Stability Results for the Wave Equation with Indefinite Damping

The one-dimensional wave equation with damping of indefinite sign in a bounded interval with Dirichlet boundary conditions is considered. It is proved that solutions decay uniformly exponentially to zero provided the damping potential is in theBV-class, has positive average, is small enough and satisfies a finite number of further constraints guaranteeing that the derivative of the real part of the eigenvalues is negative when the damping vanishes. This sharp result completes a previous one by the first author showing that an indefinite sign damping always produces unstable solutions if it is large enough and it answers by the afirmative to a conjecture concerning small damping terms. The method of proof relies on the developments by S. Cox and the second author on the high frequency asymptotic expansion of the spectrum for damped wave equations and on the characterization of the decay rate in terms of the spectral abscissa.