Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback

We study the stabilization problem for the wave equation with localized Kelvin--Voigt damping and mixed boundary condition with time delay. By using a frequency domain approach we show that, under an appropriate condition between the internal damping and the boundary feedback, an exponential stability result holds. In this sense, this extends the result of [19] 
where, in a more general setting, the case of distributed structural damping is considered.


SERGE NICAISE AND CRISTINA PIGNOTTI
We are interested in giving an exponential stability result for such a problem under a suitable relation between the function a(x) and the constant k.
We will show that, under some geometrical assumptions described below and the condition a 0 > |k|C P , (1.6) where C P is a sort of Poincaré constant, the energy of the solutions of system (1.1)-(1.5) satisfies a uniform exponential decay estimate.
Time delay effects are often present in applications and physical models and it is well-known that a delay arbitrarily small may induce instability phenomena in several evolution problems, which are uniformly stable in absence of delay. In particular, this is the case of wave type equations (see e.g. [5,6]). Therefore, it is important to look for feedback laws which are robust with respect to (small) delays. We refer to [18,21] for stability results for wave equations with delay in the case of frictional damping or standard boundary dissipative damping.
The problem (1.1)-(1.5), in the particular case of a distributed damping, i.e. a(x) = a 0 a.e. x ∈ Ω, with a 0 > 0, has been investigated in [19] (see also Morgul [17] who proposed a class of dynamic boundary controllers in the 1-d case). More precisely, there we consider abstract second-order evolution equations with (dynamic) boundary feedback laws with delay and distributed structural damping and prove an exponential stability result under a suitable condition between the internal damping and the boundary conditions. The problem at hand enters in that abstract framework in the case of a(x) constant. We refer also to [1] and [2] for the analysis of related problems for wave equations with distributed Kelvin-Voigt damping and time delay effects.
The proof in [19] relies mainly on multipliers arguments allowing to obtain appropriate estimates for suitable Lyapunov functionals. The case of a local structural damping is more difficult to deal with, due to the unboundedness of the damping. Then, the analysis of [19] cannot be performed here. Our stability result will be now obtained by using a frequency domain approach introduced by Huang [8] and Prüss [20].
First of all we still consider the problem without time delay and extend the analysis of Liu and Rao [16] to the case of a mixed boundary condition, Dirichlet on the part Γ 0 of the boundary and dynamic boundary condition on Γ 1 . Then, we use the stability result for the undelayed problem to obtain the exponential stability of the problem with delay, under the condition (1.6). Note that, as suggested from some counterexamples of [19], such a condition seems to be optimal in order to have stability.
The paper is organized as follows. In sect. 2 we give a well-posedness result by using semigroup theory; in sect. 3 we prove the stability for the delay problem by using the stability of the undelayed one which will be proved in sect. 4.

2.
Well-posedness of the problem. In this section we will give well-posedness results for problem (1.1)-(1.5) using semigroup theory.

SERGE NICAISE AND CRISTINA PIGNOTTI
Denote by H the Hilbert space Assuming that (compare with (1.6)) |k| ≤ a 0 C P , (2.13) we will show that A generates a C 0 semigroup on H.
Let ξ be a positive real number such that (2.14) Note that, from (2.13), such a constant ξ exists. Let us define on the Hilbert space H the inner product Proof. Take U = (u, v, z) ∈ D(A). Then So, by Green's formula and using the definition of D(A) we get (2.16) Using the Cauchy-Schwarz and the Young inequalities we find |z(x, 1)| 2 dΓ. (2.17) Using the definition (1.7) of C P , we deduce that (2.18) Now, observing that from (2.14), we obtain AU, U H ≤ 0, which means that the operator A is dissipative. Now, we will show that A is surjective. (2.20) The first equation of (2.20) gives v. From the third equation we deduce and, in particular, The second equation of (2.20) can be reformulated as Then, integrating by parts, and so, from (2.5), Therefore, the Lax-Milgram Theorem ensures the existence of a unique solution u ∈ H 1 Γ0 (Ω) of (2.24). If we consider w ∈ D(Ω), then from (2.24) we deduce ∇u + a∇v = ∇u + a∇f ∈ H(div , Ω) .
So, we have found a triple (u, v, z) ∈ D(A) solution of (2.20).
From (2.20), (2.23) and (2.24) we easily deduce Then, 0 ∈ ρ(A). Therefore, by the contraction principle we obtain R(λI − A) = H for λ > 0 sufficiently small. Thus, applying the Lumer-Phillips Theorem we conclude that the operator A generates a C 0 semigroup of contractions on H.
Let us introduce the energy of the system which is the standard energy for wave equation plus an integral term due to the presence of a time delay, where ξ > 0 is the parameter fixed above.

