Simon Gautier

Gaëlle Orsal

Tarik Moudakir

N Maloufi

François Jomard

Marc Alnot

Zakaria Djebbour

Andrei A Sirenko

M Abid

K Pantzas

I T Ferguson

P L Voss

A Ougazzaden

Metal-organic vapor phase epitaxy of BInGaN quaternary alloys and characterization of boron content

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The InGaN system covers a broad range of wavelengths from UV (3.4 eV for GaN) [1] to infrared (0.7 eV for InN) [2]. As a result, InGaN has been extensively studied and commercialized for solidstate light sources in the green/blue/violet spectral regions. InGaN will also extend the functionality of a wide range of optoelectronic devices for photovoltaics [3], medical diagnostics, and full color displays [START_REF] Kam Ur A | T heBlueLas erD io de,S pringe r[END_REF]. Many groups have studied InGaN structures including quantum wells and quantum dots to realize such devices. However, for high indium content, the lattice mismatch between the InGaN layer and the GaN template substrates causes high misfit dislocation densities, limiting the range of indium composition.

One solution to this problem is the addition of boron in order to decrease the lattice mismatch of InGaN with respect to a GaN substrate. For instance, an alloy containing 3% boron decreases the lattice mismatch of In 0.15 Ga 0.85 N/GaN by 0.6%. BInGaN quaternary alloys offer the possibility to optimize the bandgap energy and lattice parameter independently of each other without inducing strain into the layers. This is highly desirable for the bandgap engineering of advanced optoelectronic heterostructures [5]. In addition, the incorporation of boron into GaN-based devices will potentially allow the development of a new class of neutron detectors [6]. In this paper we report on the epitaxial growth and characterization of BInGaN materials for optoelectronic and neutron detection applications.

Experiment

Growth was performed in a low pressure MOVPE T-shape reactor using only nitrogen as a carrier gas [7]. Trimethylgallium (TMGa), trimethylindium (TMIn), triethylboron (TEB) III-group precursors were used as gallium, indium, and boron sources, respectively. For the MOVPE growth of InGaN-based materials, a relatively low growth temperature is essential to ensure In incorporation and due to the inefficient decomposition of NH 3 a n Corresponding author. Tel.: + 33 3 87 37 85 40, fax: +33 3 87 37 85 59.

E-mail address: simon.gautier@metz.supelec.fr (S. Gautier).

1 very large flow rate of NH 3 is required [8]. In this study, dimethylhydrazine (DMHy) precursor was added because of its low decomposition temperature which enhances the concentration of atomic nitrogen in the gas phase, allowing the growth at a relatively low flow rate of NH 3 at the growth temperature of 730 1C [9]. However, adding high flow rates of DMHy leads to undesirable carbon deposition in the growth chamber. In this study, the optimum value of the DMHy/V ratio was around 4%. As a result, no parasitic reaction in the vapour phase between DMHY and (TMIn, TMGa) was observed. The In incorporation in the solid phase was found to be proportional to the TMIn/III ratio in the vapour phase and the growth rate values were as expected.

A large number of BInGaN samples have been grown, however this work focuses on a systematic series of three samples in which the flow of TMGa and TMIn was constant to investigate boron incorporation in these samples. The TEB flow was varied between 0, 3.27, and 6.54 sccm leading to a TEB/III molar ratio in the vapour phase of 0%, 2%, and 4%, respectively.

Our growth chamber has been designed to provide good thickness uniformity over 2'' wafers. Strain uniformity across the wafers has been investigated for each boron concentration given above. So far, BInGaN alloys have been grown on multipieces of 2cm 2 template substrates in the same run and no significant shift of the HR-XRD peaks has been observed on samples from the same run.

A typical structure consists of a 40 nm GaN buffer layer grown on GaN template substrate, followed by a 140 nm BInGaN layer.

Results and discussion

Lattice parameters of BInGaN layers were measured by high resolution X-ray diffraction (HR-XRD) including symmetrical and asymmetrical 2y-o scans, and both symmetrical and asymmetrical reciprocal space mapping (RSM). Surface topography and RMS roughness were investigated by atomic force microscopy (AFM) in intermittent contact mode. Secondary ion mass spectroscopy (SIMS) was used to obtain the depth concentration profile for Ga, In, B, N, and impurities such as O, etc. The bandgap energy was obtained by optical absorption measurements. The composition measurement of quaternary BInGaN is not straightforward because of the relatively low boron content in these samples. Therefore a baseline was created using ternary alloys of InGaN and BGaN of various indium and boron contents, measured with a high accuracy by HR-XRD, and used as reference to calibrate SIMS for quantitative measurements of boron and indium in the BInGaN layers. The BInGaN composition results were later confirmed by X-ray photoelectron spectroscopy (XPS) measurements.

Fig. 1 shows (0 0 .2) 2y-o scans of BInGaN layers grown with various TEB flows. The peak at lower angle and of lower intensity corresponds to the BInGaN film whereas the other peak at higher angle corresponds to diffraction from the GaN template. The signal from the GaN template dominates the X-ray diffraction profile since the GaN layer is 3.5 mm thick compared to 140 nm for the BInGaN. At increased TEB flow, the peak corresponding to BInGaN shifts towards the GaN peak indicating a reduction of the misfit strain. The RSMs of the asymmetric (1 1 .4) plane show fully relaxed BInGaN layers, as presented in Fig. 2(a andb). At boron percentage of 0% and based on Vegard's law the average In content in the ternary InGaN reference layer was equal to 19.5%. Atomic force microscopy showed that the BInGaN surface exhibits morphology similar to that reported for InGaN [10] and RMS roughness was between 3 and 5 nm.

The SIMS profile exhibits a uniform boron distribution (Fig. 3) in the BInGaN layer, along the growth direction, while the Ga concentration varies anti-phase with the In and B. This is an indication that the mixing of In, Ga, and B atoms occurs on the sub-lattice of III sites to form an alloy. Depth concentration profile analysis of In, Ga, B and N concentrations were also performed by SIMS for all samples. As mentioned above, several BGaN and InGaN layers were used as reference in order to calibrate SIMS analysis for quantitative measurements of boron and indium content in the BInGaN layers. It was found that an increase of the TEB flow in the growth chamber from 0 to 6.54 sccm lead to a B and In content variation in the solid phase from 0% to 2% and from 19.5% to 14%, respectively.

XPS analysis was performed for all samples to further understand these results. Broad range and elemental XPS spectra were measured using Al Ka X-ray and Mg Ka X-ray sources to avoid the overlaps with Auger lines. The binding energies were chargecorrected by referencing the C 1s to 285.00 eV. The composition, both boron and indium, obtained by XPS and SIMS are in good agreement.

Optical spectroscopy was utilized to investigate the bandgap variation in the BInGaN layers. Transmission T(l) and reflection R(l) spectra versus the wavelength were measured at room temperature with a Perkin Elmer LAMBDA 950 spectrophotometer. Absorbance spectra a 2 (l) are shown in Fig. 4 Fig. 6 shows calculated bandgap values for quaternary B x In y Ga 1 À x À y N compound, which have been obtained using equations and the bandgap values of the endpoint binaries (GaN, InN, BN) from Refs. [START_REF] Rumyantsev | Properties of Advanced Semiconductor Materials GaN, AlN, InN, BN, SiC, SiGe[END_REF][START_REF] Bougrov | Properties of Advanced Semiconductor Materials GaN, AlN, InN, BN, SiC, SiGe[END_REF][START_REF] Vurgaftman | [END_REF][14][15]: E g (GaN) =3.39 eV, E g (BN) = 5.5 eV, and E g (InN)= 0.7 eV. The bowing parameter C BGaN = 9.2 eV of the BGaN ternary compound is from Ref. [16]. To accommodate our experimental value of 2.53 eV for the bandgap of In 0.195 Ga 0.805 N sample, the bowing parameter of InGaN was chosen to be C InGaN =2.1 eV, which is 10% less than the recommended value of 2.3 eV from Ref. [14]. Since our experimental data and the {x,y} range for the calculated bandgap values are close to the GaN binary endpoint, the bowing parameter of BInN has practically no influence on the calculated values shown in Fig. 6.W ew e r eu s i n g C BInN = 14 eV, which was obtained by extrapolating the dependence of the bowing parameters in ternary nitride compounds as a function of the lattice mismatch between the binary endpoints. Note that BInN should have the highest value of the bowing parameters among all known ternary nitrides since boron and indium have the highest contrast the III-nitrogen bonding length [16].

Thickness (nm) BInGaN layer

GaN buffer

GaN template

Two experimental values for B 0.01 In 0.16 Ga 0.83 N and B 0.02 In 0.14 Ga 0.84 N samples are shown in Fig. 6. Within the error bars of the boron composition, the measured bandgaps of B x In y Ga 1 À x À y N compounds are in a perfect agreement with the calculated values, thus confirming the strong effect of the bandgap bowing in this quaternary system. It is notable that a combination of the two bowing parameters for ternary compounds of InGaN and BGaN results in a decrease of the bandgap for BInGaN quaternary compound, which is lattice matched to GaN for a broad range of the x and y compositions (see straight line in Fig. 6). In other words, the addition of boron to InGaN has an effect similar to that of increased indium for the bandgap decrease. At the same time, in terms of the lattice parameter this quaternary compound demonstrates a decrease of the lattice mismatch with respect to GaN. It opens a possibility to design, for example, light emitters for the green and yellow parts of the visible spectrum with the B x In y Ga 1 À x À y N active region that is lattice matched to conventional GaN substrates.

Conclusion

We demonstrate the growth of a new quaternary alloys BInGaN, with boron content up to 2% at indium content of 14%. These materials have the potential to allow independent choice of bandgap and lattice parameter for bandgaps between 0.7 and 3.4 eV, enabling advanced III-N heterostructure designs. Our experimental results confirm a strong bowing for the bandgap of the quaternary system of BInGaN with C InGaN = 2.1 eV and C BGaN = 9.2 eV.

Fig. 1 .

 1 Fig. 1. 2y-o symmetric HRXRD scans showing the GaN, InGaN and BInGaN (0 0 .2) diffraction peaks for various TEB flows.

Fig. 2 .

 2 Fig. 2. Reciprocal space maps of the (1 1 .4) diffraction of the InGaN and BInGaN samples grown on GaN template for (a) TEB= 0 sccm and (b) TEB =3.27 sccm. Both samples demonstrate the complete relaxation of strain between the grown layer and the GaN substrate.

  . They have been obtained from transmission and reflection data as a(l)=(1/d) Â ln{[1 ÀR(l)]/T(l)}, d being the thickness of the BInGaN layers. The extrapolation of the linear section of a 2 to zero absorption corresponds to the optical bandgap, which is illustrated in the same figure for these three samples. These bandgap values are shown as data points (a), (b), and (c) in Fig. 5, which plots bandgap versus the out-of-plane lattice parameter.

Fig. 3 .Fig. 4 .

 34 Fig. 3. SIMS elemental concentration depth profiles for the B, In, Ga and N signals for the BInGaN on GaN template.

Fig. 5 .

 5 Fig. 5. Bandgap energy versus out-of-plane lattice parameter c for nitride compounds. Experimental data are shown with squares.

Fig. 6 .

 6 Fig. 6. Calculated bandgap values for quaternary B x In y Ga 1 À x À y N compound. The low-left corner corresponds to the binary endpoint of GaN. The straight line that originates from the GaN corner corresponds to the {x,y} compositions that are lattice matched to the in-plane lattice parameter of GaN. The white curve corresponds to the B x In y Ga 1 À x À y N bandgap of 3 eV and divides the UV and visible parts of the corresponding optical spectrum. Experimental points for three B x In y Ga 1 À x À y N samples are shown with squares. The size of the squares in vertical direction corresponds to the uncertainty of the indium composition. The horizontal error bars indicate the uncertainty in the measured boron composition.


