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,∂tu)|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.