STABILIZATION OF MULTIDIMENSIONAL WAVE EQUATION WITH LOCALLY BOUNDARY FRACTIONAL DISSIPATION LAW UNDER GEOMETRIC CONDITIONS

. In this paper, we consider a multidimensional wave equation with boundary fractional damping acting on a part of the boundary of the domain. First, combining a general criteria of Arendt and Batty with Holmgrens theorem we show the strong stability of our system in the absence of the compact- ness of the resolvent and without any additional geometric conditions. Next, we show that our system is not uniformly stable in general, since it is the case of the interval. Hence, we look for a polynomial decay rate for smooth initial data for our system by applying a frequency domain approach combining with a multiplier method. Indeed, by assuming that the boundary control region satisfy some geometric conditions and by using the exponential decay of the wave equation with a standard damping, we establish a polynomial energy decay rate for smooth solutions, which depends on the order of the fractional derivative.

The fractional derivative operator of order α, 0 < α < 1, is defined by The fractional differentiation is inverse operation of fractional integration that is defined by From equations (5) and (6), clearly that Now, we present marginally distinctive forms of (5) and (6). These exponentially modified fractional integro-differential operators and will be denoted as follows and Note that the two operators D α and D α,η differ just by their Kernels. D α,η is merely Caputo's fractional derivative operator, expect for its exponential factor. Thus, similar to identity (7), we do have [D α,η f ] = I 1−α,η Df.
The order of our derivatives is between 0 and 1. The boundary fractional damping of the type ∂ α,η t u where 0 < α < 1, η ≥ 0 arising from the material property has been used in several applications such as in physical, chemical, biological, ecological phenomena. The order of our derivatives is between 0 and 1. Very little attention has been paid to this type of feedback. In addition to being nonlocal, fractional derivatives involve singular and nonintegrable kernels (t −α , 0 < α < 1). This leads to substantial mathematical difficulties because all the previous methods developed for convolution terms with regular and/or integrable kernels are no longer valid.
In the last year, fractional differential equations have become popular among scientists in order to model various stable physical phenomena with a slow decay rate, say that are not uniformly stable (i.e., are not of exponential type) .
It has been shown (see [24] and [26]) that, as ∂ t , the fractional derivative ∂ α t forces the system to become dissipative and the solution to approach the equilibrium state. Therefore, when applied on the boundary, we can consider them as controllers which help to reduce the vibrations. Boundary dissipations of fractional order or, in general, of convolution type are not only important from the theoretical point of view but also for applications. They naturally arise in physical, chemical, biological, ecological phenomena see for example [29], [33] and references therein.
They are used to describe memory and hereditary properties of various materials and processes. For example, in viscoelasticity, due to the nature of the material micro-structure, both elastic solid and viscous fluid like response qualities are involved. Using Boltzman assumption, we end up with a stress-strain relationship defined by a time convolution. Viscoelastic response occurs in a variety of materials, such as soils, concrete, rubber, cartilage, biological tissue, glasses, and polymers (see [5]- [6]- [7] and [23]). In our case, the fractional dissipations may come from a viscoelastic surface of the beam or simply describe an active boundary viscoelastic damper designed for the purpose of reducing the vibrations (see [25] and [26]). In [15], Zhang and Dai considered the multidimensional wave equation with boundary source term and fractional dissipation defined by where m > 1. They proved by Fourrier transforms and the Hardy-Littelwood-Sobolev inequality the exponential stability for sufficiently large initial data.
In [2], Benaissa and al. considered the Euler-Bernoulli beam equation with boundary dissipation of fractional derivative type defined by where 0 < α < 1, η ≥ 0 and γ > 0. They proved, under the condition η > 0, by a spectral analysis, the non uniform stability. On the other hand, for η > 0, they also proved that the energy of system (12) decay as time goes to infinity as t − 1 1−α . In [25], B. Mbodje investigate the asymptotic behavior of solutions with the system x ∈ (0, 1), He proved that the associated semigroup is not exponentially stable, but only strongly asymptotically and the energy of this system will decay, as time goes to infinity, as t −1 . In [3], Alabau-Boussouira and al. have studied the exponential and polynomial stability of a wave equation for boundary memory damping with singular kernels. This paper is organized as follows: In Subsection 2.1, we reformulate the system (1)-(4) into an augmented system by coupling the wave equation with a suitable diffusion equation and we prove the well-posedness of our system by semigroup approach. In the Subsection 2.2, combining a general criteria of Arendt and Batty with Holmgren's theorem we show that the strong stability of our system in the absence of the compactness of the resolvent and without any additional geometric conditions. In Subsection 2.3, We show that our system is not uniformly stable in general, since it is the case of the interval, more precisely we show that an infinite number of eigenvalues approach the imaginary axis. In Section 3, we look for a polynomial decay rate for smooth initial data for our system by applying a frequency domain approach combining with a multiplier method. Indeed, by assuming that the boundary control region satisfy some geometric conditions (see Remark 3.1 below) and by using the exponential decay of the wave equation with a standard damping, we establish a polynomial energy decay for smooth solution as type t − 1 1−α .