Serge Nicaise

Cristina Pignotti

Stabilization of second-order evolution equations with time delay

Keywords: 2000 Mathematics Subject Classification: 35L05, 93D15 second-order evolution equations, delay feedbacks, stabilization

We consider second-order evolution equations in an abstract setting with damping and time delay and give sufficient conditions ensuring exponential stability. Our abstract framework is then applied to the wave equation, the elasticity system and the Petrovsky system.

Introduction

Let H be a real Hilbert space with norm and inner product denoted respectively by • H and •, • H and let A : D(A) → H be a positive self-adjoint operator with a compact inverse in H. Denote by V := D(A 1 2 ) the domain of A 1 2 . Moreover, for i = 1, 2, let U i be real Hilbert spaces with norm and inner product denoted respectively by • U i and •, • U i and let B i : U i → V ′ be linear operators. In this setting we consider the problem

u tt (t) + Au(t) + B 1 B * 1 u t (t) + B 2 B * 2 u t (t -τ ) = 0 t > 0, (1.1)
u(0) = u 0 and u t (0) = u 1 , (1.2)
B * 2 u t (t) = f 0 (t) t ∈ (-τ, 0), (1.3) where the constant τ > 0 is the time delay. We assume that the delay feedback operator B 2 is bounded, that is B 2 ∈ L(U 2 , H), while the standard one B 1 ∈ L(U 1 , V ′ ) may be unbounded. Time delays are often present in applications and practical problems and it is by now wellknown that even an arbitrarily small delay in the feedback may destabilize a system which is uniformly exponentially stable in absence of delay. For some examples in this sense we refer to [START_REF] Datko | Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks[END_REF][START_REF] Datko | An example on the effect of time delays in boundary feedback stabilization of wave equations[END_REF][START_REF] Nicaise | Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks[END_REF][START_REF] Xu | Stabilization of wave systems with input delay in the boundary control[END_REF]. [START_REF] Ammari | Feedback boundary stabilization of wave equations with interior delay[END_REF] We are interested in giving stability results for the above problem under a suitable assumption on the "size " of the feedback operator B 2 , when the feedback B 1 is a stabilizing one. More precisely, we will show that for a system which is exponentially stable in absence of time delay, i.e. for B 2 = 0, the exponential stability is preserved if B * 2 is sufficiently small. In this sense this paper extends and generalizes the result of [START_REF] Pignotti | A note on stabilization of locally damped wave equations with time delay[END_REF] for wave equation with local damping and time delay. On the other hand it completes the analysis of [START_REF] Nicaise | Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks[END_REF][START_REF] Nicaise | Stabilization of second order evolution equations with unbounded feedback with delay[END_REF]. Indeed here we do not assume [START_REF] Nicaise | Stabilization of second order evolution equations with unbounded feedback with delay[END_REF] (cfr. assumption (1.8) of [START_REF] Nicaise | Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks[END_REF] for the wave equation).

∃ α < 1 such that B * 2 u U 2 ≤ α B * 1 u U 1 , ∀ u ∈ V ; as in
Assuming that an observability inequality holds for the system (1.1), (1.2) when B 2 = 0, through the definition of a suitable energy (see (3.2)) and the use of a perturbation argument as in [START_REF] Pignotti | A note on stabilization of locally damped wave equations with time delay[END_REF], we obtain sufficient conditions ensuring exponential stability. Our abstract framework is then applied to some concrete examples, namely the wave equation, the elasticity system and the Petrovsky system.

The paper is organized as follows. In section 2 a well-posedness result of the abstract system is proved. In section 3 we obtain exponential stability results for the abstract system under suitable conditions. Finally, in sections 4, 5 and 6 we apply our abstract results to the wave equation with local and boundary dampings, the elasticity system and the Petrovsky system respectively. Other examples (like wave or beam equations on networks) could be given, we skip them for shortness.

Well-posedness

In this section we will give well-posedness results for problem (1.1)-(1.3) using semigroup theory.

As in [START_REF] Nicaise | Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks[END_REF] we introduce the function z(ρ, t) := B * 2 u t (tτ ρ), ρ ∈ (0, 1), t > 0.

(2.1)

Then, problem (1.1)-(1.3) can be rewritten as

u tt (t) + Au(t) + B 1 B * 1 u t (t) + B 2 z(1, t) = 0 t > 0, (2.2)
τ z t (ρ, t) + z ρ (ρ, t) = 0, ρ ∈ (0, 1), t > 0, (2.3)
u(0) = u 0 and u t (0) = u 1 , (2.4)
z(ρ, 0) = f 0 (-τ ρ) ρ ∈ (0, 1), (2.5) z(0, t) = B * 2 u t (t), t > 0. (2.6)
If we denote U := (u, u t , z) T , then U ′ = (u t , u tt , z t ) T and U satisfies

U ′ = AU U (0) = (u 0 , u 1 , f 0 (-τ •)) T , (2.7)
where the operator A is defined by

A   u v z   :=   v -Au -B 1 B * 1 v -B 2 z(1) -τ -1 z ρ   , (2.8)
with domain

D(A) := (u, v, z) T ∈ V × V × H 1 ((0, 1); U 2 ) : Au + B 1 B * 1 v ∈ H and z(0) = B * 2 v . (2.9)
Denote by H the Hilbert space

H := V × H × L 2 ((0, 1); U 2 ), (2.10)
equipped with the inner product

  u v z   ,   ũ ṽ z   H := A 1 2 u, A 1 2 ũ H + v, ṽ H + ξ 1 0 z(ρ), z(ρ) U 2 dρ, (2.11)
where ξ is any fixed positive number.

The following well-posedness result holds.

Proposition 2.1 For any initial datum U 0 ∈ H there exists a unique solution

U ∈ C([0, +∞), H) of problem (2.7). Moreover, if U 0 ∈ D(A), then U ∈ C([0, +∞), D(A)) ∩ C 1 ([0, +∞), H).
Proof. We will show that the operator A defined by (2.8), (2.9) generates a strongly continuous semigroup in the Hilbert H defined in (2.10), (2.11).

Denoting by I the identity operator, we first show that there exists a positive constant c such that A -cI is dissipative (cfr. [START_REF] Ammari | Feedback boundary stabilization of wave equations with interior delay[END_REF]). Let (u, v, z) T ∈ D(A), then

A   u v z   ,   u v z   H :=   v -Au -B 1 B * 1 v -B 2 z(1) -τ -1 z ρ   ,   u v z   H = A 1 2 v, A 1 2 u H -Au + B 1 B * 1 v + B 2 z(1), v H -ξ 1 0 z ρ (ρ), z(ρ) U 2 dρ . Since Au + B 1 B * 1 v + B 2 z(1) ∈ H ⊂ V ′ , by duality we have A   u v z   ,   u v z   H = A 1 2 v, A 1 2 u H -Au, v V ′ ,V -B 1 B * 1 v, v V ′ ,V -B 2 z(1), v V ′ ,V -ξ 1 0 z ρ (ρ), z(ρ) U 2 dρ = -B * 1 v U 1 -z(1), B * 2 v U 2 -ξ 1 0 z ρ (ρ), z(ρ) U 2 dρ .
Integrating by parts and using the relation

z(0) = B * 2 v, we get 1 0 z ρ (ρ), z(ρ) U 2 dρ = 1 2 ( z(1) 2 U 2 -B * 2 v 2 U 2 ),
thus, using also Young's inequality

A   u v z   ,   u v z   H ≤ ξ 2 + 1 2ξ B * 2 v 2 U 2 ≤ c v 2 H ,
for a suitable constant c > 0. Hence, the operator A -cI is dissipative. Now, we show that λI -A is surjective for some λ > 0. Given (f, g, h) T ∈ H we seek (u, v, z) T ∈ D(A) such that

(λI -A)   u v z   =   f g h   .
This is equivalent to

λu -v = f ;
(2.12)

λv + Au + B 1 B * 1 v + B 2 z(1) = g ; (2.13) λz + τ -1 z ρ = h . (2.14)
Analogously to [START_REF] Nicaise | Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks[END_REF], suppose that we have found u with the appropriate regularity. Then, by (2.12)

, v = λu -f ∈ V . ( 2
λ 2 u + Au + λB 1 B * 1 u + B 2 z(1) = g + λf + B 1 B * 1 f ,
and then, by (2.17),

λ 2 u + Au + λB 1 B * 1 u + λe -λτ B 2 B * 2 u = g + λf + B 1 B * 1 f -B 2 z 0 . (2.18)
We denote the right-hand side of (2.18) by w, namely

w := g + λf + B 1 B * 1 f -B 2 z 0 ∈ H ⊂ V ′ .
Then, from (2.18), we have

λ 2 u + Au + λB 1 B * 1 u + λe -λτ B 2 B * 2 u, ϕ V ′ ,V = w, ϕ V ′ ,V .
Since u ∈ V ⊂ H, we can rewrite

λ 2 u + Au + λB 1 B * 1 u + λe -λτ B 2 B * 2 u, ϕ V ′ ,V = λ 2 u, ϕ V ′ ,V + Au, ϕ V ′ ,V + λ B * 1 u, B * 1 ϕ V ′ ,V + λe -λτ B * 2 u, B * 2 ϕ V ′ ,V = λ 2 u, ϕ H + A 1 2 u, A 1 2 ϕ H + λ B * 1 u, B * 1 ϕ U 1 + λe -λτ B * 2 u, B * 2 ϕ U 2 .
Therefore, we obtain

λ 2 u, ϕ H + A 1 2 u, A 1 2 ϕ H + λ B * 1 u, B * 1 ϕ U 1 + λe -λτ B * 2 u, B * 2 ϕ U 2 = w, ϕ V ′ ,V . (2.19)
The left-hand side of (2. [START_REF] Liu | Locally distributed control and damping for the conservative systems[END_REF]) is a continuous and coercive bilinear form on V. Then, Lax-Milgram's lemma implies the existence of a unique solution u ∈ V of (2.19) that satisfies

λ 2 u + Au + λB 1 B * 1 u + λe -λτ B 2 B * 2 u = w in V ′ .
This implies that Au ∈ H and by defining v by (2.15) and z by (2.16), we have found (u, v, z) T ∈ D(A) satisfying (2.12)- (2.14). This implies that λI -A is surjective for all λ > 0 and the same holds for the operator λI -(A -cI).

Then, the Lumer-Phillips Theorem implies that A -cI generates a strongly continuous semigroup of contraction in H. Hence, the operator A generates a strongly continuous semigroup in H.

Stability result

For a fixed constant ξ satisfying ξ > 1, (

we define the energy functional for solutions to problem (1.1)-(1.3) as

E(t) := E(u, t) = 1 2 ( A 1 2 u(t) 2 H + u t (t) 2 H ) + ξ 2 t t-τ B * 2 u t (s) 2 U 2 ds . (3.2)
We can obtain a first estimate. 3)

E ′ (t) ≤ -B * 1 u t (t) 2 U 1 + 1 + ξ 2 B * 2 u t (t) 2 U 2 - ξ -1 2 B * 2 u t (t -τ ) 2 U 2 . (3.3) Proof: Differentiating E(t) we get E ′ (t) = A 1 2 u(t), A 1 2 u t (t) H + u t (t), u tt (t) H + ξ 2 B * 2 u t (t) 2 U 2 - ξ 2 B * 2 u t (t -τ ) 2 U 2 .
Hence using the definition of A and (1.1) we get successively

E ′ (t) = Au(t), u t (t) V ′ ,V -u t (t), Au(t) V,V ′ -u t (t), B 1 B * 1 u t (t) V,V ′ -u t (t), B 2 B * 2 u t (t -τ ) V,V ′ + ξ 2 B * 2 u t (t) 2 U 2 - ξ 2 B * 2 u t (t -τ ) 2 U 2 .
Then,

E ′ (t) = -B * 1 u t (t) 2 U 1 -B * 2 u t (t), B * 2 u t (t -τ ) U 2 + ξ 2 B * 2 u t (t) 2 U 2 - ξ 2 B * 2 u t (t -τ ) 2 U 2 ,
and (3.3) follows from Cauchy-Schwarz's inequality.