Proposition 2.2. For any regular solution
the energy is decreasing and there exists a positive constant C such that Proof. It suffices to notice that We then conclude owing to (2.18) and the assumption on ξ.

Stability of the delay problem.
In this section, we will prove an exponential stability result for problem (1.1)-(1.5) under the assumption (1.6) and some geometrical assumptions described below.
Our stability result is based on a frequency domain approach, namely the exponential decay of the energy is obtained by using the following result (see [20] or [8]): for some positive constants C and ω if and only if where ρ(A) denotes the resolvent set of the operator A.
From this result, we are reduced to check that the imaginary axis is included in the resolvent set (condition (3.1)) and to analyze the behaviour of the resolvent on the imaginary axis.
Then for all β ∈ IR, one has Proof. Since we have already shown in Theorem 2.1 that The first identity of (3 .17) gives Now, observe that, from the third identity in (3.17), and then Let us now introduce the auxiliary variablẽ Now, observe that and so −div (∇u + a∇v) = −∆ũ + div (v∇a) .
Then, the second identity of (3.17), can be rewritten as iβv − ∆ũ + div (v∇a) = g . and then, after integration by parts, Now observe that, from the boundary condition on Γ 1 , Hence by (3.20), we get as (3.26) Then, Let us denote V := H 1 Γ0 (Ω) and let V its dual space. We introduce the operator with (A τ,βũ )(w) = a τ,β (ũ, w), ∀ w ∈ V . Then, we rewrite identity (3.28) as To conclude the proof of Proposition 3.3 we need some preliminary results.
equipped with the inner product and let A 0 be the operator corresponding to τ = 0 and k = 1, that is We will prove in the next section that the above assumptions imply that A 0 generates an exponentially stable semigroup. Then, from [8], we know that iIR ⊂ ρ(A 0 ) and that there exists C > 0 such that From this estimate, we can deduce the estimate (3.42) below that will be useful in Proposition 3.10. Indeed, from (3.37), for every F 0 ∈ H 0 , the solution (u * , v * ) ∈ D(A 0 ) of which is equivalent to Moreover, Then, Proof. For F ∈ H and β ∈ IR , let U ∈ D(A) be a solution of (3.44) The first identity of (3.44) gives Now, observe that, from the identity (3.20), Then,  (Γ1×(0,1)) )) , and therefore , from which follows, by using Young's inequality, We have now to estimate (u, v) H0 . For this, let (u * , v * ) ∈ D(A 0 ) be the solution of which is equivalent to (3.53) We have then Thus, using the boundary condition on Γ 1 , Then, from (3.53), we have and so from which follows, by using Cauchy-Schwarz inequality, Using these last inequalities in (3.54), we obtain By recalling (3.50) and (3.51) we have then , and so, using Young's inequality, (3.55) and then, from (3.49), This proves that the resolvent of A is uniformly bounded on the imaginary axis.
The above resolvent analysis and Lemma 3.1 allow to state our stability result.

Stability of the undelayed problem.
As in section 2, we can prove that A 0 generates a C 0 -semigroup of contractions in H 0 . Hence in order to prove that this semigroup is exponentially decaying, we again use Lemma 3.1.
As before we start with condition (3.1).
Then from (4.10) we find thatũ satisfies (3.31) and (3.32) with τ = 0. Thus, if we set u =ũ 1 + iβa and v = iβ 1 + iβaũ , we find that (u, v) ∈ D(A 0 ) and that This implies u = v = 0, thenũ = 0.  Taking the inner product of (4.17) with u n in V and of (4.18) with v n in H, we obtain Summing the above identities and taking the real part, we deduce Now we introduce a cut-off function η ∈ C 1 (Ω) such that 1] elsewhere .
In summary we have obtained the next stability result.  6) is satisfied, then A 0 generates a C 0 -semigroup of contraction (T 0 (t)) t≥0 that is exponentially stable, namely there exist two constants M 0 > 0 and ω 0 > 0, such that for all U 0 ∈ H 0 we have T 0 (t)U 0 H0 ≤ M 0 e −ω0t U 0 H0 , ∀t ≥ 0 .