Indirect boundary stabilization for weakly coupled degenerate wave equations under fractional damping

In this paper, we consider the well-posedness and stability of a one-dimensional system of degenerate wave equations coupled via zero order terms with one boundary fractional damping acting on one end only. We prove optimal polynomial energy decay rate of order $1/t^{(3-\tau)}$. The method is based on the frequency domain approach combined with multiplier technique.

Degenerate equations are studied by posing two closely connected problems: 1) a demonstration of the solvability of, say, boundary value problems taking into account changes in their formulation which are a consequence of the degeneration of type; and 2) a determination of properties of the solutions which are analogous to those of non-degenerate equations (smoothness, Harnack inequalities for elliptic and parabolic equations, etc.).
We review the related papers, regarding linear degenerate wave system, from a qualitative and quantitative study.For a single degenerate wave equation, we beginning with the work treated in [3], for (x, t) ∈ (0, 1) × (0, +∞) where the goal was mainely on the equation u tt (x, t) − (a(x)u x (x, t)) x = 0 in (0, 1) × (0, ∞), together with boundary linear damping of the form where β > 0 is the given constant.m a = sup 0<x≤1 x|a ′ (x)| a(x) < 2 is the measurement of the degree of the degeneracy.Thanks to the energy multiplier method, it is proved that the total energy of the whole system decays exponentially.
Next, in a recent paper of Liu and Rao [15] general systems of coupled second order evolution equations have been studied.The system is described where Ω ⊂ IR n is a bounded domain with smooth boundary Γ of class C 2 such that Γ = Γ D ∪ Γ N and Γ D ∩ Γ N = ∅.They established, by the frequency domain approach, polynomial decay rate of order ln t t for smooth initial data, while waves propagate with equal speeds.Moreover, while waves propagate with different speeds, i.e. the case b = 1, they proved that the energy decays at a rate which depends on the arithmetic property of the ratio of the wave speeds b.
Very recently, Wehbe and Koumaiha [12] considered a one-dimensional setting of a system of wave equation coupled via zero order terms.More precisely, they studied the stabilization of the following system of partially damped coupled wave equations propagating with equal speeds, described by where γ > 0. They proved optimal polynomial energy decay rate of order 1 t , by using a frequency domain approach and Riesz basis property of the generalized eigenvector of the system.
In [2], Akil et al considered a one-dimensional coupled wave equations on its indirect boundary stabilization defined by on (0, 1).
They established a polynomial energy decay rate of type t −s(τ ) , such that In [11], kerdache et al investigate the decay rate of the energy of the coupled wave equations with a two boundary fractional dampings, that is, on (0, 1).
Using semigroup theory, they proved an optimal polynomial type decay rate.Motivated by the works [15], [5] and [12] we wonder what the asymptotic behavior of the coupled degenerate wave equations would be, considering a boundary fractional damping acting only on one equation.
This paper is divided into four sections.In section 2, we introduce the appropriate functional spaces that are naturally associated with degenerate problems and preliminary result used throughout the paper.Section 3 is devoted to the proof of the well-posedness and strong asymptotic of the considered system.In Section 4 we establish an optimal polynomial decay of type t − 2 3−τ for smooth initial data, by the frequency domain method.

Preliminary results
Let a ∈ C([0, 1] ∩ C 1 (]0, 1]) be a function satisfying the following assumptions: where [•] stands for the integer part.When m a > 1, we suppose β > 0 because if β = 0 and the feedback law only depends on velocities, we may encounter the situation where the closed-loop system is not well-posed in terms of the semigroups in the Hilbert space.
In order to express the boundary conditions of the first component of the solution of (1) in the functional setting, we define the spaces H 1 0,a (0, 1) and W 1 a (0, 1) depending on the value of m a , as follows: (i) For 0 ≤ m a < 1, we define (ii) For 1 ≤ m a < 2, we define It is easy to see that H 1 a (0, 1) when β > 0 is a Hilbert space with the scalar product Let us also set ∀u ∈ H 1 a (0, 1).
3 Well-posedness and strong stability

Augmented model
In this section we reformulate (P ) into an augmented system.For that, we need the following proposition.
Proposition 3.1 (see [11]) Let ϑ be the function: Then the relationship between the 'input' U and the 'output' O of the system where U ∈ C 0 ([0, +∞)), is given by where Lemma 3.1 (see [11]) We are now in a position to reformulate system (P ).Indeed, by using Proposition 3.1, system (P ) may be recast into the augmented model: We define the energy associated to the solution of the problem (P ′ ) by the following formula: Lemma 3.2 Let (u, v, ϕ) be a regular solution of the problem (P ′ ).Then, the energy functional defined by (12) satisfies In this section, we give an existence and uniqueness result for problem (P ′ ) using the semigroup theory.Introducing the vector function U = (u, ũ, v, ṽ, ϕ) T , where ũ = u t , ṽ = v t , system (P ′ ) can be treated as a Cauchy evolution problem where Θ 0 = (u 0 , u 1 , v 0 , v 1 , ϕ 0 ) T and is the operator given by We introduce the following phase space (the energy space): that is a Hilbert space with the following inner product We have the following existence and uniqueness result.

