The Rate of Convergence to the Asymptotics for the Wave Equation in an Exterior Domain

where Ω = R \ O, and O is a bounded open set in R with smooth boundary. Throughout this paper, we suppose that n is an odd integer with n ≥ 3. We assume that Ω is connected and that the initial data ~ f = (f0, f1) belongs to the associated energy space HD(Ω). Here and in the following, for an open set Y ⊂ R,HD(Y ) stands for the completion of (C 0 (Y )) 2 with respect to ‖~ f‖HD(Y ) = ‖∇f0‖L2(Y ) + ‖f1‖L2(Y ). U(t) denotes the propagator of the mixed problem (1.1) with (1.2) and (1.3); in other words, we define