Exponential stability of a coupled system with Wentzell conditions

A coupled system of hyperbolic equations in Maxwell/wave with Wentzell conditions in a bounded domain of $\mathbb{R}^3$ 
is considered. Under suitable assumptions, we show the exponential stability of the system. Our method is based on an identity 
with multipliers that allows to show an appropriate stability estimate.

1. Introduction. In this paper, we shall be concerned with the problem of exponential stabilization of a coupled Maxwell/wave system with Wentzell conditions by linear boundary feedbacks.
The Wentzell condition on Γ in (1) arises when modelling vibrating bodies with a thin boundary layer of high rigidity. Such boundary conditions are characterized by the presence of tangential differential operators of the same order as the interior operator and appear in several fields of applications such as physics, in diffusion processes [16,22], in mechanics [4,15], as well as in wave phenomena [6].
Here Ω is a bounded domain of R 3 with a boundary Γ = ∂Ω of class C 2 . In the above system we denote by u(x, t) ∈ R 3 the displacement vector, E(x, t), H(x, t) ∈ R 3 represent respectively the electric and magnetic field at x = (x 1 , x 2 , x 3 ) ∈ Ω and t is the time variable, and µ denote the electric permittivity and magnetic permeability, respectively, and we will assume that they are positive real numbers. Also, in system (1) ξ is the coupling constant, a is a positive real number, and ∆ T 236 HICHEM KASRI AND AMAR HEMINNA means the tangential Laplace operator on Γ. As usual ν is the unit normal of Γ pointing toward the exterior of Ω.
For ξ = 0, problem (1) decouples into two independent mixed problems for the wave equation with Wentzell boundary conditions in Ω, (2) and for the Maxwell's system in Ω. ( There is a large literature devoted to the stabilization of hyperbolic problems set in bounded domains of R d , d ≥ 1. Let us mention some of them: Boundary stabilization of the system (2) with nonlinear boundary feedback has been considered in [6] (here a ∈ C 1 (Γ) is a nonnegative function), the author showed that the energy tends to zero as t → +∞ in a bounded smooth domain Ω. However, when a ≡ 0, A. Heminna [7,8] proved that this system is not uniformly exponentially stable. Later on, K. Laoubi and S. Nicaise [14] studied the same problem on the unit square of the plane Ω = (0, 1) 2 and obtained a polynomial rate of the decay of the energy. For Maxwell's equations with Silver-Müller boundary condition and = µ = 1, V. Komornik [11] showed, under suitable geometrical conditions that the energy tend to zero exponentially as time goes to infinity with a precise decay rate estimate. His proof is based on the "standard" identity with multiplier, assuming that Ω is strictly star-shaped with respect to the origin. Recently, when and µ are piecewise constant with Lipschitz polyhedra Ω, M. Eller et al. [2] gave a necessary and sufficient condition ensuring that the energy of the solutions of (3) decays exponentially. Their method of proof is based on the validity of a kind of stability estimate which is obtained using the multiplier method but leading to a relatively strong assumption on the permittivity and permeability coefficients. This result was later generalized by S. Nicaise [18] in what concerns the electromagneto-elastic system with Dirichlet-Neumann boundary conditions by nonlinear boundary feedbacks. He proved that the energy decays exponentially in the case of linear feedbacks if Ω is strictly star shaped with respect to the origin, and assuming that , µ are constants in the whole domain Ω. As mentioned before, A. Heminna [7,8] proved that the natural feedback is not sufficient to guarantee the exponential decay of the energy of the system (2) even if the dissipative feedback is applied to the entire boundary of the domain. A question thus arises of whether the energy of the solutions of the linear coupled system (1) decays exponentially. In other words: are there positive constants M , ω such that Our main interest in this article is exactly to give an answer to this question. Namely, we show that the exponential decay of the energy of the solutions of (1) is equivalent to the validity of a stability estimate, estimate that can be obtained in some particular cases using the identity with multiplier.
The approach adopted in this paper was inspired in [18], an approach which in turn originated in [2] adapted to our case.
The rest of this paper is organized as follows: Section 2 describes the problem and we introduces some notations and lemmas needed for our work, the main result is stated in Section 3, whereas Section 4 is devoted to the proof of the main result.
2. Statement of problem.
2.1. Notations and results. In this subsection we introduce some few formulas to be invoked in the sequel.
We define Ω, Γ, ν as above. For all x ∈ Γ, we denote by π(x) the orthogonal projection on tangent plane T x (Γ) and for a given vector field v : Ω → R 3 , we will write (see for instance [5,4,15]) We further denote by ∂ T (resp. ∂ ν ) the tangential (resp. normal) derivative, (∂ T ν) the curvature operator on Γ and π∂ T v T the covariant derivative of the field v T . If v is some regular function, the transposed vector of ∂ T v is the tangential gradient of v and is denoted by ∇ T v. We have The proof of the following lemmas can be found in [5], [6], [16] and references therein.
Lemma 2.1. Let f be a function of class C 2 defined on Γ; we have where the bar denotes the transposed of a vector.
Lemma 2.2. Let f be a function of class C 2 and q T a tangent field of class C 1 defined on Γ; then where i 2 is the identity of the tangent plane, and "tr" means the trace of a matrix.

