CUBO A Mathematical Journal
Vol.14, No¯ 02, (153–173). June 2012

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

Yavar Kian

Centre de Physique Théorique CNRS-Luminy,

Case 907, 13288 Marseille, France.

email: Yavar.Kian@cpt.univ-mrs.fr

ABSTRACT

Consider the mixed problem with Dirichelet condition associated to the wave equation

∂2tu − divx(a(t,x)∇xu) = 0, where the scalar metric a(t,x) is T-periodic in t and

uniformly equal to 1 outside a compact set in x, on a T-periodic domain. Let U(t,0)

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

U(T,0) and we establish sufficient conditions for local energy decay.

RESUMEN

Considere el problema mixto con condiciones de Dirichlet asociadas a la ecuación de

onda ∂2tu − divx(a(t,x)∇xu) = 0, donde la métrica 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-periódico. Sea U(t,0) el propagador asociado. Asumiendo que las perturbaciones

son non-trapping, probamos la continuación meromorfa de la resolvente de corte del

operador de Floquet U(T,0) y establecemos condiciones suficientes para la decadencia

local de enerǵıa.

Keywords and Phrases: time-dependent perturbation, moving obstacle, local energy decay,

wave equation.

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



154 Yavar Kian CUBO
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1 Introduction

Let Ω be an open domain in R1+n, n ≥ 3 with C∞ boundary ∂Ω. Introduce the sets

Ω(t) = {x ∈ Rn : (t,x) ∈ Ω}, O(t) = Rn \ Ω(t), t ∈ R.

We assume that there exists ρ1 > 0 such that for all t ∈ R

O(t) ⊂ {x : |x| ≤ ρ1}. (1.1)

Moreover there exists T > 0 such that

O(t + T) = O(t), t ∈ R. (1.2)

For each (t,x) ∈ ∂Ω, let ν(t,x) = (νt(t,x),νx(t,x)) be the exterior unit normal vector to ∂Ω at

(t,x) ∈ ∂Ω pointing into Ω. Then, we assume that there exists 0 < c < 1 such that

|νt| < c|νx|. (1.3)

Consider the following mixed problem






utt − divx(a(t,x)∇xu) = 0, (t,x) ∈ Ω,

u|∂Ω = 0

(u,ut)(s,x) = (f1(x),f2(x)) = f(x), x ∈ Ω(s),

(1.4)

where the perturbation a(t,x) ∈ C∞(Rn+1) is a scalar function which satisfies the conditions:

(i) C ≥ a(t,x) ≥ c > 0, (t,x) ∈ Rn+1,

(ii) there exists ρ > ρ1 such that a(t,x) = 1 for |x| ≥ ρ,

(iii) there exists T > 0 such that a(t + T,x) = a(t,x), (t,x) ∈ Rn+1.

(1.5)

Throughout this paper we assume n ≥ 3. Consider the set H(t) which is the closure of the space

C∞0 (Ω(t)) × C
∞

0 (Ω(t)) with respect to the norm

‖f‖
H(t)

=







∫

Ω(t)

(

|∇xf1|
2
+ |f2|

2
)

dx







1

2

, f = (f1,f2) ∈ C
∞

0 (Ω(t)) × C
∞

0 (Ω(t)).

Let us introduce some general properties of solutions of (1.4). We show, in Section 1, that for

f ∈ H(s) there exists a unique solution of (1.4) and we introduce the propagator

U(t,s) : H(s) ∋ (f1,f2) = f 7→ U(t,s)f = (u,ut)(t,x) ∈ H(t) (1.6)

with u the solution of (1.4). Moreover, we prove that U(t,s) is a bounded operator satisfying the

following estimate

‖U(t,s)‖L(H(s),H(t)) ≤ Be
A|t−s|. (1.7)



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Local energy decay for the wave equation with a time-periodic ... 155

The goal of this paper is to establish sufficient conditions for a local energy decay taking the

form

‖χU(t,s)χ‖L(H(s),H(t)) ≤ Cχp(t − s), t ≥ s, (1.8)

with p(t) ∈ L1(R+) and χ ∈ C∞0 (|x| ≤ ρ + 1).

We study problem (1.4) under a non-trapping condition. More precisely, let U(t,s,x,x0) be

the kernel of the propagator U(t,s) and consider the following

(H1) For all r > 0, there exists T1(r) > 0 such that

U(t,s,x,x0) ∈ C
∞ ({(t,s,x,x0) : |x| ≤ r, |x0| ≤ r, |t − s| ≥ T1(r)}) .

From [15], we know that singularities propagate along null-bicharacteristics (with consideration of

their reflections from ∂Ω). Thus, one can show that condition (H1) is equivalent to the requirement

that all null-bicharacteristics of (1.4) with consideration of reflections from ∂Ω go out to infinity

as |t − s| → +∞. Let us recall that the non-trapping condition (H1) is necessary for (1.8) since

for some trapping perturbations we may have solutions with exponentially increasing energy (see

[7] for Ω = R1+n and [22] for a(t,x) = 1). On the other hand, even for non-trapping periodic per-

turbations some parametric resonances could lead to solutions with exponentially growing energy

(see [6] for time-periodic potentials). To exclude the existence of such solutions we must consider

a second assumption.

Many authors have investigated the local energy decay of wave equations. The main

hypothesis is that the perturbations are non-trapping. For a(t,x) = a0(x) independent of

time and fixed obstacles, the meromorphic continuation and estimates of the cut-off resolvent

χ
(

−divx(a0(x)∇x.) − λ
2
)−1

χ, where χ ∈ C∞0 (R
n) and λ ∈ C, are the main arguments for estimate

(1.8) (see [24], [25], [27] and [28]). From these results, by considering the connection between the

Fourier transform in time of the solutions and the stationary operator −divx(a0(x)∇x.) − λ
2, one

can deduce (1.8). For time dependent metric a(t,x) or moving obstacle, since the domain or the

Hamiltonian −divx(a(t,x)∇x.) are time-dependent, we cannot apply these arguments. However,

the analysis of the Floquet operator U(T,0) makes it possible to obtain (1.8) with T-periodic

perturbations and moving obstacle. In [8] the authors have extended the Lax-Phillips theory to

problem (1.4) with a(t,x) = 1 and they have established a local energy decay (1.8). By using the

compactness of the local evolution operator, deduced from a propagation of singularities, and the

RAGE theorem of Georgiev and Petkov (see [9]), Bachelot and Petkov have shown in [1] that in

the case of odd dimensions, the decay of the local energy associated to the wave equation with

time periodic potential is exponential for initial data with compact support included in a subspace

of finite codimension. Petkov has extended this result to even dimensions (see [21]), by using

the meromorphic continuation of the cut-off resolvent of the Floquet operator associated to this

problem.

Let us introduce the cut-off resolvent, associated to the Floquet operator U(T,0), defined by

Rψ1,ψ2(θ) = ψ1(U(T,0) − e
−iθ

)
−1ψ2 : H(0) → H(0), ψ1,ψ2 ∈ C

∞

0 (R
n
).



