c 2006 Society for Industrial and Applied Mathematics

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1 SIAM J. CONTROL OPTIM. Vol. 45, No. 5, pp c 6 Society for Industrial and Applied Mathematics STABILITY AND INSTABILITY RESULTS OF THE WAVE EQUATION WITH A DELAY TERM IN THE BOUNDARY OR INTERNAL FEEDBACKS SERGE NICAISE AND CRISTINA PIGNOTTI Abstract. In this paper we consider, in a bounded and smooth domain, the wave equation with a delay term in the boundary condition. We also consider the wave equation with a delayed velocity term and mixed Dirichlet Neumann boundary condition. In both cases, under suitable assumptions, we prove exponential stability of the solution. These results are obtained by introducing suitable energies and by using some observability inequalities. If one of the above assumptions is not satisfied, some instability results are also given by constructing some sequences of delays for which the energy of some solutions does not tend to zero. Key words. wave equation, delay feedbacks, stabilization AMS subject classifications. 35L5, 93D15 DOI / Introduction. We investigate the effect of time delay in boundary or internal stabilization of the wave equation in domains of R n. Such effects arise in many practical problems, and it is well known, at least in one dimension, that they can induce some instabilities; see [4, 5, 6, 17]. To our knowledge, analysis in higher dimensions has not yet been done. In this paper, we give some stability results under a sufficient condition, and we further show that if this condition is not satisfied, then there exist some delays for which the system is destabilized. So, in a certain sense, our sufficient condition is also necessary in order to have a general stability result. Let R n be an open bounded set with a boundary Γ of class C. We assume that Γ is divided into two parts Γ D and, i.e., Γ = Γ D, with Γ D = and Γ D. In this domain, we consider the initial boundary value problem (1.1) (1.) (1.3) (1.4) (1.5) u tt (x, t) Δu(x, t) = in (, + ), u(x, t) = on Γ D (, + ), ν (x, t) = μ 1u t (x, t) μ u t (x, t τ) on (, + ), u(x, ) = u (x) and u t (x, ) = u 1 (x) in, u t (x, t τ) =f (x, t τ) in (,τ), where ν(x) denotes the outer unit normal vector to the point x Γ and ν is the normal derivative. Moreover, τ > is the time delay, μ 1 and μ are positive real numbers, and the initial data (u,u 1,f ) belong to a suitable space. We are interested in giving an exponential stability result for such a problem. Received by the editors January 3, 6; accepted for publication (in revised form) May 4, 6; published electronically November 14, 6. Université de Valenciennes et du Hainaut Cambrésis, MACS, Institut des Sciences et Techniques de Valenciennes, Valenciennes Cedex 9, France (snicaise@univ-valenciennes.fr). Dipartimento di Matematica Pura e Applicata, Università di L Aquila, Via Vetoio, Loc. Coppito, 671 L Aquila, Italy (pignotti@univaq.it). 1561

2 156 SERGE NICAISE AND CRISTINA PIGNOTTI Let us denote by v, w or, equivalently, by v w the Euclidean inner product between two vectors v, w R n. We assume that there exists a scalar function v C () such that (i) v is strictly convex in ; that is, there exists α> such that (1.6) D (v)(x)ξ,ξ α ξ x, ξ R n, where D (v) denotes the Hessian matrix of v; (ii) the vector field H := v verifies (1.7) H(x) ν(x) x Γ D. For the above assumptions see [14], where some observability estimates for secondorder hyperbolic equations are given. It is well known that if μ =, that is, in absence of delay, the energy of problem (1.1) (1.5) is exponentially decaying to zero. See for instance Chen [, 3], Lagnese [11, 1], Lasiecka and Triggiani [13], Komornik and Zuazua [1], and Komornik [8, 9]. On the contrary, if μ 1 =, that is, if we have only the delay part in the boundary condition on, system (1.1) (1.5) becomes unstable. See, for instance Datko, Lagnese, and Polis [6]. Although these examples involve only one space dimension, we can expect a similar phenomenon to occur in higher space dimensions. So, it is interesting to seek a stabilization result when both μ 1 and μ are nonzero. In this case, the boundary feedback is composed of two parts and only one of them has a delay. This problem has been studied in one space dimension by Xu, Yung, and Li [17]. After a spectral analysis these authors proved a stability result for the case when μ <μ 1. In their paper it is also shown that if μ >μ 1, the system is unstable and if μ 1 = μ, some instabilities may occur. Here, in agreement with [17] and assuming that (1.8) μ <μ 1, we obtain a stabilization result in a general space dimension by using a suitable observability estimate. This is done by applying inequalities obtained from Carleman estimates for the wave equation by Lasiecka, Triggiani, and Yao in [14] and by using compactness-uniqueness arguments. If μ 1 = μ, we further show that there exists a sequence of arbitrary small (and large) delays such that instabilities occur. In the case μ >μ 1, we also obtain delays which destabilize the system. More precisely, we show the next results. Under assumption (1.8) let us define the energy of a solution of problem (1.1) (1.5) as (1.9) E(t) := 1 {u t (x, t)+ u(x, t) }dx + ξ 1 u t (x, t τρ)dρdγ, where ξ is a positive constant verifying (1.1) τμ <ξ<τ(μ 1 μ ). Clearly this energy is larger than the standard energy 1 {u t (x, t)+ u(x, t) }dx and contains an additional term that comes from the delay term.

