Energy Decay for a Dissipative Wave Equation with Compactly Supported Data

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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 datafrom the energy space, then there exists a domain Xm in RN suchthat {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, ifu0 + u1 = 0, then∫Xm|u|2 dx ≤ C(1 + t)-m for t ≥ 0.

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