Quasilinear equation in moving domain

In this article we are concerned with the existence and uniqueness of global weak solutions of a mixed problem associated with one-dimensional damped elastic stretched string equation when the supports of the ends have small displacements. In addition, we show that the energy decays exponentially. In previous investigations about string equation in moving domain, local or global solutions for increasing domain with the growth of the time has been shown. Here, thanks to the internal strong damping we eliminate this hypothesis.