3 STABILIZATION OF THE WAVE EQUATION WITH DELAY 1563 Theorem 1.1. Assume that (1.8) holds. There exist positive constants C 1,C such that, for any solution of problem (1.1) (1.5), (1.11) E(t) C 1 E()e Ct t. Theorem 1.. If (1.8) does not hold, there exist a sequence of delays, and solutions of problem (1.1) (1.5) corresponding to these delays, such that their standard energy is constant. In this paper we also study the problem for the wave equation with internal feedback. In particular, we consider the system (1.1) (1.13) (1.14) (1.15) (1.16) u tt (x, t) Δu(x, t)+a(x)[μ 1 u t (x, t)+μ u t (x, t τ)] = in (, + ), u(x, t) = on Γ D (, + ), ν (x, t) = on (, + ), u(x, ) = u (x) and u t (x, ) = u 1 (x) in, u t (x, t τ) =g (x, t τ) in (,τ), where a L () is a function such that (1.17) a(x) a. e. in, and (1.18) a(x) >a > a. e. in ω, where ω is an open neighborhood of. Exponential stability results for the above problem in the case of μ =, that is, without delay, have been obtained by several authors. See for instance Zuazua [18] and Liu [16]. On the contrary, at least for the one-dimensional case, Datko [4] has shown that the wave equation with a velocity term and mixed Dirichlet Neumann boundary condition is destabilized by a time delay in the velocity term. In this paper, in the case μ <μ 1, we show that the energy is exponentially decaying to zero. This is done, as for the problem with boundary feedback, by using a suitable observability estimate. If μ μ 1, we obtain an explicit sequence of arbitrary small delays that destabilize the system. As before, under assumption (1.8) let us define the energy of a solution of (1.1) (1.16) as (1.19) F(t) := 1 {u t (x, t)+ u(x, t) }dx + ξ 1 a(x) u t (x, t τρ)dρdx, where ξ is a positive constant verifying (1.1). Again F is larger than the standard energy and contains an extra term due to the delay. Theorem 1.3. Let assumption (1.8) be satisfied. Then there exist positive constants C 1,C such that, for any solution of problem (1.1) (1.16), (1.) F(t) C 1 F()e Ct t. Theorem 1.4. If (1.8) does not hold, there exist a sequence of arbitrary small (or large) delays, and solutions of problem (1.1) (1.16) corresponding to these delays, such that their standard energy does not tend to.

4 1564 SERGE NICAISE AND CRISTINA PIGNOTTI Remark 1.5. In [14], in order to deal with variable coefficients, the authors assume that there exists a scalar function v strictly convex with respect to the Riemannian metric induced by the spatial operator. Here, we are principally interested in the effect of the delay term in the boundary or internal feedback. So, in order to avoid technicalities, we consider constant coefficients. Actually, our stability results hold even for variable coefficients under the assumption of [14]. The paper is organized as follows. Well-posedness of the problems is analyzed in section using semigroup theory. In subsection.1 we study the well-posedness of problem (1.1) (1.5), while in subsection. we concentrate on problem (1.1) (1.16). In sections 3 and 4 we prove the exponential stability of the problem with boundary and internal feedbacks, respectively. Finally, section 5 is devoted to some instability examples.. Well-posedness of the problems. In this section we will give well-posedness results for problem (1.1) (1.5) and problem (1.1) (1.16) using semigroup theory..1. Boundary feedback. Let us set (.1) z(x, ρ, t) =u t (x, t τρ), x,ρ (, 1), t>. Then, problem (1.1) (1.5) is equivalent to (.) (.3) (.4) (.5) (.6) (.7) (.8) u tt (x, t) Δu(x, t) = in (, + ), τz t (x, ρ, t)+z ρ (x, ρ, t) = in (, 1) (, + ), u(x, t) = on Γ D (, + ), ν (x, t) = μ 1u t (x, t) μ z(x, 1,t) on (, + ), z(x,,t)=u t (x, t) on (, ), u(x, ) = u (x) and u t (x, ) = u 1 (x) in, z(x, ρ, ) = f (x, ρτ) in (, 1). If we denote U := (u, u t,z) T, then U := (u t,u tt,z t ) T = ( u t, Δu, τ 1 z ρ ) T. Therefore, problem (.) (.8) can be rewritten as (.9) { U = AU, U()=(u,u 1,f (, τ)) T, where the operator A is defined by A u v z := v Δu τ 1 z ρ,

