Energy decay for wave equations with a potential and a localized damping

We consider the total energy decay together with the \(L^{2}\)-bound of the solution itself of the Cauchy problem for wave equations with a short-range potential and a localized damping, where we treat it in the one-dimensional Euclidean space \(\textbf{R}\). To study these, we adopt a simple multiplier method. In this case, it is essential that compactness of the support of the initial data not be assumed. Since this problem is treated in the whole space, the Poincaré and Hardy inequalities are not available as have been developed for the exterior domain case with \(n \ge 1\). However, the potential is effective for compensating for this lack of useful tools. As an application, the global existence of a small data solution for a semilinear problem is demonstrated.