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

在这个项目中，我们处理了线性和非线性波动方程在非圆柱域的周期或准周期的时间。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)考虑了时间周期非圆柱形域一维非线性波动方程的BVP问题。如果非线性强迫项、边界函数和边界数据在时间上是周期的且具有相同的周期，则BVP具有时间周期解。这个问题一直被认为是难题之一。(2)考虑了准周期非圆柱域一维线性波动方程的IBVP问题。方程的非齐次项和边界数据也是时间准周期的。正如我们在之前的研究项目中所展示的，解通常在时间上几乎是周期性的，因此在时间上是有界的。我们对这一现象进行了更深入的研究，发现存在着时间无界波的更多叠加态的解。(3)考虑了三维径向对称线性波动方程在时间-准周期非圆柱形域上的IBVP问题。我们证明了解在时间上几乎是周期性的。(4)考虑了空间域为球的时间周期非圆柱形域上三维径向对称非线性波动方程的BVP问题。在与(1)相似的假设下，BVP具有时间周期解。结果似乎很有趣。为了解决这些问题，我们研究了一些有用的方法。该方法包括将波动方程的BVP变换为某些泛函方程和将非柱域变换为柱域的域变换。前者由山口先生创立，后者由山口先生和吉田先生创立。该方法基于M. Herman和J. Yoccoz在周期情况下和M. Yamaguchi在准周期情况下的约简定理。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