Note that, from (3.3), the energy of solutions to problem (1.1)-(1.3) is not decreasing in general. Indeed the second term in the right-hand side of (3.3), coming from the delay term in (1.1), is non negative. We now consider, as in [START_REF] Pignotti | A note on stabilization of locally damped wave equations with time delay[END_REF], the next auxiliary problem which is close to the first one but whose energy is decreasing.

ϕ tt (t) + Aϕ(t) + B 1 B * 1 ϕ t (t) + B 2 B * 2 ϕ t (t -τ ) + ξB 2 B * 2 ϕ t (t) = 0, t > 0, (3.4
)

ϕ(0) = ϕ 0 , ϕ t (0) = ϕ 1 , (3.5)
B * 2 ϕ t (t) = g 0 (t), t ∈ (-τ, 0), (3.6)
where ξ is the same constant as in (3.2).

The well-posedness of system (3.4)-(3.6) can be proved using standard semigroup theory as in Proposition 2.1. Analogously to above we introduce the function

η(ρ, t) = B * 2 ϕ t (t -τ ρ), ρ ∈ (0, 1), t > 0;
and we rewrite the problem in the abstract form

Φ ′ = A 0 Φ , Φ(0) = (ϕ 0 , ϕ 1 , g 0 (-τ •)) T , (3.7)
where the operator A 0 is defined by (2.9)) in the Hilbert space H defined by (2.10) and (2.11).

A 0   ϕ ψ η   :=   ψ -Aϕ -B 1 B * 1 ψ -B 2 η(1) -ξB 2 B * 2 ψ -τ -1 η ρ   , with domain D(A 0 ) = D(A) (see

Proposition 3.2 For any initial datum

Φ 0 ∈ H there exists a unique solution Φ ∈ C([0, +∞), H) of problem (3.7). Moreover, if Φ 0 ∈ D(A 0 ), then Φ ∈ C([0, +∞), D(A 0 )) ∩ C 1 ([0, +∞), H).
For solutions of problem (3.4)-(3.6) the energy F (•),

F (t) := F (ϕ, t) = 1 2 ( A 1 2 ϕ(t) 2 H + ϕ t (t) 2 H ) + ξ 2 t t-τ B * 2 ϕ t (s) 2 U 2 ds , (3.8)
with ξ satisfying (3.1), is decreasing in time.

More precisely, we have the following result.

F ′ (t) ≤ -B * 1 ϕ t (t) 2 U 1 - ξ -1 2 B * 2 ϕ t (t) 2 U 2 - ξ -1 2 B * 2 ϕ t (t -τ ) 2 U 2 . (3.9)
Then, if ξ satisfies (3.1), the energy F (•) is decreasing.

Proof. In order to have (3.9) we differentiate (3.8). Hence, using the definition of A and (3.4), we obtain

F ′ (t) = A 1 2 ϕ, A 1 2 ϕ t H + ϕ t , ϕ tt H + ξ 2 B * 2 ϕ t (t) 2 U 2 - ξ 2 B * 2 ϕ t (t -τ ) 2 U 2 = Aϕ(t), ϕ t (t) V ′ ,V -ϕ t (t), Aϕ(t) V,V ′ -ϕ t (t), B 1 B * 1 ϕ t (t) V,V ′ -ξ ϕ t (t), B 2 B * 2 ϕ t (t) V,V ′ -ϕ t (t), B 2 B * 2 ϕ t (t -τ ) V,V ′ + ξ 2 B * 2 ϕ t (t) 2 U 2 - ξ 2 B * 2 ϕ t (t -τ ) 2 U 2 .
Then,

F ′ (t) = -B * 1 ϕ t (t) 2 U 1 -ξ B * 2 ϕ t (t) 2 U 2 -B * 2 ϕ t (t), B * 2 ϕ t (t -τ ) U 2 + ξ 2 B * 2 ϕ t (t) 2 U 2 - ξ 2 B * 2 ϕ t (t -τ ) 2 U 2 ,
and therefore (3.8) follows from Cauchy-Schwarz's inequality.

Consider now the following damped system associated with (1.1) and (1.2),

w tt (t) + Aw(t) + B 1 B * 1 w t = 0 t > 0 (3.10) w(0) = w 0 and w t (0) = w 1 (3.11)
with (w 0 , w 1 ) ∈ V × H. For our stability result we need that this system is exponentially stable or equivalently that the next observability inequality holds (see Lemma 3.2 of [START_REF] Nicaise | Stability and controllability of an abstract evolution equation of hyperbolic type and concrete applications[END_REF]). Namely we assume that there exists a time T > 0 such that for every time T > T there is a constant c, depending on T but independent of the initial data, such that

E S (0) ≤ c T 0 B * 1 w t (t) 2 U 1 dt, (3.12)
for every weak solution of problem (3.10), (3.11) with initial data (w 0 , w 1 ) ∈ V × H.

Here E S (•) denotes the standard energy for wave type equations, that is

E S (t) = E S (w, t) := 1 2 ( A 1 2 w(t) 2 H + w t (t) 2 H ).
For shortness let us denote by C 2 the norm of B 2

B 2 = B * 2 = C 2 . (3.13)
We can prove an exponential stability result for the perturbed problem (3.4)-(3.6).

Theorem 3.4 Assume that (3.1) holds and that the observability estimate (3.12) holds for problem (3.10) -(3.11). Then, there are two positive constants K, μ such that

F (t) ≤ Ke -μt F (0), t > 0, (3.14)
for any solution of problem (3.4) -(3.6). In particular,

K = C 0 + 1 C 0 , (3.15) μ = 1 2T ln C 0 + 1 C 0 , (3.16)
with T any fixed time satisfying T > max {T , τ }, T being an observability time for (3.12), and

C 0 = max 2c, 32cT C 2 2 + ξ ξ -1 , 32cC 2 2 T ξ 2 ξ -1 , (3.17)
where C 2 is as in (3.13) and c := c(T ) is the observability constant in (3.12).

Proof. Following a classical argument (see [START_REF] Zuazua | Exponential decay for the semi-linear wave equation with locally distributed damping[END_REF]) we can decompose the solution ϕ of (3.4)-(3.6) as ϕ = w + w where w is the solution of system (3.10), (3.11) with w 0 = ϕ 0 , w 1 = ϕ 1 ; while w solves

wtt (t) + A w(t) + B 1 B * 1 wt (t) = -ξB 2 B * 2 ϕ t (t) -B 2 B * 2 ϕ t (t -τ ) t > 0 (3.18)
w(0) = 0 and wt (0

) = 0 (3.19)
By (3.8),

F (0) = E S (w, 0) + ξ 2 0 -τ B * 2 ϕ t (s) 2 U 2 ds = E S (w, 0) + ξ 2 τ 0 B * 2 ϕ t (t -τ ) 2 U 2 dt .
Therefore, from (3.12), if T > max{T , τ } we obtain

F (0) ≤ c T 0 B * 1 w t (t) 2 U 1 dt + ξ 2 T 0 B * 2 ϕ t (t -τ ) 2 U 2 dt ≤ 2c T 0 ( B * 1 ϕ t (t) 2 U 1 + B * 1 wt (t) 2 U 1 )dt + ξ 2 T 0 B * 2 ϕ t (t -τ ) 2 U 2 dt, (3.20)
where c is the observability constant for the damped system (3.10), (3.11). Now, observe that from (3.18),

d dt 1 2 ( wt (t) 2 H + A 1 2 w(t) 2 H ) + B * 1 wt 2 U 1 = wt , wtt + A w + B 1 B * 1 wt = wt , -ξB 2 B * 2 ϕ t (t) -B 2 B * 2 ϕ t (t -τ ) H .
Integrating in time from 0 to t, for t ∈ (0, 2T ], and using (3.19) we have

1 2 ( wt (t) 2 H + A 1 2 w(t) 2 H ) + t 0 B * 1 wt (s) 2 U 1 ds = t 0 wt , -ξB 2 B * 2 ϕ t (s) -B 2 B * 2 ϕ t (s -τ ) H ds ,
and then

wt (t) 2 H + 2 t 0 B * 1 wt (s) 2 U 1 ds ≤ 1 8T C 2 2 t 0 B * 2 wt (s) 2 U 2 ds + 8T C 2 2 ξ 2 t 0 B * 2 ϕ t (s) 2 U 2 ds + 1 8T C 2 2 t 0 B * 2 wt (s) 2 U 2 ds + 8T C 2 2 t 0 B * 2 ϕ t (s -τ ) 2 U 2 ds , (3.21)
where C 2 was defined in (3.13). This estimate directly implies that for all t ∈ [0, 2T ], one has

wt (t) 2 H + 2 t 0 B * 1 wt (s) 2 U 1 ds ≤ 1 4T C 2 2 2T 0 B * 2 wt (s) 2 U 2 ds + 8T C 2 2 ξ 2 2T 0 B * 2 ϕ t (s) 2 U 2 ds +8T C 2 2 2T 0 B * 2 ϕ t (s -τ ) 2 U 2 ds , (3.22
) and so, integrating in [0, 2T ],

2T 0 wt (t) 2 H dt + 2 2T 0 t 0 B * 1 wt (s) 2 U 1 ds dt ≤ 1 2C 2 2 2T 0 B * 2 wt (s) 2 U 2 ds +16T 2 C 2 2 ξ 2 2T 0 B * 2 ϕ t (s) 2 U 2 ds + 16T 2 C 2 2 2T 0 B * 2 ϕ t (s -τ ) 2 U 2 ds . (3.23)
Therefore,

1 2 2T 0 wt (t) 2 H dt + 2 2T 0 t 0 B * 1 wt (s) 2 U 1 ds dt ≤ 16T 2 C 2 2 ξ 2 2T 0 B * 2 ϕ t (s) 2 U 2 ds +16T 2 C 2 2 2T 0 B * 2 ϕ t (s -τ ) 2 U 2 ds , from which follows 2T 0 t 0 B * 1 wt (s) 2 U 1 ds dt ≤ 8T 2 C 2 2 ξ 2 2T 0 B * 2 ϕ t (s) 2 U 2 ds +8T 2 C 2 2 2T 0 B * 2 ϕ t (s -τ ) 2 U 2 ds .
Using the fact that

2T 0 t 0 B * 1 wt (s) 2 U 1 ds dt = 2T 0 B * 1 wt (s) 2 U 1 (2T -s) ds ≥ T 0 B * 1 wt (s) 2 U 1 (2T -s) ds ≥ T T 0 B * 1 wt (s) 2 U 1 ds,
we deduce that

T 0 B * 1 wt (s) 2 U 1 ds ≤ 8T C 2 2 ξ 2 2T 0 B * 2 ϕ t (s) 2 U 2 ds +8T C 2 2 2T 0 B * 2 ϕ t (s -τ ) 2 U 2 ds . (3.24)
Using (3.24) in (3.20) we obtain

F (0) ≤ 2c T 0 B * 1 ϕ t (t) 2 U 1 dt + (16cT C 2 2 + ξ 2 ) 2T 0 B * 2 ϕ(t -τ ) 2 U 2 dt +16cC 2 2 T ξ 2 2T 0 B * 2 ϕ t (t) 2 U 2 dt, (3.25)
that we rewrite as

F (0) ≤ 2c 2T 0 B * 1 ϕ t (t) 2 U 1 dt + 32cT C 2 2 + ξ ξ -1 ξ -1 2 2T 0 B * 2 ϕ(t -τ ) 2 U 2 dt + 32cC 2 2 T ξ 2 ξ -1 ξ -1 2 2T 0 B * 2 ϕ(t) 2 U 2 dt ≤ -C 0 2T 0 F ′ (t)dt, (3.26)
with C 0 as in (3.17). Therefore, from (3.26), using also that F (•) is decreasing we obtain

F (2T ) ≤ F (0) ≤ C 0 (F (0) -F (2T )).
Then, For all ξ > 1 in the definition (3.2), there is β > 0 depending on T , τ , ξ and on the operator B 1 , such that if the delay feedback satisfies B * 2 < β, then there exist positive constants K, µ for which we have

F (2T ) ≤ C 0 C 0 + 1 F (
E(t) ≤ Ke -µt E(0), t > 0, (3.27)
for any solution of (1.1) -(1.3).

