Energy decay for the linear and semilinear wave equations in exterior domains with some localized dissipations

We derive the total energy decay and boundedness for the solutions to the initial boundary value problem for the wave equation in an exterior domain : with , where and a(x) is a nonnegative function which is positive near some part of the boundary and near infinity. We apply these estimates to prove the global existence of decaying solutions for semilinear wave equations with nonlinearity f(u) like . We note that no geometrical condition is imposed on the boundary . Received: 16 June 1999; in final form: 13 March 2000 / Published online: 4 May 2001