On the exponential time-decay for the one-dimensional wave equation with variable coefficients

We consider the initial-value problem for the one-dimensional, time-dependent wave equation with positive, Lipschitz continuous coefficients, which are constant outside a bounded region. Under the assumption of compact support of the initial data, we prove that the local energy decays exponentially fast in time, and provide the explicit constant to which the solution converges for large times. We give explicit estimates of the rate of this exponential decay by two different techniques. The first one is based on the definition of a modified, weighted local energy, with suitably constructed weights. The second one is based on the integral formulation of the problem and, under a more restrictive assumption on the variation of the coefficients, allows us to obtain improved decay rates.