Proof. We can see problem (1.1)-(1.3) as a perturbation of the auxiliary one. Therefore,

A   u v z   = (A 0 + B)   u v z   with B   u v z   =   0 -ξB 2 B * 2 v 0   .
From Theorem 3.4 and Theorem 3.5, we know that if

-μ + K B < 0, (3.28)
where μ and K are defined by (3.15), (3.16) and (3.17), then

E(t) ≤ Ke -µt E(0), with µ = μ -K B ,
< μ K , that is ξC 2 2 < 1 2T C 0 C 0 + 1 ln C 0 + 1 C 0 . (3.29)
The difficulty is that the constant C 0 (defined by (3.17)) appearing in the right-hand side of this estimate depends on ξ and C 2 as well. So let us consider the continuous function h : (0, +∞) → (0, +∞),

h(s) := s s + 1 ln s + 1 s .
Then, h tends to zero for s → 0 + and for s → +∞. Moreover, h assumes the maximal value 1/e at 1 e-1 , is increasing before 1 e-1 and decreasing after. Now it is easy to check that ξ ξ-1 > 1 e-1 . Considering that ξ is fixed > 1 and T is fixed as well, we consider C 0 as a function of C 2 ≥ 0, that we write C 0 (C 2 ). But we remark that from its definition, C 0 is non decreasing (in C 2 ) and

C 0 (0) = max 2c, ξ ξ -1 > 1 e -1 .
Hence h(C 0 (C 2 )) is non increasing as a function of C 2 with h(C 0 (0)) > 0 and since the left-hand side of (3.29) is increasing in C 2 and is zero at C 2 = 0, there exists a point β > 0 such that

2ξβ 2 T = h(C 0 (β)),
and for which (3.29) holds for all C 2 ∈ [0, β).

Obviously β depends on T (and then on T and τ ), on ξ and, through the observability constant c and the time T, on the feedback operator B 1 .

Remark 3.7 If B 1 is bounded, namely if B 1 ∈ L(U 1 , H),
E S (0) = 1 2 ( A 1 2 w 0 2 H + w 1 2
H ) ≤ c T 0 B * 1 ϕ t (t) 2 U 1 dt, (3.30)
holds for some T > 0 and c > 0, for every weak solution ϕ of the conservative system

ϕ tt (t) + Aϕ(t) = 0 t > 0 (3.31) ϕ(0) = w 0 and ϕ t (0) = w 1 (3.32)
with initial data (w 0 , w 1 ) ∈ V × H. Let us consider the initial boundary value problem

u tt (x, t) -∆u(x, t) + b 1 (x)u t (x, t) + b 2 (x)u t (x, t -τ ) = 0 in Ω × (0, +∞), (4.1) u(x, t) = 0 on ∂Ω × (0, +∞), (4.2) u(x, 0) = u 0 (x) and u t (x, 0) = u 1 (x) in Ω, (4.3) b 2 u t (x, t) = f 0 (x, t) in ω 2 × (-τ, 0), (4.4) (4.5) with initial data (u 0 , u 1 , f 0 ) ∈ H 1 0 (Ω) × L 2 (Ω) × L 2 ((-τ, 0); L 2 (ω 2 )), where ω i = {x ∈ Ω : b i (x) > 0} is the support of b i , i = 1 or 2.
This problem enters into our previous framework, if we take H = L 2 (Ω) and the operator A defined by

A : D(A) → H : u → -∆u,
where D(A) = {u ∈ H 1 0 (Ω) : ∆u ∈ L 2 (Ω)}. This operator A is a self-adjoint and positive operator with a compact inverse in H and is such that V = D(A 1/2 ) = H 1 0 (Ω). We then define U 1 = L 2 (ω 1 ), U 2 = L 2 (ω 2 ) and the operators B i , i = 1, 2, as

B i : U i → H : v → b i (x)ṽ, (4.6)
where ṽ ∈ L 2 (Ω) is the extension of v by zero outside ω i . It is easy to verify that In this setting, the energy functional is

B * i ϕ = b i ϕ |ω i for ϕ ∈ H. As B i B * i ϕ = b i ϕ,
E(t) = 1 2 Ω {u 2 t (x, t) + |∇u(x, t)| 2 }dx + ξ 2 t t-τ Ω b 2 (x)u 2 t (x, s)dxds, (4.7)
which is the standard energy for wave equation

E S (t) = E S (w, t) := 1 2 Ω (w 2 t + |∇w| 2 )dx,
plus an integral term due to the presence of a time delay.

Since B 1 is bounded, according to Remark 3.7, our main assumption concerns the existence of an observability estimate for the standard wave equation:

ϕ tt (x, t) -∆ϕ(x, t) = 0 in Ω × (0, +∞) (4.8) ϕ(x, t) = 0 on ∂Ω × (0, +∞) (4.9)
ϕ(x, 0) = w 0 (x) and ϕ t (x, 0) = w 1 (x) in Ω (4.10)

with (w 0 , w 1 ) ∈ H 1 0 (Ω) × L 2 (Ω). We then assume that there exists a time T > 0 such that for every time T > T there is a constant c, depending on T but independent of the initial data, such that

E S (0) ≤ c T 0 Ω b 1 (x)ϕ 2 t (x, s)dxds, (4.11)
for every weak solution of problem (4.8) -(4.10).

According to (3.13) we have

B 2 = b 2 1/2 ∞ , (4.12)
where, for

v ∈ L ∞ (Ω), we denote v ∞ = sup x∈Ω |v(x)|, the L ∞ norm of v.
Therefore, according to Theorem 3.6, we have the next result:

. For all ξ > 1 in the definition (4.7), there is β > 0 depending on T , τ , ξ and b 1 such that if b 2 ∞ < β, then there exist positive constants K, µ for which we have

E(t) ≤ Ke -µt E(0), t > 0,
for any solution of (4.1) -(4.4).

Remark 4.2 1. From Lemma VII.2.4 of [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation des systèmes distribués[END_REF] (see also [START_REF] Komornik | Exact controllability and stabilization, the multiplier method[END_REF][START_REF] Lagnese | Control of wave processes with distributed control supported on a subregion[END_REF][START_REF] Lasiecka | Uniform exponential decay in a bounded region with L 2 (0, T ; L 2 (Σ))-feedback control in the Dirichlet boundary conditions[END_REF][START_REF] Liu | Locally distributed control and damping for the conservative systems[END_REF][START_REF] Zuazua | Exponential decay for the semi-linear wave equation with locally distributed damping[END_REF]), the observability estimate (4.11) holds for the wave equation (4.8)

-(4.10) if the boundary of Ω is of class C 2 , if T is bigger than the diameter of Ω and if b 1 (x) ≥ b 0 > 0, a.e. x ∈ ω, (4.13)
when the open subset ω of Ω is a neighborhood of Γ0 , where

Γ 0 = { x ∈ ∂Ω : (x -x 0 ) • ν(x) > 0 }, (4.14)
for some x 0 ∈ R n and ν(x) is the outer unit normal vector at x ∈ ∂Ω.

Internal and boundary dampings

We assume here that the boundary ∂Ω of Ω is splitted up as ∂Ω = Γ 0 ∪ Γ 1 , where Γ 0 , Γ 1 are closed subsets of ∂Ω with Γ 0 ∩ Γ 1 = ∅. Moreover we assume that Γ 0 and Γ 1 have an non empty interior (on ∂Ω). We suppose given k ∈ L ∞ (Γ 0 ) and b ∈ L ∞ (Ω) such that b(x) ≥ 0 a.e. x ∈ Ω and k(x) ≥ k 0 > 0 a.e. x ∈ Γ 0 .

We here consider the problem

u tt (x, t) -∆u(x, t) + b(x)u t (x, t -τ ) = 0, x ∈ Ω, t > 0, (4.15)
u(x, t) = 0, x ∈ Γ 1 , t > 0 (4.16) ∂u ∂ν (x, t) = -k(x)u t (x, t), x ∈ Γ 0 , t > 0 (4.17) u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), x ∈ Ω, (4.18) √ bu t (x, t) = f 0 (x, t), x ∈ Ω, t ∈ (-τ, 0), (4.19)
with initial data in a suitable space. This problem enters into our previous framework, if we take H = L 2 (Ω) and the operator A defined by

A : D(A) → H : u → -∆u, where D(A) := { u ∈ H 1 Γ 1 (Ω) : ∆u ∈ L 2 (Ω)
and

∂u ∂ν = 0 on Γ 0 }, with H 1 Γ 1 := { u ∈ H 1 (Ω) : u = 0 on Γ 1 }.
We then define U 1 := L 2 (Γ 0 ), U 2 := L 2 (ω 2 ) (ω 2 being the support of b) and the operators B 1 , B 2 as

B 2 ∈ L(U 2 ; H), B 2 u = b(x) ũ, ∀ u ∈ L 2 (ω 2 ),
and

B 1 ∈ L(U 1 ; V ′ ), B 1 u = √ k A -1 N u, ∀ u ∈ L 2 (Γ 0 ), B * 1 w = √ kw |Γ 0 , ∀ w ∈ V := D(A 1/2 ),
where A -1 is the extension of A to H, namely for all h ∈ H and ϕ ∈ D(A), A -1 h is the unique element in (D(A)) ′ (the duality is in the sense of H), such that (see for instance [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF])

A -1 h; ϕ (D(A)) ′ ,D(A) = Ω hAϕ dx.
Here and below N ∈ L(L 2 (Γ 0 ); L 2 (Ω)) is defined as follows: for all v ∈ L 2 (Γ 0 ), N v is the unique solution (transposition solution) of

∆N v = 0, N v |Γ 1 = 0, ∂N v ∂ν |Γ 0 = v.
With these definitions, we can show that problem (4. w(x, 0) = w 0 (x) and w t (x, 0) = w 1 (x) in Ω (4. [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF] with (w 0 , w 1 ) ∈ H 1 Γ 1 (Ω) × L 2 (Ω). Hence our main assumption will be: There exists a time T > 0 such that for every time T > T there is a constant c, depending on T but independent of the initial data, such that

E S (0) ≤ c T 0 Γ 0 k(x)w 2 t (x, s)dxds, (4.25)
for every weak solution of problem (4.21) -(4.24). Then, our previous results apply also to this model and we can restate Theorem 3.6. Remark 4.5 1. From Theorem 1 and Remark 1 of [START_REF] Komornik | A direct method for the boundary stabilization of the wave equation[END_REF] (see also [START_REF] Chen | Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain[END_REF][START_REF] Chen | Control and stabilization for the wave equation in a bounded domain I-II[END_REF][START_REF] Chen | A note on the boundary stabilization of the wave equation[END_REF][START_REF] Lagnese | Decay of solutions of wave equations in a bounded region with boundary dissipation[END_REF]), the observability estimate (4. for "small" internal delay feedback, has been first proved in [START_REF] Ammari | Feedback boundary stabilization of wave equations with interior delay[END_REF], for b and k constant and Γ 0 given by (4.14), by constructing a suitable Lyapunov functional and using the multiplier method. We give here a simpler proof by using a more general method, allowing to weaken the assumptions on b, k and Γ 0 .

with initial data (w 0 , w 1 ) ∈ H 1 0 (Ω) n × L 2 (Ω) n . If such an estimate holds, the stability result from section 3 can be applied to the above system.