2.
Well-posedness -strong stability and non uniform stability. In this section, we will study the strong stability of system (1)-(4) in the absence of the compactness of the resolvent and without any additional geometric conditions on the domain Ω. First, we will study the existence, uniqueness and regularity of the solution of our system.

2.1.
Augmented model and well-Posedness. Firstly, we reformulate system (1)-(4) into an augmented system and we generalize the Theorem 2 in [25] from dimension 1 to dimension d. For this aim, we need the following results Proposition 1. Let µ be the function defined by Then the relation between the 'input' U and the 'output' O of the following system is given by where D α,η and I 1−α,η are given by (8) and (9) respectively.
Proof. We distinguish two cases: Step 1. The case η = 0. Using (15) and (16), we get It follows, from (17), that Using the fact π −1 sin(απ) = and using step 1, we get (18). Now, using Proposition 1, system (1)-(4) may be recast into the following augmented model: where γ is a positive constant, η ≥ 0 and κ = . Finally, system (25) is considered with the following initial conditions Our main interest is the existence, uniqueness and regularity of the solution to this system. We define the Hilbert space equipped with the following inner product The energy of the solution of system (22)-(26) is defined by: Lemma 2.1. Let U = (u, u t , ω) be a regular solution of problem (22)- (26). Then, the functional energy defined in equation (28) satisfies Proof. Multiplying equation (22) byū t , using integration by parts over Ω and equation (25), we get Multiplying equation (24) by γκω(x, t, ξ) and integrating over Γ 1 × R d , we get Combining equations (30)-(31), we obtain d dt This completes the proof.
Hence, from Lemma 2.1, system (22)- (26) is dissipative in the sense that its energy is a non-increasing function of the time variable t. Now, we define the linear unbounded operator A by and By denoting v = u t and U 0 = (u 0 , v 0 , w 0 ) , system (22)-(26) can be written as an abstract linear evolution equation on the space H It is known that operator A is m-dissipative on H and consequently, generates a C 0semigroup of contractions e tA following Lumer-Phillips theorem (see [21,30]). Then the solution to the evolution equation (33) admits the following representation: which leads to the well-posedness of (33). Hence, semigroup theory allows to show the next existence and uniqueness results: For any initial data U 0 ∈ H, the problem (33) admits a unique weak solution Moreover, if U 0 ∈ D(A), then the problem (33) admits a unique strong solution

