Energy Decay for a Dissipative Wave Equation with Compactly Supported Data

Consider the Cauchy problem for the dissipative wave equation : utt − Δu + u = 0, u = u(x; t) in RN × (0,∞) with u(x,0) = u0(x) and ut(x,0) = u1(x). If {u0,u1} are compactly supported data from the energy space, then there exists a domain Xm in RN such that {x ∈ RN ||x| ≥ t1/2+δ} ⊊ Xm for large t ≥ 0 and ∫ Xm (|ut| + |∇u|2) dx ≤ C(1 + t)-m with m > 0 for t ≥ 0, and moreover, if u0 + u1 = 0, then ∫ Xm |u|2 dx ≤ C(1 + t)-m for t ≥ 0