Remarks on weak stabilization of semilinear wave equations

Université Pierre et Marie Curie, Analyse Numérique, Tour 55-65 5 étage, 4 place Jussieu, 75252 Paris Cedex 05, France; haraux@ann.jussieu.fr.

Received: 19 June 2000
Revised: 11 November 2000
Revised: 4 May 2001

Abstract

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 non negligible sub-region for at least one sign of the velocity, all solutions of the perturbed system converge weakly to 0 as time tends to infinity. We present here a simple and natural method of proof of this kind of property, implying as a consequence some recent very general results of Judith Vancostenoble.