L1 estimates for dissipative wave equations in exterior domains

We study the exterior initial–boundary value problem for the linear dissipative wave equation ( + ∂t )u = 0 in Ω × (0,∞) with (u, ∂t u)|t=0 = (u0,u1) and u|∂Ω = 0, where Ω is an exterior domain in N-dimensional Euclidean space RN. We first show higher local energy decay estimates of the solution u(t), and then, using the cut-off technique together with those estimates, we can obtain the L1 estimate of the solution u(t) when N 3, that is, u(t) L1(Ω) C( u0 Hn(Ω) + u1 Hn−1(Ω) + u0 Wn,1(Ω) + u1 Wn−1,1(Ω)) for t 0, where n = [N/2] is the integer part of N/2. Moreover, by induction argument, we derive the higher energy decay estimates of the solution u(t) for t 0. © 2006 Elsevier Inc. All rights reserved