权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Behavior of solutions of IBVP witrh periodically moving boundary conditions of evolution equations

具有周期性移动演化方程边界条件的 IBVP 解的行为

基本信息

批准号：
09640223
负责人：
YAMAGUCHI Masaru
金额：
$ 1.92万
依托单位：
Tokai University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1999
项目状态：
已结题

项目摘要

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. … More 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

本文讨论了一类一维或二维波动方程的IBVP问题，该方程定义在具有周期性或拟周期性运动边界和边界函数的区域上。我们的研究目的是研究IBVP解的行为，主要研究结果如下.由两个边界函数定义的简单复合函数A对解的行为起着至关重要的作用。由反射特性定义的一个重要映射是由组合函数定义的。利用反射特性，明确了问题的几何结构.考虑上述A定义周期动力系统（PDS）的情况。如果PDS的旋转数满足数论中的丢番图不等式，则每个解在空间和时间上都是拟周期的。这一成果在这一学科中具有划时代的意义。也就是说，这种情况下的问题几乎完全解决了。从这个结果中，J.库珀的有趣结果是例外情况。 ...更多信息利用数论中的一些结果，考虑了其必要条件，得到了一些结果.我们发现了一种新的有趣的区域映射，它将周期性非圆柱形区域变换到圆柱形区域上。重要的事实是，波算子被保存由这个映射，不同于其他结果。由此，我们可以证明非线性波动方程边值问题周期解的存在性，这在以往的研究中是很难证明的.同时，我们也明确了非齐次波动方程边值问题周期解的性质.一般来说，上述A定义准周期动力系统的情况是困难的。为了得到周期情形的相应结果，我们定义了上（下）旋数的新概念。利用这一点，我们证明了重要的归约定理，并应用这一定理，我们得到了结果。与PDS不同的是，在这种情况下，边界函数是摄动形式的。在这种意义下，结果不是整体的，而周期的情况是整体的，在膜振动中，我们只考虑了边界是圆的情况，解应该是球对称的。一般情况下是非常困难的。少