Research on the Behavior of Solutions Evolution Equations in Time-Almost Periodic Noncylindrical Domains

时间类周期非圆柱域中解演化方程的行为研究

基本信息

批准号：
12640220
负责人：
YAMAGUCHI Masaru
金额：
$ 2.3万
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：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

项目摘要

In this project we dealt with linear and nonlinear wave equations in noncylindrical domain periodic or quasiperiodic in time. We studied the qualitative behavior of solutions of the initial boundary value problems (IBVP) and the boundary value problems (BVP). Our results are as follows.(1) We considered BVP for 1-D nonlinear wave equations in time-periodic noncylindrical domains. If the nonlinear forcing term, the boundary functions and the boundary data are periodic in time with same period, BVP have time-periodic solutions. This problem had been regarded as one of the difficult problems.(2) We considered IBVP for 1-D linear wave equations in time-quasiperiodic noncylindrical domains. The nonhomogeneous terms of the equations and the boundary data are also time quasiperiodic. As we showed in the previous Research Project, the solutions are generally almost periodic in time, hence bounded in time. We studied this phenomena more deeply, and found that there exist solutions which are the … More superpositions of time-unbounded waves.(3) We considered IBVP for 3-D radially symmetric linear wave equations in time-quasiperiodic noncylindrical domains whose space-domains are surrounded by two balls. We showed that the solutions are generally almost periodic in time.(4) We considered BVP for 3-D radially symmetric nonlinear wave equations in time-periodic noncylindrical domains whose space-domains are balls. Under the similar assumptions to those of (1) BVP have time-periodic solutions. The results seem to be interesting.In order to solve the problems, we developed some useful method. This method consists of a transformation of BVP for wave equations to some functional equations and domain transformations that transform the noncylindrical domains to cylindrical domains. The former was established by M. Yamaguchi and the latter by M. Yamaguchi and H. Yoshida. This method is based on the Reduction Theorems by M. Herman and J. Yoccoz in periodic case and by M. Yamaguchi in quasiperiodic case. Less

在这个项目中，我们处理非圆柱域周期或准周期时间中的线性和非线性波动方程，我们研究了初始边值问题（IBVP）和边值问题（BVP）的解的定性行为，我们的结果如下。 (1) 我们考虑时间周期非圆柱域中的一维非线性波动方程的 BVP 如果非线性强迫项、边界函数和边界数据在时间上具有相同的周期， BVP具有时间周期解，这一问题一直被认为是难题之一。(2)考虑了时间准周期非圆柱域中的一维线性波动方程的IBVP，方程的非齐次项和边界数据为。正如我们在之前的研究项目中所示，解通常在时间上几乎是周期性的，因此我们更深入地研究了这种现象，并发现存在解。 …更多时间无界波的叠加。(3) 我们考虑了时间准周期非圆柱域中的 3-D 径向对称线性波动方程的 IBVP，其空间域被两个球包围。我们证明了解通常几乎是周期性的。 (4) 我们考虑了空间域为球的时间周期非圆柱域中的 3-D 径向对称非线性波动方程的 BVP。与 (1) BVP 类似的假设有时间周期解。结果似乎很有趣。为了解决这些问题，我们开发了一些有用的方法，该方法包括将波动方程的 BVP 变换为某些泛函。前者由 M. Yamaguchi 建立，后者由 M. Yamaguchi 和 H. Yoshida 建立。该方法基于 M. Yamaguchi 的约简定理。 Herman 和 J. Yoccoz 的周期情况以及 M. Yamaguchi 的准周期情况 Less