On multiplicative time-dependent perturbations of semigroups and cosine families generators

In this work we aim to investigate a second order PDE modelling a vibrating string. Our strategy consists in transforming the PDE problem into a semilinear second order ODE in a suitable infinite dimensional space. Since the tension coefficient of the PDE may vary with time, the linear operator of the ODE depends on time. We therefore provide sufficient conditions guaranteeing that a suitable family of unbounded linear operators generates a fundamental system.