156 Yavar Kian CUBO
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According to (1.7), Rψ1,ψ2(θ) is a family of bounded operators analytic with respect to θ on

{θ ∈ C : I(θ) > AT}. Applying some arguments of [26], in Section 2, we show the meromorphic

continuation of Rψ1,ψ2(θ) to C for n odd and to {θ ∈ C : θ /∈ 2πZ+iR
−} for n even. Let us recall

that the meromorphic continuation of Rψ1,ψ2(θ) is closely related to the asymptotic expansion of

χU(t,0)χ, χ ∈ C∞0 (R
n), as t → +∞ (see Section 2 and the main theorem in [26]). Consequently, it

seems natural to consider the meromorphic continuations of Rψ1,ψ2(θ) that imply (1.8). Consider

the following assumption.

(H2) There exist φ1,φ2 ∈ C
∞

0 (R
n), satisfying φ1(x) = φ2(x) = 1 for |x| ≤ ρ + T + 2, such that

the operator Rφ1,φ2(θ) admits an analytic continuation from {θ ∈ C : Im(θ) ≥ A > 0}

to {θ ∈ C : Im(θ) ≥ 0}, for n ≥ 3, odd, and to {θ ∈ C : Im(θ) > 0} for n ≥ 4, even.

Moreover, for n even, Rφ1,φ2(θ) admits a continuous continuation from {θ ∈ C : Im(θ) > 0}

to {θ ∈ C : Im(θ) ≥ 0,θ 6= 2kπ,k ∈ Z} and we have

lim sup
λ→0

Im(λ)>0

‖Rφ1,φ2(λ)‖ < ∞.

Assuming (H1) and (H2) fulfilled, we obtain the following.

Theorem 1. Assume (1.1), (1.2), (1.3), (1.5), (H1) and (H2) fulfilled. Then, estimate (1.8) is

fulfilled with 




p(t) = e−δt for n ≥ 3 odd,

p(t) =
1

(t + 1) ln2(t + e)
for n ≥ 4 even.

(1.9)

Let us remark that, assuming (H1) fulfilled, (H2) is a necessary and sufficient condition for

estimate (1.8) with p(t) satisfying (1.9). Moreover, if (H2) is not fulfilled, even the uniform estimate

in time of the local energy ‖χU(t,0)χ‖L(H(0),H(t)) may not hold. For example, if Rφ1,φ2(θ) has a

pole θ0 ∈ C with I(θ0) > 0, one can establish the estimate

‖χU(t,0)χ‖L(H(0),H(t)) ≥ Ce
I(θ

0
)

T
t

and deduce existence of a solution with compactly supported initial data and exponentially growing

local energy. It has been established in [6] that these phenomenon can occur even with a non-

trapping condition. The goal of (H2) is to avoid existence of such solutions.

Remark 1. Let the metric (aij(t,x))1≤i,j≤n be such that for all i, j = 1 · · ·n we have

(i) there exists ρ > 0 such that aij(t,x) = δij, for |x| ≥ ρ, with δij = 0 for i 6= j and δii = 1,

(ii) there exists T > 0 such that aij(t + T,x) = aij(t,x), ∀(t,x) ∈ R
n+1,

(iii)aij(t,x) = aji(t,x),∀(t,x) ∈ R
n+1,

(iv) there exist C > c > 0 such that C|ξ|2 ≥

n∑

i,j=1

aij(t,x)ξiξj ≥ c|ξ|
2, ∀(t,x) ∈ R1+n, ξ ∈ Rn.



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Local energy decay for the wave equation with a time-periodic ... 157

If we replace a(t,x) in (1.4) we get the following mixed problem






utt −

n∑

i,j=1

∂

∂xi

(

aij(t,x)
∂

∂xj
u

)

= 0, (t,x) ∈ Ω,

u|∂Ω = 0,

(u,ut)(s,x) = (f1(x),f2(x)) = f(x), x ∈ Ω(s).

(1.10)

All the results of this paper remain valid for the mixed problem (1.10) and their proofs follow from

the same arguments.

Notice that the estimate

‖ψ1U(NT,0)ψ2‖L(H(0)) ≤
Cψ1,ψ2

(N + 1) ln2(N + e)
, N ∈ N, (1.11)

implies (1.8). On the other hand, if (1.11) is valid, the assumption (H2) for n even is fulfilled.

Indeed, for large A >> 1 and Im(θ) ≥ AT we have

Rψ1,ψ2(θ) = −e
iθ

∞∑

N=0

ψ1U(NT,0)ψ2e
iNθ

and applying (1.11), we conclude that Rψ1,ψ2(θ) admits an analytic continuation from

{θ ∈ C : Im(θ) ≥ A > 0} to {θ ∈ C : Im(θ) > 0}. Moreover, Rψ1,ψ2(θ) is bounded for θ ∈ R. In

Section 4, we give some examples of metrics a(t,x) and moving obstacle O(t) such that (1.11) is

fulfilled.

2 General properties

The purpose of this section is to establish some general properties of solutions of problem (1.4).

We will study the global well posedness of (1.4) and we will prove estimate (1.7). We start by

fixing the notion of solutions of (1.4).

Definition 2.1. A distribution u(t,x) ∈ D′(Ω) is called a solution of (1.4) if the following conditions

hold:

(i) (u(t, .),ut(t, .)) ∈ H(t) for each t ∈ R; extended inside O(t) by setting u(t,x) = 0, the

functions

t 7−→ ∇xu(t, .), t 7−→ ut(t, .)

are continuous with values in L2(Rn),



158 Yavar Kian CUBO
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(ii) (u(s, .),ut(s, .)) = (f1,f2) = f

(iii) ∂2tu − divx(a(t,x)∇xu) = 0 in Ω in the sense of distributions.

In the next result we obtain the existence and uniqueness of solutions of (1.4).

Theorem 2. Assume (1.1), (1.2), (1.3) and (1.5) fulfilled. Then, for each f ∈ H(s) there exists a

unique solution u(t, .) of (1.4) with the property that for each t > 0

sup
|t−s|≤D
|s|≤2D

‖(u(t, .),ut(t, .))‖H(t) ≤ CD ‖f‖H(s) (2.1)

Proof. First we treat the existence and uniqueness of the solution for small |t − s|. Given z ∈ Ω(s),

consider the cone

Cz,s = {(t,x) ∈ R
1+n : |x − z| ≤ |t − s|}.

For |t − s| small enough and for z outside a small neighborhood of ∂Ω(s) we obtain Cz,s ⊂ Ω.

Consequently, for (t,x) ∈ Cz,s the solution u(t,x) of the mixed problem coincides with the solution

of the Cauchy problem

{
utt − divx(a(t,x)∇xu) = 0, (t,x) ∈ R × R

n,

(u,ut)(s,x) = (f1(x),f2(x)) = f(x), x ∈ R
n,

(2.2)

with f extended by 0 for x ∈ O(s). Thus, for |t − s| ≤ ǫ and ǫ sufficiently small, we will determine

u(t,x) in some small neighborhood of ∂Ω∩ {|t − s| ≤ ǫ}. Given (s,z) with z ∈ ∂Ω(s), we establish

the existence and uniqueness of u(t,x) in some space-time neighborhood of (s,z). Covering the

compact set {s} × ∂Ω(s) by a finite number of such neighborhoods and using the local uniqueness

result for the points where these neighborhoods overlap, we deduce the existence and uniqueness

for small |t − s|. Introduce in a neighborhood of (s,z), z ∈ ∂Ω(s), local coordinates (t,y), y′ =

(y1, . . . ,yn−1), so that (s,z) is transformed into (0,0), while the boundary ∂Ω is given by yn =

g(t,y′) with g a C∞ function such that ∇y′g(0,0) = 0. Since

ν(t,y′,g(t,y′)) =
1

√

1 + |gt(t,y′)|
2
+ |∇y′g(t,y′)|

2
(−gt(t,y

′
),−∇y′g(t,y

′
),1),

statement (1.3) implies that

|gt(t,y
′)| < c(|∇y′g(t,y

′)| + 1) .

