130 research outputs found
On the meromorphic continuation of the resolvent for the wave equation with time-periodic perturbation and applications
Consider the wave equation , where
with and is -periodic in time and decays
exponentially in space. Let be the associated propagator and let
be the resolvent of
the Floquet operator defined for \im(\theta)>BT with
sufficiently large. We establish a meromorphic continuation of from
which we deduce the asymptotic expansion of
, where , as with a remainder term whose energy decays
exponentially when is odd and a remainder term whose energy is bounded with
respect to , with , when is even. Then,
assuming that has no poles lying in $\{\theta\in\C\ :\
\im(\theta)\geq0\}\theta\to0\partial_t^2u-\Delta_xu+V(t,x)u=F(t,x)$
Local energy decay for the wave equation with a time-periodic non-trapping metric and moving obstacle
Consider the mixed problem with Dirichelet condition associated to the wave
equation \partial_t^2u-\Div_{x}(a(t,x)\nabla_{x}u)=0, where the scalar metric
is -periodic in and uniformly equal to 1 outside a compact set
in , on a -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 establish sufficient conditions for local energy decay.Comment: Corrections of some misprint
Local energy decay in even dimensions for the wave equation with a time-periodic non-trapping metric and applications to Strichartz estimates
We obtain local energy decay as well as global Strichartz estimates for the
solutions of the wave equation $\partial_t^2 u-div_x(a(t,x)\nabla_xu)=0,\
t\in{\R},\ x\in{\R}^n,a(t,x)1xR_\chi(\theta)=\chi(\mathcal U(T, 0)-e^{-i\theta})^{-1}\chi\mathcal U(T, 0)Ta(t,x)\{\theta\in\mathbb{C}\ :\
\textrm{Im}(\theta) \geq 0\}n \geq 3\{
\theta\in\mathbb C\ :\ \textrm{Im}(\theta)\geq0,\ \theta\neq 2k\pi-i\mu,\
k\in\mathbb{Z},\ \mu\geq0\}n \geq4n \geq4R_\chi(\theta)\theta=0$
Stability in the determination of a time-dependent coefficient for wave equations from partial data
We consider the stability in the inverse problem consisting of the
determination of a time-dependent coefficient of order zero , appearing in a
Dirichlet initial-boundary value problem for a wave equation
in with a
bounded domain of , , from partial observations on
. The observation is given by a boundary operator associated to the
wave equation. Using suitable complex geometric optics solutions and Carleman
estimates, we prove a stability estimate in the determination of from the
boundary operator
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