5 STABILIZATION OF THE WAVE EQUATION WITH DELAY 1565 with domain (.1) { D(A) := (u, v, z) T ( E(Δ,L ()) HΓ 1 D () ) H 1 () L ( ; H 1 (, 1)) : } ν = μ 1v μ z(, 1) on ; v = z(, ) on, where, as usual, and H 1 Γ D () = { u H 1 () : u = on Γ D }, E(Δ,L ()) = {u H 1 ():Δu L ()}. Recall that for a function u E(Δ,L ()), ν belongs to H 1/ ( ) and the next Green formula is valid (see section 1.5 of [7]) (.11) u wdx = Δuwdx + ν ; w w HΓ 1 D (), where ; ΓN means the duality pairing between H 1/ ( ) and H 1/ ( ). Note further that for (u, v, z) T D(A), ν belongs to L ( ) since z(, 1) is in L ( ). Denote by H the Hilbert space (.1) H := H 1 Γ D () L () L ( (, 1)). Assuming that (.13) μ μ 1, we will show that A generates a C semigroup on H. Let ξ be a positive real number such that (.14) τμ ξ τ(μ 1 μ ). Note that, from (.13), such a constant ξ exists. Let us define on the Hilbert space H the inner product (.15) u v z, ũ ṽ z H := { u(x) ũ(x)+v(x)ṽ(x)}dx+ξ 1 z(x, ρ) z(x, ρ)dρdγ. Theorem.1. For any initial datum U Hthere exists a unique solution U C([, + ), H) of problem (.9). Moreover, if U D(A), then U C([, + ), D(A)) C 1 ([, + ), H). Proof. Take U =(u, v, z) T D(A). Then = v u (AU, U) Δu, v τ 1 z ρ z H = { v(x) u(x)+v(x)δu(x)}dx ξτ 1 1 z ρ (x, ρ)z(x, ρ)dρdγ.

6 1566 SERGE NICAISE AND CRISTINA PIGNOTTI So, by Green s formula, (.16) (AU, U) = Integrating by parts in ρ, weget 1 z ρ (x, ρ)z(x, ρ)dρdγ = (x)v(x)dγ ξτ 1 ν 1 1 z ρ (x, ρ)z(x, ρ)dρdγ. z ρ (x, ρ)z(x, ρ)dρdγ+ {z (x, 1) z (x, )}dγ, that is (.17) 1 z ρ (x, ρ)z(x, ρ)dρdγ = 1 {z (x, 1) z (x, )}dγ. Therefore, from (.16) and (.17), ξτ 1 (AU, U) = (x)v(x)dγ {z (x, 1) z (x, )}dγ ν = (μ 1 v(x)+μ z(x, 1))v(x)dΓ ΓN ξτ 1 {z (x, 1) z (x, )}dγ = μ 1 v Γ (x)dγ μ z(x, 1)v(x)dΓ N ΓN ξτ 1 z ΓN (x, 1)dΓ+ ξτ 1 v (x)dγ, from which follows, using the Cauchy Schwarz inequality, (.18) ( (AU, U) μ 1 + μ ) ( ) + ξτ 1 v μ (x)dγ+ ξτ 1 z (x, 1)dΓ. Now, observe that from (.14), μ 1 + μ + ξτ 1, μ ξτ 1. Then, (AU, U), which means that the operator A is dissipative. Now, we will show that λi A is surjective for a fixed λ>. Given (f,g,h) T H, we seek a U =(u, v, z) T D(A) solution of (λi A) u v z = f g h, that is, verifying (.19) λu v = f, λv Δu = g, λz + τ 1 z ρ = h. Suppose that we have found u with the appropriate regularity. Then, (.) v := λu f

