One dimensional wave equations in domain with quasiperiodically moving boundaries and quasiperiodic dynamical systems

Abstract

We shall deal with IBVP for a linear one-dimensional wave equation in domain with time-quasiperiodically oscillating boundaries. We shall show that for any given initial data and almost all boundary data, every solution is quasiperiodic in $t$, provided that the basic frequencies of timequasiperiodic data of IBVP satisfy the number-theoretic Diophantine conditions. In order to solve this problem, we shall show the reduction theorem of one-dimensional quasiperiodic dynamical systems. To prove the reduction theorem, we shall define upper and lower rotation numbers of dynamical systems and apply the rapidly iteration method to the related dynamical system defined by the boundary functions. Also we shall construct a class of time-quasiperiodic boundary data of IBVP and the basic frequencies such that IBVP has quasiperiodic solutions that are the superposition of the sequentially time-unbounded forward and backward waves.