Local energy decay of the wave equation in an exterior problem and without resonance in the neighborhood of the real line. (Décroissance de l’énergie locale de l’équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel.) (French) Zbl 0918.35081

Burq, Nicolas

Acta Math. 180, No. 1, 1-29 (1998).

The aim of this paper is to estimate the rate of decay of the local energy of the wave equation for the exterior problem and without resonance in the neighbourhood of the real line. The author shows that the decay is logarithmic in the case when the initial data have compact support. The proof is based on the existence of a region of the form \(\{\lambda\in\mathbb{C}: \text{Im }\lambda< Ce^{-\varepsilon| \lambda|}\}\) which does not contain poles of the scattering matrix. This last fast follows by using Carleman’s inequalities for Helmholtz operator.

Reviewer: C.Popa (Iaşi)

MSC:

35L20	Initial-boundary value problems for second-order hyperbolic equations
35B40	Asymptotic behavior of solutions to PDEs
35L05	Wave equation

Keywords:

rate of decay of the local energy; scattering matrix; Carleman’s inequalities for Helmholtz operator

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