2.2.
Strong stability of the system. In this part, we study the strong stability of system (22)- (26) in the sense that its energy converges to zero when t goes to infinity for all initial data in H. It is easy to see that the resolvent of A is not compact, then the classical methods such as Lasalle's invariance principle [34] or the spectrum decomposition theory of Benchimol [11] are not applicable in this case. We use then a general criteria of Arendt-Battay [4], following to which a C 0semigroup of contractions e tA in a Banach space is strongly stable, if A has no pure imaginary eigenvalues and σ(A) ∩ iR contains only a countable number of elements. We will prove the following stability result For the proof of Theorem 2.3, we need the following lemmas: Then, for all λ ∈ R, we have Proof. Let U ∈ D(A) and λ ∈ R, such that Next, a straightforward computation gives Then, using (34) and (38), we deduce that It follows, from (32) and (37), that ∂u ∂ν = 0 and v = 0 on Γ 1 .
Proof. First, let ϕ k ∈ H 1 Γ0 (Ω) be an eigenfunction of the following problem and Note that ϕ k | Γ1 = 0 because, if not a unique continuation result allows to deduce ϕ k = 0 in Ω. Now, from (45), we deduce that ω(·, ξ) = |ξ| So, the assumption of the existence of U is false and consequently the operator −A is not invertible. This completes the proof.
Lemma 2.7. If η > 0, for all λ ∈ R, we have Proof. We give the proof in the case η > 0, the proof of the second statement is fully similar. Let λ ∈ R and F = (f, g, h) ∈ H, then we look for U = (u, v, ω) ∈ D(A) solution of (iλI − A)U = F. (54) Equivalently, we have As before, by eliminating v and ω from the above system and using the fact that we get the following system: in Ω, where c 1 , c 2 is defined in equation (48) and I 1 h , I 2 h are given by It is easy to see that, for h ∈ L 2 Γ 1 × R d and α ∈]0, 1[, the integrals Therefore, setting χ = u − θ in (55), we get Using Lemma 2.6, problem (57) has a unique solution χ ∈ H 1 Γ0 (Ω) and therefore problem (55) has a unique solution u ∈ H 1 Γ0 (Ω). By defining v = iλu − f in Ω and we deduce that U = (u, v, ω) belongs to D(A) and is solution of (54). This completes the proof.
Proof of Theorem 2.3. Following a general criteria of Arendt-Batty see [4], the C 0 -semigroup of contractions e tA is strongly stable, if σ (A)∩iR is countable and no eigenvalue of A lies on the imaginary axis. First, from Lemma 2.4 we directly deduce that A has non pure imaginary eigenvalues. Next, using Lemmas 2.5 and 2.7, we conclude, with the help of the closed graph theorem of Banach, that σ(A)∩iR = {∅} if η > 0 and σ(A) ∩ iR = {0} if η = 0. The proof is thus completed.
Remark 1. We mention [32] for a direct approach of the strong stability of Kirchhoff plates in the absence of compactness of the resolvent. This result is due to the fact that a subsequence of eigenvalues of A is close to the imaginary axis. For this aim, let λ ∈ C and U = (u, v, ω) ∈ D(A) be such that AU = λU . Equivalently we have Since A is dissipative, we study the asymptotic behavior of the large eigenvalues λ of A in the strip α 0 ≤ (λ) ≤ 0, for some α 0 > 0 large enough. By eliminating v and ω from the above system and using the fact that and Lemma 4.1 we get the following system: We have the following asymptotic behaviour Proposition 2. There exist k 0 ∈ N * and a sequence (λ k ) |k|≥k0 of simple eigenvalues of A and satisfying the following asymptotic behavior: for k large enough.
Note that f 0 and f 1 remain bounded in the strip α 0 ≤ (λ) ≤ 0. It is easy to check that the roots of f 0 are given by where µ k = i k + 1 2 π. Using Rouché's theorem, we deduce that f (λ) admits an infinity of simple roots in the strip α 0 ≤ (λ) ≤ 0 denoted by λ k , with |k| ≥ k 0 , for k 0 large enough, such that Using (63), we get Next, by inserting (66)-(68) in the identity f (λ) = 0 and keeping only the terms of order 1 k 1−α , we find after a simplification From equation (69), we have Inserting equation (69) in (65), we get the desired equation (59). This implies that the C 0 -semigroup of contractions e tA is not uniformly stable in the energy space H.
Numerical Validation. By using Wolfram Mathematica, the asymptotic behavior λ k in equation (59) can be numerically validated. For instance, with α = 0.5, η = 1 and γ = 1 then from equation (59) we have The