Thus, we have |gt(0,0)| < c. If we choose a sufficiently small neighborhood of (0,0) we can assume

that |gt(t,y
′)| < c. Changing variables

xj = yj, j = 1, . . . ,n − 1, xn = yn − g(t,y
′)

we transform

∂2t − divx(a(t,x)∇x·)



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Local energy decay for the wave equation with a time-periodic ... 159

into the operator P(t,x,∂t,∂x) with principal symbol

σ(P(t,x,∂t,∂x)) = − τ
2
+ 2gtτξn − 2ξnb(t,x)ξ

′ · ∇x′g + b(t,x) |ξ
′|
2

+
(

b(t,x) |∇x′g|
2
− g2t + b(t,x)

)

ξ2n,

where b(t,x) = a(t,y). Here (τ,ξ′,ξn) are the variable dual to (t,x
′,xn). Statement (1.3) and

property (1.5) imply that

b(t,x) |∇x′g|
2
− g2t + b(t,x) > 0. (2.3)

Consider the problem






P(t,x,∂t,∂x)u = 0 in Rt × R
n−1
x′ × R

+
xn
,

u(t,x′,0) = 0 in Rt × R
n−1
x′ ,

(u(0,x),ut(0,x)) = f(x).

(2.4)

We suitably extend the coefficients of P(t,x,∂t,∂x) to R
1+n preserving the strict hyperbolicity of

P(t,x,∂t,∂x) with respect to t. For the mixed problem (2.4) we can apply the results of Miyatake

[18] and Hörmander [10], Chapter XXIV. Notice that the inequality (2.3) guarantees that the

boundary xn = 0 is timelike in the sense of Hörmander [10]. The result of Miyatake [18] says that

if ∇xf1, f2 ∈ L
2
loc

(

R
n−1
x′

× R+xn

)

with f1 = f2 for xn = 0, then for |t| ≤ δ there exists a unique

solution u(t,x) ∈ H1
loc

(

R
n−1
x′ × R

+
xn

)

of (2.4) satisfying the estimate

∑

j+|β|

∥

∥

∥
∂
j
t∂
β
xu(t,x)

∥

∥

∥

L2
loc

(

R
n−1

x ′
×R+xn

) ≤ Cδ
∑

j+|β|

∥

∥

∥
∂
j
t∂
β
xu(0,x)

∥

∥

∥

L2
loc

(

R
n−1

x ′
×R+xn

)

with a constant Cδ depending on δ. Notice that (1.5) implies that the boundary xn = 0 is non-

characteristic for P(t,x,∂t,∂x). So u(t,x) ∈ C
∞

(

R
+
xn;D

′(Rn)
)

(see Theorem B.2.9 in Hörmander

[10]) and the trace u|xn=0 is meaningful. The same argument shows that ∇xu(t, .) and ut(t, .)

depend continuously on t. Thus we obtain the existence and uniqueness of the solution of (1.4) in

Ω∩ {|t − s| ≤ ǫ}. We can determine ǫ > 0 uniformly with respect to s, provided |s| ≤ 2D. Making

a construction by steps of length ǫ, we cover the interval |t − s| ≤ D and the proof is complete.

Following Theorem 2, we can introduce the propagator U(t,s) defined by (1.6). Combining

the results of Theorem 2 and the periodicity of O(t) and a(t,x), we deduce the following.

Proposition 1. Assume (1.1), (1.2), (1.3) and (1.5) fulfilled. Then, we have

U(t + T,s + T) = U(t,s), (2.5)

‖U(t,s)‖L(H(s),H(t)) ≤ Be
A|t−s|. (2.6)

Proof. The proof of (2.5) is trivial. Let us show estimate (2.6). Applying (2.1), we obtain

sup
|s|,|t|≤T

‖U(t,s)‖L(H(s),H(t)) = C < ∞.



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Let t,s ∈ R and let 0 ≤ t′,s′ < T be such that t = lT + t′ and s = kT + s′ with k,l ∈ Z. Then,

applying (2.5), we obtain

U(t,s) = U(t′,0)U((k − l)T,0)U(s′,0) = U(t′,0)U(T,0)k−lU(s′,0).

It follows that

‖U(t,s)‖L(H(s),H(t)) ≤ C
2(1 + C)|k−l| ≤ C2eln(1+C)|k−l| ≤ C2eln(1+C)|t−s|

and we obtain (2.6) with A = ln(1 + C).

Notice that, combing the arguments used in the proof of Theorem 2 with estimate (2.6), we

can show that the Duhamel’s principal holds. Let P1 and P2 be the projectors of C
2 defined by

P1(h) = h1, P2(h) = h2, h = (h1,h2) ∈ C
2

and let P1,P2 ∈ L(C,C2) be defined by

P1(h) = (h,0), P2(h) = (0,h), h ∈ C.

Denote by V(t,s) : L2(Ω(s)) → Ḣ1(Ω(t)) the operator defined by

V(t,s) = P1U(t,s)P
2.

Notice that for h ∈ L2(Ω(s)), w = V(t,s)h is the solution of






∂2t(w) − divx(a(t,x)∇xw) = 0,

w|∂Ω = 0,

(w,∂tw)|t=s = (0,h).

Let g(t,x) be a function defined on Ω such that, for A1 > A (with A the constant of (2.6)),

e−A1tg(t,x) ∈ L2(Ω) and g(t,x) = 0 for |x| ≥ b with b ≥ ρ + 1. Then there exists a unique

solution v of





∂2t(v) − divx(a(t,x)∇xv) = g(t,x),

v|∂Ω = 0,

(v,∂tv)|t=s = (0,0).

Moreover, this solution can be written in the following way

v(t, .) =

∫t

s

V(t,τ)g(τ, .)dτ. (2.7)



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Local energy decay for the wave equation with a time-periodic ... 161

3 The meromorphic continuation of the cut-off resolvent

Rψ1,ψ2(θ)

The goal of this section is to prove the meromorphic continuation of Rψ1,ψ2(θ), assuming (H1)

fulfilled. The main result of this section is the following.

Theorem 3. Assume (H1), (1.1), (1.2), (1.3) and (1.5) fulfilled. Let ψ1, ψ2 ∈ C
∞

0 (R
n). Then,

Rψ1,ψ2(θ) admits a meromorphic continuation from {θ ∈ C : I(θ) > AT} to C for n ≥ 3 odd and

to C′ = {θ ∈ C : θ /∈ 2πZ + iR−} for n ≥ 4 even. Moreover, for n ≥ 4 even, there exists ǫ0 > 0

such that, for |θ| ≤ ǫ0, we have

Rψ1,ψ2(θ) =
∑

k≥−m

∑

j≥−mk

Rkjθ
k(logθ)−j. (3.1)

Here Rk,j ∈ L(H(0)) and, for k < 0 or j > 0, Rk,j is a finite rank operator.