Theorem 3.1 (Existence and uniqueness)
(1) If U 0 ∈ D(A), then system (14) has a unique strong solution with the following regularity, (2) If U 0 ∈ H, then system ( 14) has a unique weak solution such that

Strong stability of the system
In this part, we use a general criteria of Theorem 3.2 to show the strong stability of the C 0semigroup e tA associated to the wave system (P ) in the absence of the compactness of the resolvent of A.
To state and prove our stability results, we need some results from semigroup theory.

Theorem 3.2 ([4])
Let A be the generator of a uniformly bounded C 0 -semigroup {S(t)} t≥0 on a Hilbert space X .If: (i) A does not have eigenvalues on iIR.
(ii) The intersection of the spectrum σ(A) with iIR is at most a countable set, then the semigroup {S(t)} t≥0 is asymptotically stable, i.e, S(t)z X → 0 as t → ∞ for any z ∈ X .
Our main result is the following theorem: The C 0 -semigroup e tA is strongly stable in H; i.e, for all U 0 ∈ H, the solution of ( 14) satisfies lim t→∞ e tA U 0 H = 0.
For the proof of Theorem 3.3, we need the following two lemmas.
Lemma 3.3 A does not have eigenvalues on iIR. Proof.
We will argue by contraction.Let U ∈ D(A) and let λ ∈ IR, such that Then, we get •Case 1: If λ = 0, then, from (18) we have From (30) 3 , we have ũ(1) = 0. (32) Hence, from (30) 1 we obtain u(1) = 0 and u x (1) = 0. (33) Eliminating ũ and ṽ in equations (30) 1 and (30) 3 in equations (30) 2 and (30) 4 , we obtain the following system On the other hand, multiplying (34) 1 by v, (34) 2 by u and using the boundary condition (34) 3 , we get Multiplying equation (34) 1 by u, using Green formula, (33) and the boundary conditions, we get Multiplying equation (34) 1 by xu x , we get U ∈ D(A), then the regularity is sufficiently for applying an integration on the second integral in the left hand side in equation (37).Then we obtain Using Green formula, Proposition 2.2-(ii) and the boundary conditions, we get Multiplying equations (36) by −m a /2, and tacking the sum of this equation and (39), we get By definition of m a , we have Then using the Cauchy-Schwartz and Poincaré's inequalities, we deduce from (40) and ( 35) that there exists a positive constant C > 0, which yields u=0 for α small enough.It then follows from (35) that v = 0, and from (30) 1 and (30) 3 that ũ = ṽ = 0. Consequently, we obtain U = 0, which contradict the hypothesis U = 0.The proof has been completed.
•Case 1: λ = 0. We will prove that the operator iλI − A is surjective for λ = 0.For this purpose, let G = (g 1 , g 2 , g 3 , g 4 , g 5 ) T ∈ H, we seek X = (u, ũ, v, ṽ, ϕ) T ∈ D(A) solution of the following equation Equivalently, we have (45) Inserting (45) 1 and (45) 3 into (45) 2 and (45) 4 , we get for all w ∈ W 1 a (0, 1) and y ∈ H 1 0,a (0, 1).Then, we get We can rewrite (48) as Let (W 1 a × H 1 0,a (0, 1)) ′ be the dual space of W 1 a × H 1 0,a (0, 1).Let us define the following operators ( * * ) We need to prove that the operator B is an isomorphism.For this aim, we divide the proof into three steps: Step 1.In this step, we want to prove that the operator B 1 is an isomorphism.For this aim, it is easy to see that B 1 is sesquilinear, continuous form on W 1 a × H 1 0,a (0, 1).Furthermore ), where we have used the fact that Thus B 1 is coercive.Then, from ( * * ) and Lax-Milgram theorem, the operator B 1 is an isomorphism.
Step 2. In this step, we want to prove that the operator B 2 is compact.For this aim, from ( * ) and ( * * * ), we have and consequently, using the compact embedding from W 1 a × H 1 0,a (0, 1) to L 2 (0, 1) × L 2 (0, 1) we deduce that B 2 is a compact operator.Therefore, from the above steps, we obtain that the operator B = B 1 + B 2 is a Fredholm operator of index zero.Now, following Fredholm alternative, we still need to prove that the operator B is injective to obtain that the operator B is an isomorphism.
Taking account of Lemmas 3.3, 3.4 and from Theorem 3.2 the C 0 -semigroup e tA is strongly stable in H. ✷ 3.3 Optimal condition for strong stability of the system in the case a(x) = x γ Theorem 3.4 The C 0 -semigroup e tA is strongly stable in H if and only if the coefficient α satisfies where ν γ = |1 − γ|/(2 − γ) and j ν,1 < j ν,2 < . . .< j ν,k < . . .denote the sequence of positive zeros of the Bessel function of first kind and of order ν.
The solution of the equation ( 55) is given by togheter with the boundary conditions where where J ν and J −ν are Bessel functions of the first kind of order ν and −ν.