7 STABILIZATION OF THE WAVE EQUATION WITH DELAY 1567 and we can determine z. Indeed, by (.1), (.1) z(x, ) = v(x) for x, and, from (.19), (.) λz(x, ρ)+τ 1 z ρ (x, ρ) =h(x, ρ) for x,ρ (, 1). Then, by (.1) and (.), we obtain So, from (.), ρ z(x, ρ) =v(x)e λρτ + τe λρτ h(x, σ)e λστ dσ. ρ (.3) z(x, ρ) =λu(x)e λρτ f(x)e λρτ +τe λρτ h(x, σ)e λστ dσ on (, 1), and, in particular, (.4) z(x, 1) = λu(x)e λτ + z (x), x, with z L ( ) defined by 1 (.5) z (x) = f(x)e λτ + τe λτ h(x, σ)e λστ dσ, x. By (.) and (.19), the function u verifies that is, λ(λu f) Δu = g, (.6) λ u Δu = g + λf. Problem (.6) can be reformulated as (.7) (λ u Δu)wdx = (g + λf)wdx w HΓ 1 D (). Integrating by parts, (λ u Δu)wdx = (λ uw + u w)dx ν wdγ = (λ uw + u w)dx + (μ 1 vw + μ z(x, 1))wdΓ = (λ uw + u w)dx + {μ 1 (λu f)w + μ (λue λτ + z )w}dγ, where we have used (.) and (.4). Therefore, (.7) can be rewritten as (λ uw + u w)dx + (μ 1 + μ e λτ )λuwdγ Γ (.8) N = (g + λf)wdx + μ 1 fwdγ μ z wdγ w HΓ Γ 1 D (). N

8 1568 SERGE NICAISE AND CRISTINA PIGNOTTI As the left-hand side of (.8) is coercive on H 1 Γ D (), the Lax Milgram lemma guarantees the existence and uniqueness of a solution u H 1 Γ D () of (.8). If we consider w D() in (.8), we have that u solves in D () (.9) λ u Δu = g + λf, and thus u E(Δ,L ()). Using Green s formula (.11) in (.8) and using (.9), we obtain (μ 1 + μ e λτ )λuwdγ+ ν ; w = μ 1 fwdγ μ z wdγ, from which follows (.3) ν +(μ 1 + μ e λτ )λu = μ 1 f μ z on. Therefore, from (.3), ν = μ 1v μ z(, 1) on, where we have used (.) and (.4). So, we have found (u, v, z) T D(A), which verifies (.19). Now, the well-posedness result follows from the Hille Yosida theorem... Internal feedback. Setting (.31) z(x, ρ, t) =u t (x, t τρ), x, ρ (, 1), t>, problem (1.1) (1.16) is equivalent to (.3) (.33) (.34) (.35) (.36) (.37) (.38) u tt Δu + a(x)[μ 1 u t (x, t)+μ u t (x, t τ)] = in (, + ), τz t (x, ρ, t)+z ρ (x, ρ, t) = in (, 1) (, + ), u(x, t) = on Γ D (, + ), ν (x, t) = on (, + ), z(x,,t)=u t (x, t) on (, + ), u(x, ) = u (x) and u t (x, ) = u 1 (x) in, z(x, ρ, ) = g (x, ρτ) in (, 1). If we denote by U := (u, u t,z) T, then U := (u t,u tt,z t ) T = ( u t, Δu a(μ 1 u t + μ z(, 1, )), τ 1 z ρ ) T. Therefore, problem (.3) (.38) can be rewritten as (.39) { U = A U, U()=(u,u 1,g (, τ)) T,