To prove Theorem 4, we will use some results of [26] and [13]. For this purpose, we introduce

some tools and definitions of [26].

Let γ ∈ C∞(R) be such that γ(t) = 1 for t ≥ −2T
3

− T
10

and γ(t) = 0 for t ≤ −2T
3

− 2T
10

. Set

V1(t,s) = γ(t − s)V(t,s).

We recall that the Fourier-Bloch-Gelfand transform F is defined by

F(φ)(t,θ) =

+∞∑

k=−∞

(

φ(t + kT, ·)eikθ
)

, φ ∈ C∞0 (R × R
n).

Applying (2.6), for I(θ) > AT, with A > 0 the constant of (2.6), we can define

F(χ1V1(t,s)χ2)(t,θ) =

+∞∑

k=−∞

(

χ1V1(t + kT,s)χ2e
ikθ
)

, χ1, χ2 ∈ C
∞

0 (R
n
)

and

F′(χ1V1(t,s)χ2)(t,θ) = e
itθ

T F(χ1V1(t,s)χ2)(t,θ), χ1, χ2 ∈ C
∞

0 (R
n
).

We will use the following definition of meromorphic continuation of a family of bounded operators.

Definition 1. Let H1 and H2 be Hilbert spaces. A family of bounded operators Q(t,s,θ) : H1 → H2

is said to be meromorphic with respect to θ in a domain D ⊂ C, if Q(t,s,θ) is meromorphically

dependent on θ for θ ∈ D and for any pole θ = θ0 the coefficients of the negative powers of θ−θ0

in the appropriate Laurent extension are finite-rank operators.

Denote C′ = {z ∈ C : z 6= 2kπ − iµ, k ∈ Z, µ ≥ 0} and consider the following meromorphic

continuation.



162 Yavar Kian CUBO
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Definition 2. We say that the family of operators Q(t,s,θ), which are C∞ with respect to t and

s, for t ∈ R and 0 ≤ s ≤ 2T
3
, and T-periodic with respect to t, has the property (S′) if: 1) for odd

n the operators Q(t,s,θ), θ ∈ C, and its derivatives with respect to t form a finitely-meromorphic

family; 2) For even n the operators Q(t,s,θ) and its derivatives with respect to t form a finitely-

meromorphic family for θ ∈ C′ . Moreover, in a neighborhood of θ = 0 in C′, Q(t,s,θ) has the

form

Q(t,s,θ) = θ−m
∑

j≥0

(

θ

Rt,s(logθ)

)j

Pj,t,s(logθ) + C(t,s,θ), (3.2)

where C(t,s,θ) is analytic with respect to θ, Rt,s is a polynomial, the Pj,t,s are polynomials of order

at most lj and log is the logarithm defined on C \ iR
−. Moreover, C(t,s,θ) and the coefficients of

the polynomials Rt,s and Pj,t,s are C
∞ and T-periodic with respect to t and C∞ with respect to s

for 0 ≤ s ≤ 2T
3
.

Remark 2. Notice that if Q(t,s,θ) satisfies (S′) then ∂tQ(t,s,θ) satisfies also (S
′).

In [26] Vainberg proposed a general approach to problems with time-periodic perturbations

including potentials, moving obstacles and high order operators, provided that the perturbations

are non-trapping. One of the main results of [26] is the following.

Theorem 4. (Theorem 10, [26]) Assume that the mixed problem (1.4) is well posed, the Duhamel’s

principal holds and let (2.6) and (H1) be fulfilled. Then, for all b ≥ ρ+1, there exists T2(b) > T1(b)

and an operator

R(t,s) : L2(Ω(s)) → Ḣ1(Ω(t))

such that the following conditions are fulfilled:

(i) R(t + T,s + T) = R(t,s),

(ii) R(t,s) is bounded,

(iii) for all χ1, χ2 ∈ C
∞

0 (|x| ≤ b), F
′(χ1R(t,s)χ2)(t,θ) admits a meromorphic continuation to the

lower half plane satisfying property (S′) and χ1R(t,s)χ2 = χ1V(t,s)χ2 for t − s ≥ T2(b).

In [26] Vainberg established the result of Theorem 4 for s = 0. In [13] it has been proven that

this result can be generalized to 0 ≤ s ≤ 2T
3
. Combining these results with the properties estab-

lished in Section 1, we obtain a meromorphic continuation of the Fourier-Bloch-Gelfand transform

of the solutions of (1.4) with initial data (0,g) and their derivatives of order 1 with respect to t.

Lemma 3.1. Assume (H1), (1.1), (1.2), (1.3) and (1.5) fulfilled. Then, for all ψ1, ψ2 ∈ C
∞

0 (R
n)

and all 0 ≤ s ≤ 2T
3
,

F′(ψ1V1(t,s)ψ2)(T,θ) and F
′(ψ1∂tV1(t,s)ψ2)(T,θ)

admit a meromorphic continuation with respect to θ, continuous with respect to s ∈
[

0, 2T
3

]

, from

{θ ∈ C : I(θ) > AT} to C for n ≥ 3 odd and to C′ = {θ ∈ C : θ /∈ 2πZ + iR−} for n ≥ 4 even.



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Local energy decay for the wave equation with a time-periodic ... 163

Moreover, for n ≥ 4 even, there exists ǫ0 > 0 such that, for |θ| ≤ ǫ0, we have

F′(ψ1V1(t,s)ψ2)(T,θ) =
∑

k≥−m

∑

j≥−mk

Qkj(s)θ
k(logθ)−j. (3.3)

F′(ψ1∂tV1(t,s)ψ2)(T,θ) =
∑

k≥−m

∑

j≥−mk

Skj(s)θ
k
(logθ)−j. (3.4)

Here Qkj(s), Skj(s) ∈ L(H(s),H(0)) and are continuous with respect to s for 0 ≤ s ≤
2T
3
.

Proof. According to Section 1, the mixed problem (1.4) is well posed, the Duhamel’s principal

holds, and (2.6), (2.7) are fulfilled. Thus, we can apply the results of Theorem 4. Choose b ≥

ρ + 1 such that suppψ1∪suppψ2 ⊂ {x : |x| ≤ b}. Take hb ∈ C
∞(R) such that hb(t) = 1 for

t ≥ T2(b) +
6T
5

and hb(t) = 0 for t ≤ T2(b) + T. Then, for all 0 ≤ s ≤
2T
3
, statement (iii) of

Theorem 4 implies

hb(t)ψ1V1(t,s)ψ2 = hb(t)ψ1R(t,s)ψ2.

Thus, F′(hb(t)ψ1V1(t,s)ψ2)(t,θ) admits a meromorphic continuation satisfying property (S
′).

