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

We consider the total energy decay together with L^2-bound of the solution itself of the Cauchy problem for wave equations with a localized damping and a short-range potential. We treat it in the one dimensional Euclidean space R. We adopt a simple multiplier method to study them. In this case, it is essential that the compactness of the support of the initial data is not assumed. Since this problem is treated in the whole space, the Poincare and Hardy inequalities are not available as is developed in the exterior domain case. For compensating such a lack of useful tools, the potential plays an effective role. As an application, the global existence of small data solution for a semilinear problem is provided.