Remark 5.1 Under the assumptions of point 1 of Remark 4.2, the observability estimate (5.5) is obtained in the proof of Theorem 3.1 of [START_REF] Cavalcanti | Exact internal controllability of the elasticity system[END_REF] (estimate (3.2) of [START_REF] Cavalcanti | Exact internal controllability of the elasticity system[END_REF]).

Internal and boundary dampings

Under the assumptions of subsection 4.2 we consider the following elastodynamic system

u tt (x, t) -µ∆u(x, t) -(λ + µ)∇ div u + b(x)u t (x, t -τ ) = 0 in Ω × (0, +∞), u(x, t) = 0, x ∈ Γ 1 , t > 0 (5.6) σ(u(x, t)) • ν(x) = -k(x)u t (x, t), x ∈ Γ 0 , t > 0 (5.7) u(x, 0) = u 0 (x) and u t (x, 0) = u 1 (x) in Ω, (5.8) √ bu t (x, t) = f 0 (t) in ω 2 × (-τ, 0), (5.9)
with initial data (u

0 , u 1 , f 0 ) ∈ H 1 0 (Ω) n × L 2 (Ω) n × L 2 ((-τ, 0); L 2 (ω 2 ) n ) and σ(u) = µ( n i=1 ∂ i (u j )ν i ) n j=1 + (λ + µ)( div u) ν on Γ 0 .
This problem enters into our abstract setting, once we take H = L 2 (Ω) n , A defined in the previous subsection, and B 1 and B 2 defined as in subsection 4.2.

As B 1 is not bounded, we need to assume that there exists a time T > 0 such that for every time T > T there is a constant c, depending on T but independent of the initial data, such that 1 2 ((w 0 , w 0 )

V + Ω |w 1 | 2 dx) ≤ c T 0 Γ 0 k(x)|w t | 2 (x, s)dxds, (5.10)
for every weak solution w of the non delayed system

w tt (x, t) -µ∆w(x, t) -(λ + µ)∇ div w = 0 in Ω × (0, +∞), w(x, t) = 0, x ∈ Γ 1 , t > 0 (5.11) σ(w(x, t)) • ν(x) = -k(x)w t (x, t), x ∈ Γ 0 , t > 0 (5.12) w(x, 0) = w 0 (x) and w t (x, 0) = w 1 (x) in Ω, , (5.13)
for initial data (w 0 , w 1 ) ∈ H 1 0 (Ω) n × L 2 (Ω) n . Again if such an estimate holds, the stability result from section 3 can be applied to the system (5.6)-(5.9). Remark 5.2 1. Under the assumptions of point 1 of Remark 4.5, the observability estimate (5.10) is proved in [START_REF] Bey | Boundary stabilization of the linear elastodynamic system by a Lyapunov-type method[END_REF]. 2. If we assume that the boundary of Ω is smooth and that (xx 0 ) • ν(x) ≤ 0 on Γ 1 , then the observability estimate (5.10) is proved in Lemma 3.2 of [START_REF] Horn | Implications of sharp trace regularity results on boundary stabilization of the system of linear elasticity[END_REF]. [START_REF] Chen | Control and stabilization for the wave equation in a bounded domain I-II[END_REF] The Petrovsky system

Hinged boundary conditions

Let Ω ⊂ IR n be an open bounded set with a boundary ∂Ω of class C 4 (as before this regularity could be weakened).

Let us consider the initial boundary value problem u tt (x, t) + ∆ 2 u(x, t) + b 1 (x)u t (x, t) + b 2 (x)u t (x, tτ ) = 0 in Ω × (0, +∞), (6.1) u(x, t) = ∆u(x, t) = 0 on ∂Ω × (0, +∞), (6.2) u(x, 0) = u 0 (x) and u t (x, 0) = u 1 (x) in Ω, (6.3) u t (x, t) = f 0 (x, t) in ω 2 × (-τ, 0), (6.4) with initial data (u 0 , u 1 , f 0 ) ∈ (H 2 (Ω)∩H 1 0 (Ω))×L 2 (Ω)×L 2 ((-τ, 0); L 2 (ω 2 )) and b 1 , b 2 satisfying the same assumptions as in subsection 4.1. Now, we take H = L 2 (Ω) and let A be the operator

A : D(A) → H : u → ∆ 2 u, (6.5)
where D(A) = {v ∈ H 1 0 (Ω) ∩ H 4 (Ω) : ∆u = 0 on ∂Ω}. The operator A is self-adjoint and positive, has a compact inverse in H and satisfies D(A 1/2 ) = H 2 (Ω) ∩ H 1 0 (Ω). We then define U i = L 2 (ω i ) and the operators B i , i = 1, 2, by (4.6). So, problem (6.1)-(6.4) enters in the abstract framework (1.1)- (1.3).

If an observability estimate of the associated conservative system holds, then the results of section 3 apply also to the plate model. Remark 6.1 Under the assumptions of point 1 of Remark 4.2 and the additional regularity of the boundary, it is well-known that an observability estimate of the associated conservative system holds, see Proposition 7.5.7 (see also Example 11.2.4) of [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF].

Clamped boundary conditions

Let Ω ⊂ IR 2 be an open bounded set with a boundary ∂Ω of class C 4 .

Here we consider the initial boundary value problem u tt (x, t) + ∆ 2 u(x, t) + b 1 (x)u t (x, t) + b 2 (x)u t (x, tτ ) = 0 in Ω × (0, +∞), (6.6) u(x, t) = ∂u ∂ν (x, t) = 0 on ∂Ω × (0, +∞), (6.7) u(x, 0) = u 0 (x) and u t (x, 0) = u 1 (x) in Ω, (6.8) u t (x, t) = f 0 (x, t) in ω 2 × (-τ, 0), (6.9)

with initial data (u 0 , u 1 , f 0 ) ∈ H 2 0 (Ω) × L 2 (Ω) × L 2 ((-τ, 0); L 2 (ω 2 )) where As usual, if an observability estimate holds for the associated conservative system, the results of section 3 can be applied to this model. Remark 6.2 Under the assumptions of point 1 of Remark 4.2 and the additional assumptions of this subsection, the observability estimate for the associated conservative system has been recently proved by the authors (see [START_REF] Nicaise | Asymptotic stability of second-order evolution equations with intermittent delay[END_REF], Theorem 6.1).

Proposition 3 . 1

 31 For any regular solution of problem (1.1) -(1.

Proposition 3 . 3

 33 For any regular solution of problem (3.4) -(3.6), we have

  then by Proposition 1 of [11], the system (3.10) -(3.11) is exponentially stable (or equivalently the observability estimate (3.12) holds for problem (3.10) -(3.11)) if and only if the observability estimate

4 The wave equation 4 . 1

 41 Internal dampings Our first application concerns the wave equation with locally distributed internal dampings. More precisely, let Ω ⊂ IR n be an open bounded domain with a Lipschitz boundary ∂Ω. We suppose given b 1 , b 2 in L ∞ (Ω) such that b 1 (x), b 2 (x) ≥ 0 a.e. x ∈ Ω.

  for any ϕ ∈ H and i = 1, 2, we deduce that problem (4.1)-(4.4) enters in the abstract framework (1.1)-(1.3).

Theorem 4 . 1

 41 Assume that the observability estimate (4.11) holds for the wave equation (4.8) -

2 .

 2 From[START_REF] Bardos | Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary[END_REF], the observability estimate (4.11) also holds for the wave equation (4.8) -(4.10) if the boundary of Ω is of class C ∞ and if(4.13) holds when the open subset ω of Ω satisfies the geometric control property.

Remark 4 . 3

 43 According to point 1 of the previous remark, Theorem 4.1 allows to recover the results from Theorem 1.2 of[START_REF] Pignotti | A note on stabilization of locally damped wave equations with time delay[END_REF] in a larger setting.

Theorem 4 . 4

 44 Assume that the observability estimate (4.25) holds for every weak solution of problem (4.21) -(4.24). For all ξ > 1 in the definition (4.20), there is β > 0 depending on T , τ , ξ and k, such that if b(x) ∞ < β, then there exist positive constants K, µ for which we haveE(t) ≤ Ke -µt E(0), t > 0,for any solution of (4.15) -(4.19).

H 2 0

 2 (Ω) := { ϕ ∈ H 2 (Ω) : u = ∂u ∂ν = 0 on ∂Ω },and b 1 , b 2 satisfy the same assumptions than in the previous subsection.

  [START_REF] Pignotti | A note on stabilization of locally damped wave equations with time delay[END_REF] holds for the damped wave equation (4.21) -(4.24) if the boundary of Ω is of class C 2 , if T is large enough and if Γ 0 is given by (4.14) for some x 0 ∈ R n . 2. From[START_REF] Bardos | Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary[END_REF], the observability estimate (4.25) also holds for the damped wave equation (4.21) -(4.24) if the boundary of Ω is of class C ∞ and if the part Γ 0 satisfies the geometric control property.3. If we suppress the assumption Γ 0 ∩Γ 1 = ∅, then Theorem 1 of[START_REF] Komornik | A direct method for the boundary stabilization of the wave equation[END_REF] shows that the observability estimate (4.25) holds for the damped wave equation (4.21) -(4.24) under the same assumptions than in point 1 but with the choice k(x) = (xx 0 ) • ν(x) and if n ≤ 3 (see also Proposition 6.4 of[START_REF] Grisvard | Contrôlabilté exacte des solutions de l'équation des ondes en présence de singularités[END_REF] in dimension 2). For this example, k is no more uniformly positive on Γ 0 , nevertheless it enters into our abstract framework.

	Remark 4.6 This result, namely exponential decay of the energy for solutions to problem
	(4.15)-(4.19)

The elasticity system 5.1 Internal dampings

Here we consider the following elastodynamic system u tt (x, t) -µ∆u(x, t) -(λ + µ)∇ div u

u(x, t) = 0 on ∂Ω × (0, +∞), (5.2)

)

) and b 1 , b 2 satisfying the same assumptions as in subsection 4.1. Note that in this case the state variable u is vector-valued and λ, µ are the Lamé coefficients that are positive real numbers.

As before this problem enters into our abstract setting, once we take H = L 2 (Ω) n , and A defined by

where

The operator A is a self-adjoint and positive operator with a compact inverse in H and is such that

We then define U i = L 2 (ω i ) n and the operators B i , i = 1, 2, as

where ṽ is the extension of v by zero outside ω i . As before

and thus B i B * i (ϕ) = b i ϕ, for any ϕ ∈ H and i = 1, 2. So, problem (5.1)-(5.4) enters in the abstract framework (1.1)- (1.3).

Therefore in order to apply the abstract results of section 3, we only need to check the observability estimate for the associated conservative system: There exists a time T > 0 and a constant c > 0 such that

for every weak solution ϕ of ϕ tt (x, t) -µ∆ϕ(x, t) -(λ + µ)∇ div ϕ(x, t) = 0 in Ω × (0, +∞), ϕ(x, t) = 0 on ∂Ω × (0, +∞), ϕ(x, 0) = w 0 (x) and ϕ t (x, 0) = w 1 (x) in Ω,


K Ammari
email: kais.ammari@fsm.rnu.tn

E Feireisl
email: feireisl@math.cas.cz.