2.2.
Well-posedness of (1). We briefly describe the function spaces where we will consider the solution (u, E, H) of problem (1). Here and in what follows we shall use the summation convention for repeated indices. Letting v = ∂ t u, we can rewrite system (1) as ∂ t U + AU = 0,
Theorem 2.4 (Well-posedness). The problem (1) is well posed in the space H. In particular: 1 Remark 2. It follows from Remark 1 that ∆ T u belongs to H − 1 2 (Γ) 3 and therefore the standard regularity results imply that u in H 2 (Ω) 3 , then we would obtain that the solution of (1) satisfying We now define the energy of the problem (1) by For every solution of (1) in the class (12) the following identity holds for all 0 ≤ S < T < ∞, and therefore the energy is a non-increasing function of the time variable t.
3. Main result. In this section we give the main result of this paper. We first recall the arguments of the beginning of Section 3 of [18] given in the case of the electromagneto-elastic system (see also Section 3 of [2]) and that can be easily extended to our system. Namely, the exponential stability of system (1) when Ω is strictly star-shaped with respect to the origin, i.e., Let us now introduce the following definition (see for instance [18]) Definition 3.1. We say that Ω satisfies the stability estimate if there exist T > 0 and two non negative constants C 1 , C 2 (which may depend on T ) with C 1 < T such that for all solution (u, E, H) of (1).

HICHEM KASRI AND AMAR HEMINNA
That property admits the following equivalent formulation for all solutions (u, E, H) of (1).
Proof. It's analogous to the proof of Lemma 3.2 in [2]. Now, we recall from [2] a necessary and sufficient condition for the exponential stability.
Proof. The proof is similar to the one in Theorem 3.3 of [2].
Applying an argument of [12], we can prove the following lemma.
Lemma 3.4. Let (u, E, H) be a strong solution of (1). Then, there exists C > 0 (depends on α, the domain, the coefficient µ and the parameter ξ) such that for all η ∈ (0, 1) Proof. We define z, depending on t, as follows This solution is characterized by z = w +u where w ∈ H 1 0 (Ω) 3 is the unique solution of This identity means that Taking v = z − u in this identity, we deduce that Using the definition of the energy function and the identity (15), we can find some positive constants c 1 , c 2 such that where c 2 = 2c1 a .

EXPONENTIAL STABILITY OF A COUPLED SYSTEM 241
For 0 < T < ∞, we set Q T := Ω × [0, T ] and Σ T := Γ × [0, T ]. Multiplying the first identity of (1) by z, invoke Green's formula and taking into account the boundary conditions in (1) and in (20), we arrive at Using (21), we get Using the third equation in (1) and integrating by parts in t, we obtain Using several times (15), (22), (23) and Young's inequality ab ≤ αa 2 + b 2 4α for all α > 0 and all real numbers a, b we can estimate the different integrals of the right-hand side of the above inequality as follows for some positive constant c 3 (depending on the domain, the µ and the parameter ξ). Using these different estimates, we arrive at the requested estimate by taking α = η 2 a + 1 + ξ 2 µ and C = 1 + c 1 4 + 2c 3 α 2 a + 1 + ξ 2 µ .
We now state our main results of this paper.
Theorem 3.5. Assume that Ω is a bounded domain of R 3 strictly star-shaped with respect to the origin and having smooth boundary Γ of class C 2 . Then Ω satisfies the stability estimate. Proof of Theorem 3.5. We note that it is sufficient to prove that the estimate (17) holds for any strong solution (u, E, H) of (1). The main tool we use is the multiplier method [12,17]. In order to prove (17) we proceed in two steps.
Step 1. We begin with the following identity Let Ω be a bounded domain of R 3 having a bounded Γ of class C 2 and q = (q 1 , q 2 , q 3 ) ∈ W 1,∞ (Ω) 3 . Then, for every strong solution (u, E, H) of (1) we have the following identity where Q = divq T − 2∇q and I is a 3 × 3 matrix. Here we have used the notation q : ∇u = q·∇u 1 , q·∇u 2 , q·∇u 3 and q T : where q T is the tangential component of q and div T q T is the tangential divergence of the field q T .
Proof. Multiply the first equation in (1) by 2 q : ∇u and integrating over Q T and Σ T by parts, we obtain On Γ, we have and In that case (26) becomes Using the second boundary condition in (1) and integrating by parts on Σ T , we get From Lemma 2.2, we deduce that Now, we observe that
Lemma 4.2. We have the following identity Proof. We multiply the equation (1) 1 and (1) 3 by 2u and 2ξu, respectively, and integrating by parts on Q T and Σ T , we obtain (31).
Combining (30) and (31) we obtain the following Lemma Lemma 4.3. The following identity holds It therefore follows from (32) that

HICHEM KASRI AND AMAR HEMINNA
where we have set where m ν = m · ν and m T = m − m ν ν.
Step 2. It remains to estimate each term I i (i = 1, .., 4). From Lemma 3.2 of [12] (see also [1]) and the identity (15), there exist two positive constants k 1 and k 2 such that As in [12,2], we can show that there exists a constant k 3 > 0 such that where k 4 is a positive constant independent of T . By Lemma 3.4, we find that where R = sup x∈Ω |m(x)|.
Then the term I 3 becomes where we recall that E ν = E · ν and E T = E − E ν ν. By Young's inequality, there exists k 5 > 0 such that for all θ > 0 Using the first and third equations in Lemma 2.3, we may rewrite I 4 as Next, we are going to estimate each term of the right-hand side of (37). From now on we will denote by C various positive constants which may be different at different occurrences and is independent of u and large enough. Since m ν and ∂ T ν are bounded (see e.g. [1]), we obtain Since u · (m T : ∇ T u) = 1 2 ∇ T (|u| 2 ) · m T , the last term of I 4 is equivalent to We set By Green's formula, we obtain and therefore The terms I 2 and I 4 being estimated with the help of Young's inequality, we get for all θ > 0. We readily check that Combining these estimates, we find that We now estimate the term Σ T m · ν|∂ ν u| 2 dσdt. We then have Since Γ is of class C 2 , then there exists a vector field h ∈ (W 1,∞ (Ω) 3 ) such that h = ν on Γ.