From now on, we assume that T2(b) = k0T with k0 ∈ N. For I(θ) > AT we have

F′(ψ1V1(t,s)ψ2)(T,θ) = F
′(hb(t)ψ1V1(t,s)ψ2)(T,θ) + F

′[(1 − hb(t))ψ1V1(t,s)ψ2](T,θ). (3.5)

Since 1 − hb(t) = 0 for t ≥ T2(b) +
6T
5

= (k0 + 1)T +
T
5
, for I(θ) > AT, we get

F′[(1 − hb(t))ψ1V1(t,s)ψ2](T,θ) = e
iθ

[

k0+1∑

k=1

(ψ1V(kT,s)ψ2e
ikθ

) + γ(−s)ψ1V(0,s)ψ2

]

.

Thus, F′[(1−hb(t))ψ1V1(t,s)ψ2](T,θ) admits an analytic continuation to C. Combining the mero-

morphic continuation of F′(hb(t)ψ1V1(t,s)ψ2)(T,θ), the analytic continuation of

F′[(1−hb(t))ψ1V1(t,s)ψ2](T,θ) and representation (3.5), we obtain the meromorphic continuation

of F′(ψ1V1(t,s)ψ2)(T,θ). It remains to prove the meromorphic continuation of F
′(ψ1∂tV1(t,s)ψ2)(T,θ).

Notice that

∂tV(t,s) = P2U(t,s)P
2

and, for I(θ) > AT, F′(ψ1V1(t,s))(t,θ) is well defined. For I(θ) > AT, we have

∂t [F
′
(hb(t)ψ1V1(t,s)ψ2)(t,θ)] =

iθ

T
F′(hb(t)ψ1V1(t,s)ψ2)(t,θ) + F

′
(h′b(t)ψ1V1(t,s)ψ2)(t,θ)

+ F′(hb(t)ψ1∂tV1(t,s))(t,θ)

Thus, for I(θ) > AT, we get

F′(hb(t)ψ1∂tV1(t,s)ψ2)(t,θ) =∂t [F
′(hb(t)ψ1V1(t,s)ψ2)(t,θ)] − F

′(h′b(t)ψ1V1(t,s)ψ2)(t,θ)

−
iθ

T
F′(hb(t)ψ1V1(t,s)ψ2)(t,θ)

(3.6)



164 Yavar Kian CUBO
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Since F′(hb(t)ψ1V1(t,s)ψ2)(t,θ) admits a meromorphic continuation satisfying property (S
′)

∂t [F
′(hb(t)ψ1V1(t,s)ψ2)(t,θ)] and

iθ

T
F′(hb(t)ψ1V1(t,s)ψ2)(t,θ)

admit also a meromorphic continuation satisfying property (S′). Moreover, since h′b(t) = 0 for

t ≥ T2(b) +
6T
5
, F′(h′b(t)ψ1V1(t,s)ψ2)(t,θ) admits an analytic continuation with respect to θ. It

follows that F′(hb(t)ψ1∂tV1(t,s)ψ2)(t,θ) admits a meromorphic continuation satisfying property

(S′). We conclude by repeating the arguments used for proving the meromorphic continuation of

F′(ψ1V1(t,s)ψ2)(T,θ).

Consider the operator defined by

U(t,s) = P1U(t,s)P
1.

For all h ∈ Ḣ1(Ω(s)), w = U(t,s)h is the solution of






∂2tw − divx(a(t,x)∇xw) = 0,

w|∂Ω = 0,

(w,wt)|t=s = (h,0).

Let γ1 ∈ C
∞(R) be such that γ1(t) = 1 for t ≥ −

T
20

and γ1(t) = 0 for t ≤ −
T
10
. Set

U1(t,s) = γ1(t − s)U(t,s).

Applying (2.6), for I(θ) > AT and ψ1 ψ2 ∈ C
∞

0 (R
n) we can define F′(ψ1U1(t,s)ψ2)(t,θ) and

F′(ψ1∂tU1(t,s)ψ2)(t,θ). From the results of Lemma 1 we obtain the following meromorphic

continuation of F′(ψ1U1(t,s)ψ2)(T,θ) and F
′(ψ1∂tU1(t,s)ψ2)(T,θ).

Lemma 3.2. Assume (H1), (1.1), (1.2), (1.3) and (1.5) fulfilled. Then, for all ψ1, ψ2 ∈ C
∞

0 (R
n),

F′(ψ1U1(t,s)ψ2)(T,θ) and F
′(ψ1∂tU1(t,s)ψ2)(T,θ)

admit a meromorphic continuation with respect to θ, continuous with respect to s ∈
[

0, 2T
3

]

, from

{θ ∈ C : I(θ) > AT} to C for n ≥ 3 odd and to C′ for n ≥ 4 even. Moreover, for n ≥ 4 even,

there exists ǫ0 > 0 such that, for |θ| ≤ ǫ0, we have

F′(ψ1U1(t,0)ψ2)(T,θ) =
∑

k≥−m

∑

j≥−mk

Mkjθ
k
(logθ)−j. (3.7)

F′(ψ1∂tU1(t,0)ψ2)(T,θ) =
∑

k≥−m

∑

j≥−mk

Nkjθ
k(logθ)−j. (3.8)

Here Mkj, Nkj ∈ L(H(0)) and, for k < 0 or j > 0, Mkj, Nkj are a finite rank operator.



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Local energy decay for the wave equation with a time-periodic ... 165

Proof. Let α ∈ C∞(R) be such that α(t) = 0 for t ≤ T
2

and α(t) = 1 for t ≥ 2T
3
. For all

h ∈ Ḣ1(Ω(0)), Z = α(t)U(t,0)h is the solution of






∂2tZ − divx(a(t,x)∇xZ) = [∂
2
t,α](t)U(t,0)h,

Z|∂Ω = 0,

(Z,∂tZ)|t=0 = (0,0).

(3.9)

We deduce from the Cauchy problem (3.9) the following representation

U(t,0) = α(t)U(t,0) =

∫t

0

V(t,s)[∂2t,α](s)U(s,0)ds, t ≥ T. (3.10)

Since [∂2t,α](t) = 0 for t >
2T
3
, the formula (3.10) becomes

U(t,0) =

∫ 2T
3

0

V(t,s)[∂2t,α](s)U(s,0)ds, t ≥ T.

Let R > 0 be such that

suppψ1 ∪ suppψ2 ⊂ {x : |x| ≤ R}.

Choose b = R + ρ + T + 1 and take χ ∈ C∞0 (|x| ≤ b) such that

χ(x) = 1 for |x| ≤ R + ρ + T.

The finite speed of propagation implies

ψ1U(t,0)ψ2 =

∫ 2T
3

0

ψ1V(t,s)χ[∂
2
t,α](s)U(s,0)ψ2ds, t ≥ T. (3.11)

Thus, for I(θ) > AT, we obtain

F′(ψ1U1(t,0)ψ2)(T,θ) =F
′

[∫ 2T
3

0

ψ1V1(t,s)χ[∂
2
t,α](s)U(s,0)ψ2ds

]

(T,θ)

−

∫ 2T
3

0

ψ1V1(0,s)χ[∂
2
t,α](s)U(s,0)ψ2ds + e

iθψ1ψ2

and it follows

F′ [ψ1U1(t,0)ψ2] (T,θ) =

∫ 2T
3

0

F′ [ψ1V1(t,s)χ] (T,θ)[∂
2
t,α](s)U(s,0)ψ2ds

−

∫ 2T
3

0

ψ1V1(0,s)χ[∂
2
t,α](s)U(s,0)ψ2ds + e

iθψ1ψ2.

