境界条件 / ディオフォントス不等式 / 準周期解 / Reduction Theorem / 回転指数 / ディオファントス近似不等式 / リサージュ境界条件 / 円形膜の振動 / 弦の挙動 / 準同期解 / 一次元波動方程式 / Lissajous境界条件 
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 epochmaking 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 noncylindrical 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
