Energy Decay Rate of Wave Equations with Indefinite Damping

We consider the one-dimensional wave equation with an indefinite sign damping and a zero order potential term. Using a shooting method, we establish the asymptotic expansion of eigenvalues and eigenvectors of the damped wave equation for a large class of coefficients. In addition, if the damping coefficient is “more positive than negative,” we prove that the energy of system decays uniformly exponentially to zero. This sharp result generalizes a previous work of Freitas and Zuazua (1996).