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Local energy decay for the wave equation with a time-periodic non-trapping metric and moving obstacle

Yavar Kian

Centre de Physique Theorique CNRS-Luminy, Case 907, 13288 Marseille, France. email: Yavar.Kiand@cpt.univ-mrs.fr

ABSTRACT

Consider the mixed problem with Dirichelet condition associated to the wave equation , where the scalar metric periodic in t and uniformly equal to 1 outside a compact set in x, on a T-periodic domain. Let be the associated propagator. Assuming that the perturbations are non-trapping, we prove the meromorphic continuation of the cut-off resolvent of the Floquet operator and we establish sufficient conditions for local energy decay.

RESUMEN

Considere el problema mixto con condiciones de Dirichlet asociadas a la ecuación de onda , donde la metrica escalar a(t; x) es T-periódica en t y uniformemente igual a 1 fuera de un conjunto compacto en x, sobre un dominio T-periodico. Sea U(t,0) el propagador asociado. Asumiendo que las perturbaciones son non-trapping, probamos la continuacióon meromorfa de la resolvente de corte del operador de Floquet U(T, 0) y establecemos condiciones suficientes para la decadencia local de energía.

2010 AMS Mathematics Subject Classification: 35B40, 35L15 .

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Received: November 2011.Revised: November 2011.