Combining this representation with the meromorphic continuation of F′(ψ1V1(t,s)χ)(T,θ) estab-

lished in Lemma 1, we prove the meromorphic continuation of F′(ψ1U(t,0)ψ2)(T,θ) as well as (3.7).



166 Yavar Kian CUBO
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It remains to prove the meromorphic continuation of F′(ψ1∂tU(t,0)ψ2)(T,θ). Let β ∈ C
∞

0 (R
n).

The formula (3.11) implies that, for t ≥ T, we have

∂tU(t,0)β =

∫ 2T
3

0

∂tV(t,s)[∂
2
t,α](s)U(s,0)βds.

By density, this leads to

ψ1∂tU(t,0)ψ2 =

∫ 2T
3

0

ψ1∂tV(t,s)χ[∂
2
t,α](s)U(s,0)ψ2ds, t ≥ T

and, for I(θ) > AT, we get

F′(ψ1∂tU1(t,0)ψ2)(T,θ) =

∫ 2T
3

0

F′(ψ1∂tV1(t,s)χ)(T,θ)[∂
2
t,α](s)U(s,0)ψ2ds

−

∫ 2T
3

0

ψ1V1(0,s)χ[∂
2
t,α](s)U(s,0)ψ2ds.

We conclude by combining this representation with the results of Lemma 1.

Proof of Theorem 4. By definition, we can write

γ1(t)ψ1U(t,0)ψ2 =

(

γ1(t)ψ1U(t,0)ψ2 γ1(t)ψ1V(t,0)ψ2

γ1(t)ψ1∂tU(t,0)ψ2 γ1(t)ψ1∂tV(t,0)ψ2

)

.

Moreover, for I(θ) > AT, we have

F′ [γ1(t)ψ1U(t,0)ψ2] (T,θ) = e
iθ

∞∑

k=0

(

ψ1U(T + kT,0)ψ2e
ikθ
)

= −e−iθRψ1,ψ2(θ) (3.12)

and we obtain

Rψ1,ψ2(θ) = −e
iθF′ [γ1(t)ψ1U(t,0)ψ2] (T,θ)

= −

(

eiθF′(ψ1U1(t,0)ψ2)(T,θ) e
iθF′(ψ1V1(t,0)ψ2)(T,θ)

eiθF′(ψ1∂tU1(t,0)ψ2)(T,θ) e
iθF′(ψ1∂tV1(t,0)ψ2)(T,θ)

)

.

Thus, combining the results of Lemma 1 and Lemma 2, we prove Theorem 4. �

4 Local energy decay

The goal of this section is to prove Theorem 1, assuming (H1) and (H2) fulfilled. For this purpose,

we show how assumption (H2) alter the meromorphic continuation of Rψ1,ψ2(θ) established in

Section 2. Then, by integrating on a suitable contour, we prove the local energy decay. We treat

separately the case of odd and even dimensions. We start with n odd.



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Local energy decay for the wave equation with a time-periodic ... 167

Lemma 4.1. Assume n ≥ 3 odd, (1.1), (1.2), (1.3), (1.5), (H1) and (H2) fulfilled. Then, for all

ψ1, ψ2 ∈ C
∞

0 (|x| ≤ ρ + 1), we get

‖ψ1U(t,s)ψ2‖L(H(s),H(t)) ≤ Ce
−δ(t−s), t ≥ s. (4.1)

Proof. Notice that, for I(θ) > AT, F′ [γ1(t)φ1U(t,0)φ2] (t,θ) is T-periodic with respect to t and

2π-periodic with respect to θ(see [26] Theorem ). Applying (3.12), we get

F′ [γ1(t)φ1U(t,0)φ2] (dT,θ) = F
′ [γ1(t)φ1U(t,0)φ2] (T,θ) = −e

−iθRφ1,φ2(θ). (4.2)

Moreover, from [26] we have the following inversion formula (see Lemma 1 of [26])

φ1U(dT,0)φ2 =
1

2π

∫

[i(A+1)T−π,i(A+1)T+π]

e−idθF′ [γ1(t)φ1U(t,0)φ2] (T,θ)dθ. (4.3)

We will show (4.1), by combining these statements with assumption (H2).

First, assumption (H2) and (4.2) imply that F′ [γ1(t)φ1U(t,0)φ2] (dT,θ) has no poles on

{θ : I(θ ≥ 0)}. It follows that there exists δ > 0 such that F′ [γ1(t)φ1U(t,0)φ2] (dT,θ) has no

poles on {θ : I(θ) ≥ −δT, −π ≤ Re(θ) ≤ π}. Consider the contour C1 defined by

C1 = [i(A+1)T+π,i(A+1)T−π]∪[i(A+1)T−π,−iδT−π]∪[−iδT−π,−iδT+π]∪[−iδT+π,i(A+1)T+π].

The Cauchy formula implies
∫

C1

e−idθF′ [γ1(t)φ1U(t,0)φ2] (T,θ)dθ = 0.

Also, since F′ [γ1(t)φ1U(t,0)φ2] (T,θ) is 2π-periodic with respect to θ we have
∫

[i(A+1)T−π,−iδT−π]

e−idθF′ [γ1(t)φ1U(t,0)φ2] (T,θ)dθ = −

∫

[−iδT+π,i(A+1)T+π]

e−idθF′ [γ1(t)φ1U(t,0)φ2] (T,θ)dθ

and we obtain
∫

[i(A+1)T−π,i(A+1)T+π]

e−idθF′ [γ1(t)φ1U(t,0)φ2] (T,θ)dθ =

∫

[−iδT−π,−iδT+π]

e−idθF′ [γ1(t)φ1U(t,0)φ2] (T,θ)dθ. (4.4)

It is obvious that
∫

[−iδT−π,−iδT+π]

e−idθF′ [γ1(t)φ1U(t,0)φ2] (T,θ)dθ = e
−δ(dT)

∫

[−π,π]

e−idθF′ [γ1(t)φ1U(t,0)φ2] (T,θ − iδT)dθ

and combining this with (4.4) and the inversion formula (4.3), we get

‖φ1U(dT,0)φ2‖L(H(0)) ≤ Ce
−δ(dT). (4.5)

Now let ψ1, ψ2 ∈ C
∞

0 (|x| ≤ ρ + 1), and let t, s ∈ R be such that t ≥ s. we write t = t
′ + mT and

s = s′ + kT with 0 ≤ t′,s′ < T and m, k ∈ N. The finite speed of propagation implies

ψ1U(t,s)ψ2 = ψ1U(t
′,0)φ1U((m − k)T,0)φ2U(0,s

′
)ψ2.



168 Yavar Kian CUBO
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Then, applying (4.5) and Theorem 2, we obtain

‖ψ1U(t,s)ψ2‖L(H(s),H(t)) ≤ C‖φ1U((m − k)T,0)φ2‖L(H(0)) ≤ C
′e−δ((m−k)T) ≤ C′e−δ(t−s).

