Local Energy Decay in Even Dimensions for the Wave Equation with a Time-Periodic Non-Trapping Metric and Applications to Strichartz Estimates
Serdica Mathematical Journal, Tome 35 (2010) no. 4, pp. 329-370
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We obtain local energy decay as well as global Strichartz estimates for the solutions u of the wave equation ∂t2 u-divx(a(t,x)∇xu) = 0, t ∈ R, x ∈ Rn, with time-periodic non-trapping metric a(t,x) equal to 1 outside a compact set with respect to x. We suppose that the cut-off resolvent Rχ(θ) = χ(U(T, 0)− e−iθ)−1χ, where U(T, 0) is the monodromy operator and T the period of a(t,x), admits an holomorphic continuation to {θ ∈ C : Im(θ) ≥ 0}, for n ≥ 3, odd, and to {θ ∈ C : Im(θ) ≥ 0, θ ≠ 2kπ − iμ, k ∈ Z, μ ≥ 0} for n ≥ 4, even, and for n ≥ 4 even Rχ(θ) is bounded in a neighborhood of θ = 0.
Keywords:
Time-Dependent Perturbation, Non-Trapping Metric, Local Energy Decay, Strichartz Estimates
@article{SMJ2_2010_35_4_a1,
author = {Kian, Yavar},
title = {Local {Energy} {Decay} in {Even} {Dimensions} for the {Wave} {Equation} with a {Time-Periodic} {Non-Trapping} {Metric} and {Applications} to {Strichartz} {Estimates}},
journal = {Serdica Mathematical Journal},
pages = {329--370},
publisher = {mathdoc},
volume = {35},
number = {4},
year = {2010},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SMJ2_2010_35_4_a1/}
}
TY - JOUR AU - Kian, Yavar TI - Local Energy Decay in Even Dimensions for the Wave Equation with a Time-Periodic Non-Trapping Metric and Applications to Strichartz Estimates JO - Serdica Mathematical Journal PY - 2010 SP - 329 EP - 370 VL - 35 IS - 4 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/SMJ2_2010_35_4_a1/ LA - en ID - SMJ2_2010_35_4_a1 ER -
%0 Journal Article %A Kian, Yavar %T Local Energy Decay in Even Dimensions for the Wave Equation with a Time-Periodic Non-Trapping Metric and Applications to Strichartz Estimates %J Serdica Mathematical Journal %D 2010 %P 329-370 %V 35 %N 4 %I mathdoc %U http://geodesic.mathdoc.fr/item/SMJ2_2010_35_4_a1/ %G en %F SMJ2_2010_35_4_a1
Kian, Yavar. Local Energy Decay in Even Dimensions for the Wave Equation with a Time-Periodic Non-Trapping Metric and Applications to Strichartz Estimates. Serdica Mathematical Journal, Tome 35 (2010) no. 4, pp. 329-370. http://geodesic.mathdoc.fr/item/SMJ2_2010_35_4_a1/