Local energy decay for the damped wave equation. (English) Zbl 1295.35078
Summary: We prove local energy decay for the damped wave equation on \(\mathbb R^d\). The problem which we consider is given by a long range metric perturbation of the Euclidean Laplacian with a short range absorption index. Under a geometric control assumption on the dissipation we obtain an almost optimal polynomial decay for the energy in suitable weighted spaces. The proof relies on uniform estimates for the corresponding “resolvent”, both for low and high frequencies. These estimates are given by an improved dissipative version of Mourre’s commutators method.
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35L15 | Initial value problems for second-order hyperbolic equations |
Keywords:
scattering theory; resolvent estimates; long range metric perturbation; short range absorption index; Mourre’s commutators methodReferences:
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