Lemma 4.2. Assume n ≥ 4 even, (1.1), (1.2), (1.3), (1.5), (H1) and (H2) fulfilled. Then, for all

ψ1, ψ2 ∈ C
∞

0 (|x| ≤ ρ + 1), we get

‖ψ1U(t,s)ψ2‖L(H(s),H(t)) ≤ Cp(t − s), t ≥ s (4.6)

with

p(t) =
1

(t + 1) ln
2
(t + e)

.

Proof. Repeating the arguments used in the proof of Lemma 3, we obtain that F′ [γ1(t)φ1U(t,0)φ2] (T,θ)

has no poles on {θ ∈ C′ : Im(θ) ≥ 0}. Moreover, representation (3.1) implies that there exists

ǫ0 > 0 such that for θ ∈ C
′ with |θ| ≤ ǫ0 we have

F′ [γ1(t)φ1U(t,0)φ2] (T,θ) =
∑

k≥−m

∑

j≥−mk

Rkjθ
k(logθ)−j (4.7)

and assumption (H2) implies that in this representation we have Rkj = 0 for k < 0 or k = 0 and

j < 0. It follows that, for θ ∈ C′ with |θ| ≤ ǫ0, we obtain the following representation

F′ [γ1(t)φ1U(t,0)φ2] (T,θ) = A(θ) + Bθ
m0 log(θ)−µ + o

θ→0

(

θm0 log(θ)−µ
)

(4.8)

with A(θ) analytic with respect to θ for |θ| ≤ ǫ0 , B a finite-dimensional operator, m0 ≥ 0 and

µ ≥ 1. Since F′ [γ1(t)φ1U(t,0)φ2] (T,θ) has no poles on {θ ∈ C
′ : Im(θ) ≥ 0}, there exists

0 < δ ≤
ǫ0

T
and 0 < ν < ǫ0 sufficiently small such that F

′ [γ1(t)φ1U(t,0)φ2] (T,θ) has no poles on

{θ ∈ C : Im(θ) ≥ −δT, −π ≤ Re(θ) ≤ −ν, ν ≤ Re(θ) ≤ π}.

Consider the contour Σ = Γ1 ∪ ω ∪ Γ2 where Γ1 = [−iδT − π,−iδT − ν], Γ2 = [−iδ + ν,−iδ + π].

The contour ω of C, is a curve connecting −iδT − ν and −iδT + ν symmetric with respect to

the axis Re(θ) = 0. The part of ω lying in {θ : Im(θ) ≥ 0} is a half-circle with radius ν,

ω∩{θ : Re(θ) < 0, Im(θ) ≤ 0} = [−ν−iδT,−ν] and ω∩{θ : Re(θ) > 0, Im(θ) ≤ 0} = [ν,ν−iδT].

Thus, ω is included in the region where we have no poles of F′ [γ1(t)φ1U(t,0)φ2] (T,θ). Consider

the closed contour

C2 = [i(A + 1)T + π,i(A + 1)T − π] ∪ [i(A + 1)T − π,−iδT − π] ∪ Σ ∪ [−iδT + π,i(A + 1)T + π].

An application of the Cauchy formula yields

∫

C2

e−idθF′ [γ1(t)φ1U(t,0)φ2] (T,θ)dθ = 0.



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Applying the same arguments as those used in the proof of Lemma 3, we obtain

∫

[i(A+1)T−π,i(A+1)T+π]

F [γ1(t)φ1U(t,0)φ2] (dT,θ)dθ =

∫

Σ

e−idθF′ [γ1(t)φ1U(t,0)φ2] (T,θ)dθ

and the inversion formula (4.3) implies

φ1U(dT,0)φ2 =
1

2π

∫

Σ

e−idθF′ [γ1(t)φ1U(t,0)φ2] (T,θ)dθ, d ∈ N. (4.9)

Combining this representation with (4.8) and applying some arguments used in Lemma 2 and

Lemma 3 of [13], we obtain (4.6).

Combining the results of Lemma 3 and Lemma 4, we prove Theorem 1.

5 Examples of metrics a(t,x) and obstacles O(t)

In this section we will apply some properties of solutions of the wave equations with non-trapping

metrics independent of t and fixed obstacle to construct time periodic metrics and moving obstacles

such that conditions (H1) and (H2) are fulfilled. For this purpose, we assume that (H1) is fulfilled

for the metrics a(t,x) and obstacle O(t) that we consider and we will establish examples for (H2).

In order to prove (H2), we will modify the size T of the period of a(t,x). This choice is justified

by the properties of U(t,s).

Let T1 > 0 and let ((aT(t,x),OT(t)))T≥T1 be a family of couples of functions and obstacles

such that the following conditions are fulfilled:

(H3i) aT(t,x) and OT(t) are T-periodic with respect to t and aT(t,x) satisfies (1.5),

(H3ii) for all T ≥ T1, if (a(t,x),O(t)) = (aT(t,x),OT(t)) then conditions (1.1), (1.2), (1.3) and

(H1) are fulfilled,

(H3iii) there exist a function a1(x) and an obstacle O independent of t such that for

(a(t,x),O(t)) = (a1(x),O)

condition (H1) is fulfilled and, for all T1 ≤ t ≤ T, we have aT(t,x) = a1(x) and OT(t) = O.

Let H be the closure of the space C∞0 (R
n \ O) × C∞0 (R

n \ O) with respect to the norm

‖f‖H =







∫

Rn\O

(

|∇xf1|
2
+ |f2|

2
)

dx







1

2

, f = (f1,f2) ∈ C
∞

0 (R
n
\ O) × C∞0 (R

n
\ O).



170 Yavar Kian CUBO
14, 2 (2012)

Consider the following Cauchy problem





vtt − divx(a1(x)∇xv) = 0, t ∈ R, x ∈ R
n \ O

v|∂O = 0, t ∈ R,

(v,vt)(0) = f,

(5.1)

and the associate propagator

V(t) : H ∋ f 7−→ (v,vt)(t) ∈ H.

Let u be solution of (2.2). For T1 ≤ t ≤ T we have

∂2tu − divx(a1(x)∇xu) = ∂
2
tu − divx(aT(t,x)∇xu) = 0.

It follows that for (a(t,x),O(t)) = (aT(t,x),OT(t)) we get

U(t,s) = V(t − s), T1 ≤ s < t ≤ T (5.2)

and

H(t) = H, T1 ≤ t ≤ T. (5.3)

The asymptotic expansion of χV(t)χ as t → +∞ has been studied by many authors (see [25],

[24] and [28]). It has been proven that, for non-trapping metrics and for n ≥ 3, the local energy

decreases. To prove (H2), we will apply the following result.

Theorem 5. Assume n ≥ 3. Let φ ∈ C∞0 (R
n). Then, we have

‖φV(t)φ‖L(H) ≤ Cφp(t) (5.4)

with {
p(t) =e−δt for n odd,

p(t) = 〈t〉
1−n

for n even.

Estimate (5.4) has been established by Vainberg in [24], [25] but also by Vodev in [27] and

[28]. For n ≥ 4 even we will use the following identity.