S Nicaise
email: snicaise@univ-valenciennes.fr

Polynomial stabilization of some dissipative hyperbolic systems

Keywords: AMS subject classification (2010): 35L04, 93B07, 93B52, 74H55 exponential stability, polynomial stability, observability inequality, resolvent estimate, dissipative hyberbolic system, acoustic equation

We study the problem of stabilization for the acoustic system with a spatially distributed damping. Imposing various hypotheses on the structural properties of the damping term, we identify either exponential or polynomial decay of solutions with growing time. Exponential decay rate is shown by means of a time domain approach, reducing the problem to an observability inequality to be verified for solutions of the associated conservative problem. In addition, we show a polynomial stabilization result, where the proof uses a frequency domain method and combines a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent.

Introduction

We consider the following system of equations:

       u t + ∇r + α u = 0, in Ω × R + , r t + div u = 0, in Ω × R + , u • n = 0, on Γ × R + , u(0, x) = u 0 (x), r(0, x) = r 0 (x), x ∈ Ω, (1.1)
where Ω is a bounded domain in R d , d = 2, 3, with a smooth boundary Γ, div = ∇• is the divergence operator and α ∈ C ∞ (Ω), with α ≥ 0 on Ω and such that ∃ α -> 0 such that α ≥ α -on ω.

(1.2)

Here ω = ∅ stands for the subset of Ω on which the feedback is active. As usual n denotes the unit outward normal vector along Γ.

The system of equations (1.1) is a linearization of the acoustic equation governing the propagation of acoustic waves in a compressible medium, see Lighthill [START_REF] Lighthill | Waves in fluids[END_REF][START_REF] Lighthill | On sound generated aerodynamically. I. General theory[END_REF][START_REF] Lighthill | On sound generated aerodynamically. II. Turbulence as a source of sound[END_REF], where α u represents a damping term of Brinkman type. This kind of damping arises also in the process of homogenization (see Allaire [START_REF] Allaire | Homogenization of the Navier-Stokes equations and derivation of Brinkman's law[END_REF]), and is frequently used as a suitable penalization in fluid mechanics models, see Angot, Bruneau, and Fabrie [START_REF] Angot | A penalization method to take into account obstacles in incompressible viscous flows[END_REF]. Our main goal is to find sufficient condition on the initial data and the function α so that the solution of (1.1) stabilizes to zero for t → ∞ and, if this occurs, we are interested in the rate of this decay.

Let L 2 (Ω) denote the standard Hilbert space of square integrable functions in Ω. To avoid abuse of notation, we shall write • for the L 2 (Ω)-norm or the L 2 (Ω) d -norm. Denoting H = (L 2 (Ω)) d × L 2 (Ω), we introduce the operator

A = 0 ∇ div 0 : D(A) = ( u, r) ∈ H, (∇r, div u) ∈ H, u.n |Γ = 0 ⊂ H → H, and
B = √ α 0 ∈ L((L 2 (Ω)) d , H), B * = √ α 0 ∈ L(H, (L 2 (Ω)) d ).
Accordingly, the problem (1.1) can be recast in an abstract form:

Z t (t) + A Z(t) + BB * Z(t) = 0, t > 0, Z(0) = Z 0 , (1.3)
where Z = ( u, r), or, equivalently,

Z t (t) = A d Z(t), t > 0, Z(0) = Z 0 , (1.4)
with A d = -A -BB * with D(A d ) = D(A).

It can be shown (see Section 2 below) that for any initial data ( u 0 , r) ∈ D(A) the problem (1.1) admits a unique solution

( u, r) ∈ C([0, ∞); D(A)) ∩ C 1 ([0, ∞); H).
Moreover, the solution ( u, r) satisfies, the energy identity

E(0) -E(t) = t 0 √ α u(s) 2 (L 2 (Ω)) d ds for all t ≥ 0 (1.5) with E(t) = 1 2 ( u(t), r(t)) 2 H , ∀ t ≥ 0, (1.6)
where we have denoted

( u, r), ( v, p) H = Ω ( u(x). v(x) + r(x)p(x)) dx, ( u, r) H = Ω | u(x)| 2 + r 2 (x) dx.
Using (1.6) and the standard density arguments, we can extend the solution operator for the data ( u 0 , r) ∈ H. Consequently, we associate to the problem (1.1) (or to the abstract Cauchy problems (1.3), (1.4)) a solution (semi)-group that is globally bounded in H.

As the energy E is nonincreasing along trajectories, we want to determine the set of initial data ( u 0 , r 0 ) for which

E(t) → 0 as t → ∞. (1.7)
Such a question is of course intimately related to the structural properties of the function α, notably to the geometry of the set ω on which the damping is effective.

In this paper, we characterize the set of initial data for which (1.7) holds in terms of the set ω, and, eventually we obtain some information on the rate of decay. In particular, we establish an observability inequality for the associated conservative system yielding exponential decay and use a frequency domain method, combined with the multiplier technique, to obtain polynomial rate of decay. It is worth-observing that the associated conservative system coincides with the standard linear wave equation, supplemented with the Neumann boundary conditions, where the asymptotic behavior of solutions is relatively well understood.

The paper is organized as follows. Section 2 summarizes some well known facts concerning the acoustic system (1.1). In section 3, we examine the spectral properties of the generator A d and establish a strong stability results. Section 4 addresses the exponential and not exponential stability results. In Section 5, we prove exponential stability for a modified system, with a slightly different damping law, by using an observability strategy. Polynomial stability of the modified system is studied in Section 6.

Preliminaries

We start with a simple observation that the problem (1.1) can be viewed as a bounded (in H) perturbation of the conservative system

u t + ∇r = 0, in Ω × R + , r t + div u = 0, in Ω × R + , (2.8)
which can be recast as the standard wave equation r t,t -∆r = 0.

Consequently, the basic existence theory for (1.1) derives from that of (2.8).

Moreover, since the boundary Γ as well as the damping coefficient α are smooth, solutions of (1.1) remain smooth as soon as we take u 0 , r 0 smooth and satisfying relevant compatibility conditions as the case may be. In what follows, we may therefore deal with smooth solution, whereas the results for data in H can be obtained by means of density arguments.

Long-time behavior

The operator A d possesses a non-trivial kernel that is left invariant by the evolution, namely, solutions of the "stationary" problem

∇r + α u = 0, ∇ • u = 0, u • n| Γ = 0.
(2.9)

Thus the stationary field u is solenoidal, and, integrating (2.9) over Ω yields u = 0 in supp α, ∇r = 0.

Accordingly, we introduce the space

E = Ker[A d ] = {( u, r) | ∇ • u = 0, u| suppα = 0, u • n| Γ = 0, r = const},
together with its orthogonal complement (in H) denoted H 0 .

It is easy to check that A d ( w, s), ( u, r) H = 0 for any ( w, s) ∈ D(A), ( u, r) ∈ E;

in particular, the solution operator associated to (1.1) leaves both E and H 0 invariant. Consequently, the decay property (1.7) may hold only for the initial data emenating from the set H 0 .

Strong stability

The following observation can be shown by a simple density argument:

Lemma 3.1. The solution ( u, r) of (1.1) with initial datum in D(A d ) satisfies

E ′ (t) = - Ω α | u| 2 dx ≤ 0. (3.10)
Therefore the energy is non-increasing and (1.5) holds for all initial datum in H.

As already shown in the previous section, the strong stability result (1.7) may hold only if we take the initial data

( u 0 , r 0 ) ∈ H 0 = Ker[A d ] ⊥ .
There are several ways how to show (1.7), here we make use of the following result due to Arendt and Batty [START_REF] Arendt | Tauberian theorems and stability of one-parameter semigroups[END_REF]: Theorem 3.2. Let (T (t)) t≥0 be a bounded C 0 -semigroup on a reflexive space X. Denote by A the generator of (T (t)) and by σ(A) the spectrum of A. If σ(A) ∩ iR is countable and no eigenvalue of A lies on the imaginary axis, then lim t→+∞ T (t)x = 0 for all x ∈ X.

In view of this theorem we need to identify the spectrum of A d lying on the imaginary axis.

First we look at the point spectrum. whence u| suppα = 0, and, consequently, u, r solve (3.11) with α = 0. In particular, we get -∆r = λ 2 r, ∇r • n| Γ = 0, r| suppα = 0, and, by unique continuation for elliptic problems, we get r = 0.

In accordance with Lemma 3.3 and the discussion in the previous section, λ = 0 is the only possibly eigenvalue of A d on the imaginary axis.

Next, we show that A d has no continuous spectrum on the imaginary axis, except eventually zero.

Lemma 3.4. Suppose that |ω| > 0. If λ is a non-zero real number, then iλ belongs to the resolvent set ρ(A d ) of A d .
Proof. In view of Lemma 3.3 it is enough to show that iλI -A d is surjective.

Hence given a vector f , p ∈ H, we look for ( u, r) ∈ D(A d ) such that

(iλI -A d ) ( u, r) = f , p . (3.12)
By the definition of A d , we obtain

iλ u + ∇r + α u = f , iλ r + div u = p.
Assuming that u and r exist we can write

u = 1 iλ + α (-∇r + f ). (3.13)
Inserting this expression in the second identity we obtain the differential equation in r:

iλ r -div 1 iλ + α ∇r = p -div 1 iλ + α f in Ω, ∇r • n| Γ = 0. (3.14)
Multiplying this identity by a test function s ∈ H 1 (Ω), integrating in Ω and using formal integration by parts we get the problem:

Ω iλrs + 1 iλ + α ∇r • ∇s dx = F (s), ∀s ∈ H 1 (Ω), (3.15)
where

F (s) = Ω ps + 1 iλ + α f • ∇s dx.
We use the Fredholm alternative by splitting the left-hand side of (3.15) into its principal part

a p (r, s) = Ω 1 iλ + α ∇r • ∇sdx,
and its lower order term

a 0 (r, s) = iλ Ω rsdx.
The principal part is a continuous sesquilinear coercive form on H 1 m (Ω),

H 1 m (Ω) = {v ∈ H 1 (Ω), Ω v dx = 0}; hence it induces an isomorphism A p from H 1 m (Ω) into (H 1 m (Ω)) ′ . The mapping A 0 from H 1 m (Ω) into (H 1 m (Ω)) ′
induces by a 0 is clearly given by iλI and is therefore compact.

Consequently by the Fredholm alternative, A

p + A 0 is an isomorphism from H 1 m (Ω) into (H 1 m (Ω)) ′ if and only if it is injective. But the injectivity of A p + A 0 is equivalent to the injectivity of iλI -A d . Indeed let r ∈ H 1 m (Ω) be such that (A p + A 0 )r = 0 or equivalently such that Ω iλrs + 1 iλ + α ∇r • ∇s dx = 0, ∀s ∈ H 1 m (Ω). (3.16)
Since r is of mean zero, this identity remains valid for all s ∈ H 1 (Ω). Hence by taking s ∈ D(Ω) we find that iλ rdiv 1 iλ + α ∇r = 0 in Ω.

(3.17)

By taking s = r in (3.16), we find

Ω iλ|r| 2 + 1 iλ + α |∇r| 2 dx = 0.
By taking the real part of this identity, we find that

α|∇r| 2 = 0 in Ω.
This implies that ∇r = 0 in ω.

Hence by (3.17), r = 0 in ω and by the Holmgren uniqueness theorem we deduce that r = 0.