3.
Polynomial stability under geometric control condition. This section is devoted to the study of the polynomial stability of system (22)- (26) in the case η > 0 and under appropriated geometric conditions. For that purpose, we will use a frequency domain approach, namely we will use Theorem 2.4 of [12] (see also [9,10,20]) that we partially recall.
Theorem 3.1. Let (T (t)) t≥0 be a bounded C 0 -semigroup on a Hilbert space H with generator A such that iR ⊂ ρ(A). Then for a fixed > 0 the following conditions are equivalent As the condition iR ⊂ ρ(A) was already checked in Theorem 2.3, it remains to prove that condition (70) holds. This is made with the help of a multiplier method under some geometric conditions on the boundary of the domain and by using the exponential decay of an auxiliary problem. Firstly, like as [1,28], we consider the following auxiliary problem, namely the wave equation with standard boundary damping on Γ 1 : Define the auxiliary space H a = H 1 Γ0 (Ω)×L 2 (Ω) and the auxiliary unbounded linear operator A a by A a (ϕ, ψ) = (ψ, ∆ϕ) and D(A a ) = Φ = (ϕ, ψ) ∈ H a : ∆ϕ ∈ L 2 (Ω); ψ ∈ H 1 Γ0 (Ω); We then introduce the following condition: (H) : the problem (72) is uniformly stable in the energy space H 1 Γ0 (Ω) × L 2 (Ω). Secondly, we recall the Geometric Control condition (GCC) introduced by Bardos, Lebeau and Rauch [8]: Definition 3.2. We say that Γ satisfies the geometric condition named (GCC), if every ray of geometrical optics, starting at any point x ∈ Ω at time t = 0, hits Γ 1 in finite time T . We also recall the multiplier control condition (MGC) in the following definition: Remark 2. In [8], Bardos and al. proved that (H) holds if Γ is smooth (of class C ∞ ), Γ 0 ∩ Γ 1 = ∅ and under the (GCC) condition. For less regular domains, namely of class C 2 , (H) holds if the vector field assumptions described in [17] (see (i),(ii),(iii) of Theorem 1 in [17]) hold. Moreover, in Theorem 1.2 of [18] the authors proved that (H) holds for smooth domains under weaker geometric conditions than in [17] (without (ii) of Theorem 1). Finally, it is easy to see that the multiplier control condition (MGC) implies that the vector field assumptions described in [17] are satisfied and therefore the condition (H) holds if (MGC) holds.
Remark 3. In Figure 1, We take an open arc Υ in the boundary that contains a half-circumference and let P denote the midpoint of Υ. For ε sufficiently small denote γ ε the closed arc centered at P with length less ε. For a ray to miss Υ\P at must hit P as does the equilateral triangle with vertex P . Let θ denote the union of two open arcs centered respectively at the antipodal of P and one of the other vertices of the equilateral triangle. Let Γ 1 = (Υ ∪ θ)\γ ε and Γ 0 = ∂Ω\Γ 1 , then the condition (GCC) holds.
Next, we present the main result of this section. Theorem 3.4. Assume that η > 0 and that the condition (H) holds. Then, for all initial data U 0 ∈ D(A), there exists a constant C > 0 independent of U 0 , such that the energy of the strong solution U of (33) satisfies the following estimation In particular, for U 0 ∈ H, the energy E(t, U ) converges to zero as t goes to infinity.
As announced in Theorem 3.1, by taking = 2−2α, the polynomial energy decay (73) holds if the following conditions and sup are satisfied. Condition (H1) is already proved in Theorem 2.3. We will prove condition (H2) using an argument of contradiction. For this purpose, suppose that (H2) is false, then there exist a real sequence (λ n ), with |λ n | → +∞ and a sequence (U n ) ⊂ D(A), verifying the following conditions and (75) For simplicity, we drop the index n. Detailing equation (75), we get Note that U is uniformly bounded in H. Then, taking the inner product of (75) with U in H, we get Inserting equation (76) in (77), we get Lemma 3.5. Assume that η > 0. Then the solution (u, v, w) ∈ D(A) of (76)-(78) satisfies the following asymptotic behavior estimation Proof. Using equations (74) and (76), we deduce directly the first estimation (81). Now, from the boundary condition by integrating over Γ 1 and using Cauchy-Schwartz inequality, we get Then, combining equation (79) and equation (84), we obtain the desired estimation (82). Finally, multiplying equation (78) by (iλ+|ξ| 2 +η) −1 µ(ξ), integrating over R d with respect to the variable ξ and applying Cauchy-Shwartz inequality, we obtain dξ, Using Holder inequality in equation (85), then integrating over Γ 1 , we get On the other hand, using Lemma 4.2, we have It follows, from (76), that equation (83) holds. The proof has been completed.
Lemma 3.7. Assume that condition (H) holds and let u be a solution of (80). Then, for any λ ∈ R, the solution ϕ u ∈ H 1 (Ω) of system satisfies the following estimate Proof. Following Huang [16] and Pruss [31], the exponential stability of system (72) implies that the resolvent of the auxiliary operator A a is uniformly bounded on the imaginary axis i.e. there exists M > 0 such that for all λ ∈ R. Now, since u ∈ L 2 (Ω), then the pair (0, u) belongs to H a , and from (92), there exists a unique (ϕ u , ψ u ) ∈ D(A a ) such that (iλI −A a )(ϕ u , ψ u ) = (0, u) i.e.
From equations (93)-(95), we deduce that ϕ u is solution of (90) and we have Therefore, multiplying the first equation of (90) by λφ u and using Green formula, we get By tacking the imaginary part of equation (97) and using Cauchy-Shwartz inequality, we deduce, from (96), that Finally, Combining equations (96) and (98) we obtain the desired equation (91). The proof is thus complete.
It follows, from (79) and (89), that U H = o(1) which is a contradiction with (74). Consequently condition (H2) holds and the energy of smooth solution of system (22)-(25) decays polynomial to zero as t goes to infinity. Finally, using the density of the domain D(A) in H, we can easily prove that the energy of weak solution of system (22)-(25) decays to zero as t goes to infinity. The proof has been completed.

Conclusion.
We have studied the stabilization of multidimensional wave equation with boundary fractional damping acting on a part of the boundary of the domain. Non uniform stability is proved and a polynomial energy decay rate of type t − 1 1−α is established. In view of the asymptotic behavior of the eigenvalues of the operator A (59), we deduce that the optimal energy decay rate is of type t − 2 1−α (see [22,27]). This question still be open. Proof. Firstly, we calculate A 1 . Using the hyper-spherical coordinates and the fact that the Jacobian J is defined by we get On the other hand, it is easy to see that

This implies that
Consequently, using the same calculation used in Lemma 4.1, we obtain where c is defined in (108).
Secondly, we calculate A 3 . Using the same calculation of A 1 , we obtain Let x = ρ 2 in (109), we get By tacking y = x |λ|+η + 1 in (110), we deduce that In addition, it is easy to check that for α ∈]0, 1[, the integral I y is well defined. Finally, from equation (110), we obtain wherec is defined in equation (108).