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

研究了在具有周期或准周期运动边界和边界函数的域上定义的一维或二维波动方程的IBVP问题。我们的研究目的是研究IBVP的解的行为，主要研究结果如下：1。由两个边界函数定义的简单组合函数A在解的行为中起着重要作用。反射特性的重要映射由组合函数定义。根据反射特性，阐明了问题的几何结构。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，则每一个解在空间和时间上都是准周期的。这一结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。从这个结果来看，J. Cooper的有趣结果是个例外。并利用数论的一些结果，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域映射，它将周期非柱域变换为柱域。重要的事实是，这种映射保留了波算符，这与其他结果不同。由此，我们证明了难以证明的非线性波动方程的周期解的存在性，并明确了非齐次波动方程的周期解的性质。一般情况下，上述A定义准周期动力系统是困难的。为了得到周期情况下的相应结果，我们定义了上（下）旋转数的新概念。利用这个，我们证明了重要的约简定理，应用这个定理，我们得到了结果。与PDS不同的是，这种情况下的边界函数是摄动形式。从这个意义上说，结果不是全局的，而周期情况具有全局特征。在膜振荡中，我们只考虑边界为圆且解应为球对称的情况。一般情况是极其困难的。少

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