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

本项目研究了非圆柱区域上的线性和非线性波动方程，时间上为周期或拟周期。研究了初边值问题和边值问题解的定性性质。我们的结果如下。(1)研究了时间周期非圆柱区域上一维非线性波动方程的边值问题。如果非线性强迫项、边界函数和边界数据都是周期的，且周期相同，则边值问题存在时间周期解。这一问题一直被视为难题之一。(2)考虑了时间拟周期非圆柱区域上一维线性波动方程的IBVP。方程的非齐次项和边界数据也是时间拟周期的。正如我们在上一个研究项目中所展示的，解通常在时间上是几乎周期的，因此在时间上是有界的。我们更深入地研究了这一现象，发现存在解决方案， ...更多信息 时间无界波的叠加。(3)本文研究了三维径向对称线性波动方程在时间拟周期非圆柱区域上的IBVP问题，其中空间区域被两个球包围。我们证明了这些解在时间上一般是概周期的。(4)研究了三维径向对称非线性波动方程在时间周期非圆柱区域上的边值问题。在与（1）类似的假设下，边值问题存在时间周期解。为了解决这些问题，我们提出了一些有用的方法。该方法包括将波动方程的边值问题转化为函数方程和将非圆柱形区域转化为圆柱形区域的区域变换。前者是由M. Yamaguchi和M. Yamaguchi和H.吉田这种方法是以M.赫尔曼和J.Yoccoz在周期情况下和M. Yamaguchi在准周期情况下。少

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