Lack of exponential stability
This section will be devoted to the study of the lack of exponential decay of solutions associated with the system (P ′ ).

Proposition 3.2
The C 0 -semigroup of contractions S(t) = e At associated with ( 14) is not exponentially stable.
Proof.Let µ n be an eigenvalue of Ku = −(au x ) x in H 1 0,a (0, 1) corresponding to the normalized eigenfunction e n , and Then a straightforward computation gives This shows that the resolvent of A is not uniformly bounded on the imaginary axis.Following [18] and [10], the system (P ′ ) is not uniformly and exponentially stable in the energy space H.
Precise spectral analysis in the case a(x) = x γ .We aim to show that an infinite number of eigenvalues of approach the imaginary axis which prevents the system (P ) from being exponentially stable.Indeed we first compute the characteristic equation that gives the eigenvalues of A. Let λ be an eigenvalue of A with associated eigenvector U = (u, v, ϕ) T .We consider only the case γ with boundary conditions Inserting (59) 1 into (59) 2 and (59) 3 into (59) 4 , we get The solution of equations ( 63) is given by where Φ + , Φ − , Φ ++ and Φ −− are defined by , where Then, using the series expansion of J να and J −να , one obtains , where we have used the following relation where System (61) admits a non trivial solution if and only if det(M) = 0. i.e., if and only if the eigenvalues of A are roots of the function f defined by Our purpose is to prove, thanks to Rouché's Theorem, that there is a subsequence of eigenvalues for which their real part tends to 0.
In the sequel, since A is dissipative, we study the asymptotic behavior of the large eigenvalues λ of A in the strip S = {λ ∈ I C : −α 0 ≤ ℜ(λ) ≤ 0}, for some α 0 > 0 large enough and for such λ, we remark that Φ + , Φ − remain bounded.Lemma 3.5 The large eigenvalues of the dissipative operator A are simple and can be split into two families (λ j,k ) k∈Z,|k|≥N , j = 1, 2, (N ∈ IN, chosen large enough).Moreover, the following asymptotic expansions for the eigenvalues hold: where where Moreover for all |k| ≥ N, the eigenvalues λ j,k are simple. Proof.
Step 2. We look at the roots of f 0 .From (86), f 0 has two families of roots that we denote λ 0 Using Rouché's Theorem, we deduce that f admits an infinity of simple roots in S denoted by λ 1,k and λ 2,k for |k| ≥ k 0 , for k 0 large enough, such that Step 3. Asymptotic behavior of ε 1,k .Using (92), we get Substituting (94) into (85), using that f (λ 1,k ) = 0, we get The previous equation has one solution Step 4. Asymptotic behavior of ε 2,k .Using (93), we get sinh( * Substituting ( 97) into (85), using that f (λ 2,k ) = 0, we get The previous equation has one solution We can write where ε2,k = o(1/k).Substituting (100) into (85), we get The previous equation gives Now, setting Ũk = (λ 0 j,k − A)U k , where U k is a normalized eigenfunction associated to λ j,k .We then have Hence, by Lemma 3.5, we deduce that So that, the semigroup e tA is not exponentially stable.Thus the proof is complete.✷ 4 Polynomial Stability (for ω = 0) To prove polynomial decay, we use the following theorem.
Theorem 4.1 ([6]) Assume that A is the generator of a strongly continuous semigroup of contractions (e tA ) t≥0 on a Hilbert space X .If iIR ⊂ ̺(A).Then for a fixed l > 0 the following conditions are equivalent 2) Theorem 4.2 The semigroup S A (t) t≥0 associated with system (P ′ ) is polynomially stable, i.e., there exists a constant C > 0 such that

Proof
In section 3, we have proved that the first condition in Theorem 4.1 is satisfied.Now,we need to show that sup where l = 3 − τ .We establish (103) by contradiction.So, if (103) is false, then there exist sequences (λ n ) n ⊂ IR and U n = (u n , ũn , v n , ṽn , ϕ n ) ∈ D(A) satisfying For simplification, we set g1 = g where R = 2ℜ Since iλv x = ṽx + g3x taking the real part in the above equality, we get Performing an integration by parts we obtain where Multiplying (107) 4 by v and integrating over (0, 1) and using integration by parts we get Performing an integration by parts we get Multiplying (107) 2 by u and integrating over (0, 1) and using integration by parts we get This, together with (132), gives Moreover the decay rate is optimal.In fact for the case a(x) = x γ , γ ∈ [0, 2[, the decay rate is consistent with the asymptotic expansion of eigenvalues which shows a behavior of the real part like k − (3−τ ) .✷