Asymptotic behavior of the energy of vibration of a moving string with varying lengths

In this paper, the author studies the initial-moving boundary value problem of a linear second-order hyperbolic equation describing the vibration of an axially moving string with time-varying lengths. The explicit expression of the solution is presented, and the stability of the system is investigated. It is shown that the amplitude of vibration is uniformly bounded for any given initial conditions; moreover, the energy of the system increases without bound with polynomial growth as the length of the string tends to zero and decreases to zero with polynomial decay when the length tends to infinity.