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

<p style='text-indent:20px;'>We consider the initial-value problem for one-dimensional, time-dependent wave equation with positive, Lipschitz continuous coefficients, which are constant outside a bounded region. Under assumption of compact support initial data, we prove that local energy decays exponentially fast in time, and provide explicit to solution converges large times. We give estimates rate this exponential decay by two different techniques. The first one is based on definition modified, weighted energy, suitably constructed weights. second integral formulation and, under more restrictive variation allows us obtain improved rates.</p>