Asymptotic behaviour of solutions for the wave equation with an effective dissipation around the boundary

We consider the initial–boundary value problem for the wave equation with a dissipation a(t, x)ut in an exterior domain, whose boundary meets no geometrical condition. We assume that the dissipation a(t, x)ut is effective around the boundary and a(t, x) decays as |x|→∞.We shall prove that the total energy does not in general decay, and the solution is asymptotically free as the time goes to infinity. Further, we shall show that the local energy decays like O(t −1) (t→∞).  2002 Elsevier Science (USA). All rights reserved.