Stability and instability of the wave equation solutions in a pulsating domain

The behavior of energy is studied for the real scalar held satisfying d'Alembert equation in a finite space interval 0 < x < a(t); the endpoint a(t) is assumed to move slower than the light and periodically in most parts of the paper. The boundary conditions are of Dirichlet and Neumann type. We give sufficient conditions for the unlimited growth, the boundedness and the periodicity of the energy E. The case of unbounded energy without infinite limit (0 < liminf(t-->+infinity)E(t) < limsup(t-->+infinity) E(t) = +infinity) is also possible. For the Neumann boundary condition, E may decay to zero as the time tends to infinity. If a is periodic, the solution is determined by a homeomorphism (F) over bar of the circle related to a. The behavior of E depends essentially on the number theoretical characteristics of the rotation number of (F) over bar.