Lemma 5.1. Let ψ ∈ C∞0 (|x| ≤ ρ+ 1+T1) be such that ψ = 1, for |x| ≤ ρ+
1
2
+ T1. Then, we have

U(T1,0) − V(T1) = ψ(U(T1,0) − V(T1)) = (U(T1,0) − V(T1))ψ. (5.5)

Proof. First, notice that (5.3) implies H(0) = H. Now, choose g ∈ H(0) = H and let w be the

function defined by (w,wt)(t) = U(t,0)(1−ψ)g. The finite speed of propagation implies that, for

0 ≤ t ≤ T1 and |x| ≤ ρ +
1
2
, we get w(t,x) = 0. Moreover, we have

divx(a1(x)∇x) = ∆x = divx(a(t,x)∇x), for |x| > ρ. (5.6)

Thus, w is solution on 0 ≤ t ≤ T1 of the problem





wtt − divx(a1(x)∇xw) = 0, t ∈ R, x ∈ R
n \ O

w|∂O = 0, t ∈ R,

(w,wt)(0) = (1 − ψ)g



CUBO
14, 2 (2012)

Local energy decay for the wave equation with a time-periodic ... 171

and it follows that

(U(T1,0) − V(T1))(1 − ψ) = 0. (5.7)

Now, let u and v be the functions defined by (u,ut)(t) = U(t,0)g and (v,vt)(t) = V(t)g with

g ∈ H. Applying (5.6), we can easily show that on (1 − ψ)u is the solution of

{
∂2t((1 − ψ)u)) − ∆x((1 − ψ)u)) = [∆x,ψ]u,

(((1 − ψ)u),((1 − ψ)u)t)(0) = (1 − ψ)g,

and (1 − ψ)v is the solution of

{
∂2t(((1 − ψ)v)) − ∆x((1 − ψ)v)) = [∆x,ψ]v,

(((1 − ψ)v),((1 − ψ)v)t)(0) = (1 − ψ)g.

We have

(1 − ψ)(U(T1,0) − V(T1)) = 0. (5.8)

Combining (5.7) and (5.8), we get (5.5).

Combining the arguments used in the proofs of Lemma 7, 8 and 9 and Theorem 14 of [13]

with the identity (5.5), we obtain the following.

Theorem 6. Assume n ≥ 3 and let ((aT(t,x),OT(t)))T≥T1 satisfy (H3i), (H3ii), (H3iii). Then, for

T large enough and for (a(t,x),O(t)) = (aT(t,x),OT(t)), assumption (H2) is fulfilled.

Received: November 2011. Revised: November 2011.

References

[1] A. Bachelot and V. Petkov, Existence des opérateurs d’ondes pour les systèmes hyperboliques

avec un potentiel périodique en temps, Ann. Inst. H. Poincaré (Physique théorique), 47 (1987),

383-428.

[2] C. O. Bloom and N. D. Kazarinoff, Energy decays locally even if total energy grows alge-

braically with time, J. Differential Equation, 16 (1974), 352-372.

[3] J-F. Bony and V. Petkov, Resonances for non trapping time-periodic perturbation, J. Phys.

A: Math. Gen., 37 (2004), 9439-9449.

[4] N. Burq, Décroissance de l’énergie locale de l’équation des ondes pour le problme extérieur et

absence de résonance au voisinage du réel, Acta Mathématica, 180 (1998), 1-29.

[5] N. Burq, Global Strichartz estimates for non-trapping geometries: About an article by

H.Smith and C.Sogge, Commun. PDE, 28 (2003), 1675-1683.

[6] F. Colombini, V. Petkov and J. Rauch, Exponential growth for the wave equation with com-

pact time-periodic positive potential, Comm. Pure Appl. Math., 62 (2009), 565-582.



172 Yavar Kian CUBO
14, 2 (2012)

[7] F. Colombini and J. Rauch, Smooth localised parametric resonance for wave equations, J.

Reine Angew. Math., 616 (2008), 1-14.

[8] J. Cooper and W. Strauss, Scattering theory of waves by periodically moving bodies, J. Funct.

Anal., 47 (1982), 180-229.

[9] V. Georgiev and V. Petkov, RAGE theorem for power bounded operators and decay of local

energy for moving obstacles, Ann. Inst. H. Poincaré Phys. Théor, 51 (1989), no.2, 155-185.

[10] L. Hörmander, The analysis of linear partial differential operators III, Springer-Verlag, 1985.

[11] Y. Kian, Strichartz estimates for the wave equation with a time-periodic non-trapping metric,

Asymptotic Analysis, 68 (2010), 41-76.

[12] Y. Kian, Cauchy problem for semilinear wave equation with time-dependent metrics, Nonlinear

Analysis, 73 (2010), 2204-2212.

[13] Y. Kian, Local energy decay in even dimensions for the wave equation with a time-periodic

non-trapping metric and applications to Strichartz estimates, Serdica Math. J., 36 (2010),

329-370.

[14] R. Melrose, Singularities and energy decay in acoustical scattering, Duke Math. J., 46 (1979),

43-59.

[15] R. Melrose and J. Sjöstrand, Singularities of boundary value problem, Comm. Pure Appl.

Math., I, 31 (1978), 593-617, II, 35 (1982), 129-168.

[16] C. Morawetz, J. Ralston, W. Strauss, Decay of solutions of wave equations outside non-

trapping obstacle, Comm. Pure Appl. Math., 30 (1977), no.4, 447-508.

[17] J. L. Metcalfe, Global Strichartz estimates for solutions to the wave equation exterior to a

convex obstacle, Trans. Amer. Math. Soc, 356 (2004), no.12, 4839-4855.

[18] S. Miyatake, Mixed problems for hyperbolic equation of second order, J. Math. Kyoto Univ.,

13 (1973), 435-487.

[19] H. F. Smith and C. Sogge, Global Strichartz estimates for non-trapping perturbations of the

Laplacian, Commun. PDE, 25 (2000), 2171-2183.

[20] V. Petkov, Scattering theory for hyperbolic operators, North Holland, Amsterdam, 1989.

[21] V. Petkov, Global Strichartz estimates for the wave equation with time-periodic potentials,

J. Funct. Anal., 235 (2006), 357-376.

[22] G. Popov and Ts. Rangelov, Exponential growth of the local energy for moving obstacles,

Osaka J. Math., 26 (1989), 881-895.



CUBO
14, 2 (2012)

Local energy decay for the wave equation with a time-periodic ... 173

[23] S-H. Tang and M. Zworski, Resonance expansions of scattered waves, Comm. Pure Appl.

Math., 53 (2000), 1305-1334.

[24] B. Vainberg, On the short wave asymptotic behavior of solutions of stationary problems and

the asymptotic behavior as t → ∞ of solutions of nonstationary problems, Russian Math.

Surveys, 30 (1975), 1-53.

[25] B. Vainberg, Asymptotic methods in Equation of mathematical physics, Gordon and Breach,

New York, 1988.

[26] B. Vainberg, On the local energy of solutions of exterior mixed problems that are periodic

with respect to t, Trans. Moscow Math. Soc. 1993, 191-216.

[27] G. Vodev, On the uniform decay of local energy, Serdica Math. J., 25 (1999), 191-206.

[28] G. Vodev, Local energy decay of solutions to the wave equation for non-trapping metrics, Ark.

Math, 42 (2004), no 2, 379-397.


	Introduction
	General properties
	The meromorphic continuation of the cut-off resolvent R1,2()
	Local energy decay
	 Examples of metrics a(t,x) and obstacles O(t)