In conclusion A p + A 0 is an isomorphism from H 1 m (Ω) into (H 1 m (Ω)) ′ which implies that there exists a unique solution r ∈ H 1 m (Ω) of Ω iλrs + 1 iλ + α ∇r • ∇s dx = F (s), ∀s ∈ H 1 m (Ω),
As F (1) = 0, we deduce that r ∈ H 1 m (Ω) is solution of (3.15) and relation (3.12) holds.

These Lemmas and Theorem 3.2 leads to Corollary 3.5. Let ( u, r) be the unique semi-group solution of the problem (1.1) emanating from the initial data ( u 0 , r 0 ) ∈ H. Let P E be the orthogonal projection onto the space E = Ker[A d ] in H, and let ( w, s) = P E ( u 0 , r 0 ).

Then ( u, r)(t, •) -( w, s) H → 0 as t → ∞.

Exponential stability

Now, we may ask, under some conditions on the damping coefficient, the convergence in Corollary 3.5 is exponential in time.

( u, r)(t, •) H ≤ exp(-Lt) ( u 0 , r 0 ) H , L > 0, whenever Ω r 0 dx = 0.
Proof. According to [START_REF] Haraux | Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps[END_REF] (see also [START_REF] Ammari | Stabilization of second order evolution equations by a class of unbounded feedbacks[END_REF]), the exponential stability of the system (1.1) is equivalent that the undamped system, i.e.

         φ t + ∇p = 0, in Ω × R + , p t + div φ = 0, in Ω × R + , φ • n = 0, on Γ × R + , φ(0, x) = φ 0 (x), p(0, x) = p 0 (x), x ∈ Ω, (4.1)
satisfies the following inequality: There exist positive real numbers T, C such that

T 0 Ω α(x) φ(x, t) 2 dxdt ≥ C ( φ 0 , p 0 ) 2 (L 2 (Ω) d ×L 2 (Ω) (4.2) ∀ ( φ 0 , p 0 ) ∈ (L 2 (Ω)) d × L 2 (Ω) such that Ω p 0 dx = 0.
Since this estimate is well-defined in the energy space, it holds if and only if it holds for strong solutions.

As the conservative system (4.2) admits solutions that are constant in time, namely,

u ∈ L 2 (Ω), ∇ • u| Γ = 0, p = 0,
it is clear that exponential stability cannot hold if inf x∈Ω α(x) = 0. Indeed, as α is smooth, we can always find for any δ > 0 small a solenoidal compactly supported function φ 0 in Ω such that α| supp φ 0 ≤ δ.

Consequently, relation (4.2) cannot holds uniformly for any choice of the data.

On the other hand, suppose that α is bounded below away from zero on the whole set Ω. Writing the vector field u as its Helmholtz decomposition

u = H[ u] + ∇ϕ, (4.3)
where H denotes the standard Helmholtz projection onto the space of solenoidal functions, it is enough to verify the observability criterion (4.2) for φ = ∇ϕ. In such a case, however, the conservative system (4.1) reduces to the standard wave equation and (4.2) is obviously satisfied as the damping acts uniformly on the whole domain Ω (see also Section 5).

Changing the damping law

As we have seen before, system (1.1) is exponentially stable if and only if α is uniformly positive definite. We will show in this section that if we change the feedback law in order to filter the divergence free vector fields, then we will get exponential stability for a quite large set of α.

In view of the Helmholtz decomposition (4.3), denote by P the orthogonal projection on the closed subspace of L 2 (Ω)

d V := ∇ϕ : ϕ ∈ H 1 (Ω) Ω ϕ dx = 0 .
Then in (1.1) we change the damping term α u by P (α P u) and consider the system

       u t + ∇r + P (α P u) = 0, in Ω × R + , r t + div u = 0, in Ω × R + , u.n = 0, on Γ × R + , u(0, x) = u 0 (x), r(0, x) = r 0 (x), x ∈ Ω.
(5.1) Accordingly, we arrive at the system

       ∇ϕ t + ∇r + P (α ∇ϕ) = 0, in Ω × R + , r t + ∆ϕ = 0, in Ω × R + , ∇ϕ • n = 0, on Γ × R + , ϕ(0, x) = ϕ 0 (x), r(0, x) = r 0 (x), x ∈ Ω, (5.2)
where

Ω ϕ 0 dx = Ω r 0 dx = 0.
By virtue of [START_REF] Haraux | Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps[END_REF] (see also [START_REF] Ammari | Stabilization of second order evolution equations by a class of unbounded feedbacks[END_REF]), the exponential stability of the system (5.2) on V × V is equivalent to the following property of solutions to the undamped system: There exist a positive real number T 0 such that for all T > T 0 , there exists C > 0 such that

T 0 Ω α(x) |∇ψ(x, t)| 2 dxdt ≥ C (∇ψ 0 , p 0 ) 2 (L 2 (Ω) d ×L 2 (Ω) (5.3)
for any ψ, p satisfying

       ∇ψ t + ∇p = 0, in Ω × R + , p t + ∆ψ = 0, in Ω × R + , ∇ψ • n = 0, on Γ × R + , ψ(0, x) = ψ 0 (x), p(0, x) = p 0 (x), x ∈ Ω, (5.4
)

ψ 0 ∈ H 1 (Ω), p 0 ∈ L 2 (Ω), Ω ψ 0 dx = Ω p 0 dx = 0.
However, the system (5.4) can be written in the form of a standard wave equation with the Neumann Laplacian: (5.6)

   ψ tt + ∆ψ = 0, in Ω × R + , ∇ψ • n = 0, on Γ × R + , ψ(0, x) = ψ 0 (x), ψ t (0, x) = ψ 1 (x) = -p 0 (x), x ∈ Ω,
Thus we have obtained the following exponential stability result:

Theorem 5.1. The system (5.1) is exponentially stable, meaning,

( u, r)(t, •) H ≤ exp(-Lt) ( u 0 , r 0 ) H for a certain L > 0, whenever u 0 = ∇ϕ 0 , ϕ 0 ∈ H 1 (Ω), Ω ϕ 0 dx = Ω r 0 dx = 0,
if and only if any solution ψ of the wave equation (5.5) satisfies the observability inequality (5.6).

Validity of the observability inequality (5.6) is related to the general discussion in Bardos et al. [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF] (see also Zuazua [START_REF] Zuazua | Stability and decay for a class of nonlinear hyperbolic problems[END_REF][START_REF] Zuazua | Exponential decay for the semilinear wave equation with localized damping in unbounded domains[END_REF]). Here we note that (5.6) holds provided ω contains a neighborhood of the whole boundary Γ, in the sense that there exists a neighborhood O of Γ in R d such that Ω ∩ O ⊂ ω. First we notice that by the arguments of section 4 of [START_REF] Martinez | A new method to obtain decay rate estimates for dissipative systems with localized damping[END_REF] with a = 0 and M (u) = m∇u + d- 1 2 u (see estimate (4.10)), where, as usual, m(x) = xx 0 for some x 0 ∈ R d , there exits C > 0 such that for all T > 0, we have

2T E(0) ≤ C(E(0) + T 0 Ω∩O ′ (|ψ t | 2 + |∇ψ| 2 + |ψ| 2 )dxdt, (5.7)
where O ′ is a sufficiently small neighbourhood of the boundary (such that O ′ ⊂ O). On the other hand, fixing a cut-off function η such that η ≡ 1 on O ′ and with a support O, we can write

T 0 Ω∩O ′ |ψ t | 2 dxdt ≤ T 0 Ω η|ψ t | 2 dxdt.
Hence by integration by parts in time, we get

T 0 Ω∩O ′ |ψ t | 2 dxdt ≤ - T 0 Ω ηψ tt ψdxdt + Ω ηψ t ψdx T 0 .
Now using the first identity in (5.5) and an integration by parts in space, we get

T 0 Ω∩O ′ |ψ t | 2 dxdt ≤ T 0 Ω ∇ψ • (∇η ψ)dxdt + Ω ηψ t ψdx T 0 .
Therefore using Leibniz's rule and Cauchy-Schwarz's inequality we obtain that

T 0 Ω∩O ′ |ψ t | 2 dxdt ≤ C 1 T 0 Ω∩O (|∇ψ| 2 + |ψ| 2 )dxdt + C 2 E(0),
for some positive constants C 1 , C 2 independent of T . Inserting this estimate in (5.7) we get

2T E(0) ≤ (C + C 2 )E(0) + C 4 T 0 Ω∩O (|∇ψ| 2 + |ψ| 2 )dxdt, (5.8)
where C 4 = max{1, C 1 }. To eliminate the last term of this right-hand side we use a compacteness/uniqueness argument. Namely assume that (5.3) does not hold. Then there exists a sequence of (ψ ℓ ) ℓ∈N solution of (5.5) with initial data ψ 0 ℓ and ψ 1 ℓ such that ∇ψ 0 ℓ + ψ 1 ℓ = 1, (5.9) and

T 0 Ω α(x) |∇ψ ℓ (x, t)| 2 dxdt = 1 ℓ .
(5.10)

Since the system (5.5) is conservative, the sequence (ψ ℓ ) is bounded in H 1 (Q T ), where (5.5) with initial data ψ 0 , ψ 1 . But thanks to (5.8) applied to ψ ℓψ ℓ ′ , we deduce that the sequence (ψ ℓ ) ℓ∈N is a Cauchy sequence in H 1 (Q T ). Hence ψ ℓ converges in H 1 (Q T ) to ψ, weak solution of (5.5) and satisfying (5.12)

Q T = Ω × (0, T )). Hence it converges strongly in L 2 (Q T ) to ψ a weak solution (in H 1 (Q T ) of
∇ψ 0 + ψ 1 = 1, ( 5
Accordingly ∇ψ = 0 in ω × (0, T ), or quivalently ψ(x, t) = ψ(t) in ω × (0, T ).
But again due to the first identity in (5.5),

ψ tt (t) = 0 in ω × (0, T ),
hence there exists two complex numbers a, b such that

ψ(x, t) = at + b in ω × (0, T ). Now consider the difference ψ = ψ -at -b.
Then we see that it a weak solution of (5.5) such that ψ = 0 in ω × (0, T ).

By using Theorem 9.1 of [START_REF] Triggiani | Carleman estimates with no lower-order terms for general Riemann wave equations. Global uniqueness and observability in one shot[END_REF], we deduce that

ψ = 0 on Q T , or equivalently ψ(x, t) = at + b on Q T .
But in our situation ψ 0 belongs to H 1 m (Ω) and therefore ψ(•, t) belongs to H 1 m (Ω), for all t > 0. As a consequence we deduce that

0 = Ω ψ(x, t) dx = (at + b)|Ω|, ∀t > 0.
This imples that a = b = 0 and contradicts (5.11).

Thus we have shown the following: Corollary 5.2. Suppose that inf x∈Γ α(x) > 0.

Then the system (5.1) is exponentially stable in the sense specified in Theorem 5.1.

Polynomial stability

In this section we show that system (1.1) is polynomially stable if ω contains a neighborhood of the whole boundary Γ as in Corollary 5.2.

This result is based on the following result stated in Theorem 2.4 of [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF] (see also [START_REF] Bátkai | Polynomial stability of operator semigroups[END_REF][START_REF] Batty | Non-uniform stability for bounded semi-groups on Banach spaces[END_REF][START_REF] Liu | Characterization of polynomial decay rate for the solution of linear evolution equation[END_REF] for weaker variants). Lemma 6.1. A C 0 semigroup e tL of contractions on a Hilbert space such that

ρ(L) ⊃ iβ β ∈ R ≡ iR, (6.1)
satisfies ||e tL U 0 || ≤ C t -1 l ||U 0 || D(L) , ∀U 0 ∈ D(L), ∀t > 1,
as well as

||e tL U 0 || ≤ C t -1 ||U 0 || D(L l ) , ∀U 0 ∈ D(L l ), ∀t > 1,
for some constant C > 0 and for some positive integer l if and only if

lim sup |β|→∞ 1 β l (iβ -L) -1 < ∞. (6.2)
Lemma 6.2. Assume that ω contains a neighborhood of the whole boundary Γ.

