If a second order semilinear conservative equation with esssentially oscillatory solutions such as the wave equation is perturbed by a possibly non monotone damping term which is effective in a nonnegligible sub-region for at least one sign of the velocity, all solutions of theperturbed system converge weakly to 0 as time tends to infinity. We present here asimple and natural method of proof of this kind of property, implying as a consequencesome recent very general results of Judith Vancostenoble.