Then the resolvent of the operator of A d satisfies condition (6.2) with l = 3.

Proof. We use a contradiction argument, i.e., we suppose that (6.2) is false with some l ≥ 0. Then there exist a sequence of real numbers β n → +∞ and a sequence of vectors

Z n = ( u n , r n ) ⊤ in D(A d ) with Z n H = 1, (6.3)
and

β l n (iβ n -A d )Z n H → 0 as n → ∞. (6.4)
From the definition of A d this last property is equivalent to

β l n (iβ n u n + ∇r n + α u n ) =: f n → 0 in L 2 (Ω) d , (6.5)
and

β l n (iβ n r n + div u n ) =: g n → 0 in L 2 (Ω). (6.6)
As usual, taking the inner product of (6.4) with Z n , using (6.3) and the dissipativeness of A d , we find

Ω α | u n | 2 dx = ℜ((iβ n -A d )Z n , Z n ) = o(β -l n ), (6.7)
where, here and hereafter,

a n = o(b n ) means that lim n→∞ a n b n = 0.
We multiply (6.5) by u n , integrate in Ω and use (6.3) to find

Ω (iβ n | u n | 2 + ∇r n • u n + α| u n | 2 ) dx = o(β -l n ).
Using (6.7) and integrating by parts we obtain

Ω (iβ n | u n | 2 -r n div u n ) dx = o(β -l n ).
Using (6.6) (and (6.3)) we get

iβ n Ω (| u n | 2 -|r n | 2 ) dx = o(β -l n ).
This shows that

u n 2 -r n 2 = o(β -(l+1) n ). (6.8)
The identity (6.5) implies that

∇r n = β -l n f n -(iβ n + α) u n , (6.9)
and therefore

∇r n = O(β n ), (6.10)
where, here and hereafter, a n = O(b n ) means that there exists C > 0 independent of n such that a n ≤ Cb n for n large enough.

By (6.9), we also have

Ω α |∇r n | 2 dx ≤ 2 Ω α|iβ n + α| 2 | u n | 2 dx + 2β -2l n Ω α | f n | 2 dx.
Hence, for l ≥ 2, by using (6.7) we find that

Ω α |∇r n | 2 dx = o(β 2-l n ). (6.11)
Now we notice that (6.5) and (6.7) imply that

iβ n u n + ∇r n = h n := β -l n f n -α u n , (6.12)
with

h n = o(β -l/2 n
). (6.13)

We use the Helmholtz decomposition

h n = ∇ϕ n + χ n , (6.14)
∇ϕ n 2 Ω + χ n 2 Ω = h n 2 to deduce that ϕ n 2 1,Ω + χ n 2 Ω = o(β -l n ). (6.15)
The identities (6.12) and ( 6 At this stage we use the multiplier method to find some properties on s n . First we multiply (6.21) by -is n and integrate in Ω to obtain due to (6.19) and (6.22)

Ω (β n s n + 1 β n div ∇s n )s n dx = o(β 1-l/2 n ).
By an integration by part and since the boundary term is zero due to (6.18), we obtain

Ω (β n |s n | 2 - 1 β n |∇s n | 2 ) dx = o(β 1-l/2 n ). (6.23)
Secondly we take a (smooth) real-valued multiplier m ∈ C 2 (Ω) d fixed later on but such that m = 0 on Γ and multiply (6.21) by -im • ∇s n and integrate in Ω to obtain due to (6.20) and (6.22)

Ω (β n s n + 1 β n div ∇s n )m • ∇s n dx = o(β 2-l/2 n ). (6.24)
Now, in a standard way, by means of Green's formula, the first term

I 1 := Ω s n m • ∇s n dx is transformed into I 1 = - Ω ∂ k (s n m k )s n dx + Γ m • n|s n | 2 dσ = -Ī1 - Ω div m|s n | 2 dx,
recalling that m = 0 on the boundary. Hence we have

2ℜI 1 = - Ω div m|s n | 2 dx. (6.25)
In the same manner we have

I 2 := Ω div ∇s n m • ∇s n dx = - Ω ∂ k s n ∂ k (m • ∇s n ) dx (6.26) = - Ω ∂ k s n ∂ k m j ∂ j sn dx -I 3 ,
where

I 3 = Ω ∂ k s n m j ∂ k ∂ j sn dx.
Again an integration by part leads to

I 3 = - Ω ∂ j (∂ k s n m j )∂ k sn dx = -Ī3 - Ω div m|∇s n | 2 dx. Consequently 2ℜI 3 = - Ω div m|∇s n | 2 dx,
and using this expression in (6.26) we find

2ℜI 2 = -2ℜ Ω ∂ k s n ∂ k m j ∂ j sn ) dx (6.27) + Ω div m|∇s n | 2 dx.
Taking the real part of (6.24) and using (6.25) and (6.27) we obtain

-β n Ω div m|s n | 2 dx + 1 β n Ω div m|∇s n | 2 dx - 2 β n ℜ Ω ∂ k s n ∂ k m j ∂ j sn dx = o(β 2-l/2 n
). (6.28) From (6.11) and (6.15), we see that

Ω α|∇s n | 2 dx = o(β 2-l n ), (6.29)
hence for l > 2, ∇s n tends to zero on ω. This means that we are mainly interested in the behaviour of s n outside ω. Therefore we take for m a function with a support far from the boundary, namely we take

m(x) = η(x)x,
where η is a smooth cut-off function such that

η = 1 on ω c and η = 0 in O ′ ,
where O ′ is a neigbourhood of Γ. Since

∂ k m j = δ jk η + x j ∂ k η, (6.28) becomes -β n Ω (dη + x • ∇η)|s n | 2 dx + 1 β n Ω ((d -2)η + x • ∇η)|∇s n | 2 dx - 2 β n ℜ Ω ∂ k s n ∂ k ηx j ∂ j sn dx = o(β 2-l/2 n ). ( 6

.30)

Assume that the next estimate

ω |s n | 2 dx 1 2 = o(β 1-l/2 n ) (6.31)
holds. Then combining this estimate with (6.29) in (6.30) we obtain

-β n d Ω η|s n | 2 dx + 1 β n (d -2) Ω η|∇s n | 2 dx = o(β 2-l/2 n ). (6.32)
Coming back to (6.23) and writting 1 = η + (1η), we get

Ω (β n η|s n | 2 - 1 β n η|∇s n | 2 ) dx = o(β 1-l/2 n ) - Ω (β n (1 -η)|s n | 2 - 1 β n (1 -η)|∇s n | 2 ) dx.
Hence (6.31) and (6.29) lead to .33) This estimate in (6.32) yields

Ω (β n η|s n | 2 - 1 β n η|∇s n | 2 ) dx = o(β 1-l/2 n ) + o(β 3-l n ). ( 6
Ω η|s n | 2 dx = o(β 1-l/2 n ) + o(β 2-l n ).
By choosing l ≥ 2 and using again (6.31), we arrive at Coming back to our original variables, for r n we have by (6.34) and (6.15) that

Ω |s n | 2 dx = o(β 1-l/2 n ). ( 6
r n 2 = o(β 1-l/2 n ),
while for u n using (6.16), (6.35) and (6.15)

u n 2 = o(β 1-l/2 n ).
In conclusion for l ≥ 3, r n and u n tend to zero which contradicts (6.3).

It remains to prove (6.31). For that purpose, we first show that the mean of s n on Γ tends to zero. Fix a function h ∈ (C 2 ( Ω)) d such that supp h ⊂ ω and h • n = 1 on Γ. Such a function exists by Lemma I.3.1 of [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF] since we assume that Ω has a C 3 -boundary. We then multiply (6.5) by h and integrate in Ω to get

Ω ((iβ n + α) u n + ∇r n )h dx = o(β -l n ).
Hence integrating by parts, we obtain

Ω ((iβ n + α) u n h -r n div h) dx + Γ r n dσ = o(β -l n ). (6.36)
We still need to transform the term Ω r n div h dx.

By using (6.6) we may write ). (6.38) This estimate and (6.15) leads to (6.31).

This Lemma and Lemmas 3.3 and 3.4 show that the restriction of the operator A d to H 0 satisfies the hypotheses of Lemma 6.1 with l = 3. Therefore we have obtained the next main result. Theorem 6.3. Assume that ω contains a neighborhood of the whole boundary Γ then the system (1.1) with an initial datum ( u 0 , r 0 ) in H 0 ∩ D(A d ), H 0 = Ker[A d ] ⊥ , is polynomially stable, namely there exists C > 0 such that E(t) ≤ Ct -2/3 ( u 0 , r 0 ) 2 D(A d ) , ∀t > 0.

Lemma 3 . 3 .

 33 Suppose that |ω| > 0. If λ is a non-zero real number, then iλ is not an eigenvalue of A d . Proof. Suppose that iλ u + ∇r + α u = 0, iλ r + div u = 0. (3.11) From (3.11) we deduce that iλ Ω (| u| 2 -|r| 2 )dx + Ω α| u| 2 dx = 0;

Theorem 4 . 1 .

 41 If the damping coefficient α is not uniformly positive definite, meaning inf x∈Ω α(x) = 0, then the system (1.1) is not exponentially stable. Conversely, if α is uniformly positive definite, meaning ω = Ω, then the system (1.1) is exponentially stable, specifically,

2 (L 2 (

 22 |∇ψ(x, t)| 2 dxdt ≥ C (∇ψ 0 , ψ 1 ) Ω) d ×L 2 (Ω) .

  |∇ψ(x, t)| 2 dxdt = 0.

  .34) Taking into account (6.23) we getΩ |∇s n | 2 dx = β 2 n Ω |s n | 2 dx + o(β 2-l/2 n ),and (6.34) finally leads toΩ |∇s n | 2 dx = o(β 3-l/2

Ω r n 2 =

 2 div h dx = -i Ω (β -(l+1) n g ndiv u n )div h dx.Again by Green's formula and the fact thatu n • n = 0 on Γ we obtain Ω r n div h dx = -i Ω (β -(l+1) n g n div h + u n • ∇div h) dx.Using this identity in (6.36) we obtainΓ r n dσ = -Ω ((iβ n + α) u n hiβ -(l+1) n g n div hi u n • ∇div h) dx + o(β -l nfor some C > 0 depending only on ω. Hence using (6.37) and (6.11) yields ω |r n | 2 dx 1 o(β 1-l/2 n

  .14) yield iβ n u n + ∇r n = ∇ϕ n + χ n ,As u n is solenoidal, s n is harmonic, and, furthermore sinceu n • n = χ n • n = 0 on Γ,

	Next, by (6.3) and (6.15), one has
	s n = O(1).		(6.19)
	while (6.10) and (6.15) lead to
	∇s n = O(β n ).	(6.20)
	Now by (6.6), (6.16) and (6.18) we see that s n satisfies
	iβ n s n -	1 iβ n	div ∇s n = gn	(6.21)
	where
	gn := β -l n g n -iβ n ϕ n .
	Due to (6.6) and (6.15) we see that
	gn = o(β 1-l/2 n	).	(6.22)
	or equivalently
	iβ n u n = -∇s n + χ n ,	(6.16)
	with
	s n = r n -ϕ n .		(6.17)
	we automatically have
	∇s n • n = 0 on Γ.	(6.18)

Supported by Grant 201/09/ 0917 of GA ČR as a part of the general research programme of the Academy of Sciences of the Czech Republic, Institutional Research Plan RVO: 67985840.