9 STABILIZATION OF THE WAVE EQUATION WITH DELAY 1569 where the operator A is defined by A u v z := v Δu aμ 1 v aμ z(, 1) τ 1 z ρ, with domain { D(A ):= (u, v, z) T ( H () HΓ 1 D () ) H 1 () L (; H 1 (, 1)) : (.4) } ν = on ; v = z(, ) in. Denote by H the Hilbert space (.41) H := H 1 Γ D () L () L ( (, 1)), equipped with the inner product (.4) u v z, ũ ṽ z := H { u(x) ũ(x)+v(x)ṽ(x)}dx+ξ 1 z(x, ρ) z(x, ρ)dρdx, where ξ is a fixed positive number satisfying (.14). Arguing analogously to the previous case, we can show that the operator A generates a C semigroup on H. Consequently we have the following well-posedness result. Theorem.. For any initial datum U H there exists a unique solution U C([, + ), H ) of problem (.39). Moreover, if U D(A ), then U C([, + ), D(A )) C 1 ([, + ), H ). 3. Boundary stability result. In this section, in order to prove an exponential stability result for problem (1.1) (1.5), we assume (1.8). Let E( ) be the energy defined by (1.9) and (1.1). We have the following result. Proposition 3.1. For any regular solution of problem (1.1) (1.5), the energy is decreasing and there exists a positive constant C such that (3.1) E (t) C {u t (x, t)+u t (x, t τ)}dγ. Proof. Differentiating (1.9), we obtain E (t) = {u t u tt + u u t }dx + ξ and then, applying Green s formula, (3.) E (t) = u t ν dγ+ξ Now, observe that 1 1 u t (x, t τρ)u tt (x, t τρ)dρdγ, u t (x, t τρ)u tt (x, t τρ)dρdγ. u t (x, t τρ)= τ 1 u ρ (x, t τρ)

10 157 SERGE NICAISE AND CRISTINA PIGNOTTI and u tt (x, t τρ)=τ u ρρ (x, t τρ). Therefore, (3.3) 1 u t (x, t τρ)u tt (x, t τρ)dρdγ = τ 3 1 u ρ (x, t τρ)u ρρ (x, t τρ)dρdγ. Integrating by parts in ρ, we obtain 1 1 u ρ (x, t τρ)u ρρ (x, t τρ)dρdγ = u ρ (x, t τρ)u ρρ (x, t τρ)dρdγ + {u ρ(x, t τ) u ρ(x, t)}dγ, that is, (3.4) 1 u ρ (x, t τρ)u ρρ (x, t τρ)dρdγ = 1 Then, from (3.3) and (3.4), (3.5) 1 u t (x, t τρ)u tt (x, t τρ)dρdγ = τ 3 {u ρ(x, t τ) u ρ(x, t)}dγ. 1 u ρ (x, t τ)u ρρ (x, t τρ)dρdγ = τ 3 {u ΓN ρ(x, t) u ρ(x, t τ)}dγ = τ 1 {u t (x, t) u t (x, t τ)}dγ. Using (3.), (3.5), and the boundary condition (1.3) on, we have E (t) = μ 1 u Γ t (x, t)dγ μ u t (x, t)u t (x, t τ)dγ N (3.6) + ξτ 1 u ΓN t (x, t)dγ ξτ 1 u t (x, t τ)dγ. From (3.6), applying the Cauchy Schwarz inequality we obtain ( E (t) μ 1 + μ ) ( ) + ξτ 1 u μ t (x, t)dγ+ ξτ 1 u t (x, t τ)dγ, which implies E (t) C {u t (x, t)+u t (x, t τ)}dγ, with { ( C = min μ 1 μ ) ( ξτ 1, μ ) } + ξτ 1. Since ξ is chosen satisfying assumption (1.1), the constant C is positive.

11 We can write STABILIZATION OF THE WAVE EQUATION WITH DELAY 1571 E(t) =E(t)+E N (t), where E(t) is the standard energy for the wave equation (3.7) E(t) := 1 {u t (x, t)+ u(x, t) }dx, and (3.8) E N (t) := ξ 1 u t (x, t τρ)dρdγ. With a change of variable we can rewrite (3.9) E N (t) = ξ t u t (x, s)dsdγ. τ t τ We can now give a boundary observability inequality which we will use to prove the exponential decay of the energy E(t). Proposition 3.. There exists a time T > such that for all times T > T, there exists a positive constant C (depending on T ) for which T (3.1) E() C {u t (x, t)+u t (x, t τ)}dγdt for any regular solution u of problem (1.1) (1.5). Proof. From Proposition 6.3 of [14], for T greater than a sufficiently large time T, and any ε>, we have T (3.11) E() c {( ) } + u ν t dγdt + c u H 1/+ε ( (,T )) for a suitable constant c (depending on T ). Estimate (3.11) is obtained by Carleman estimates under the assumption that there exists a function v of class C satisfying (1.6) and (1.7). The function v is needed to construct a suitable weight function for Carleman estimates (see the proof of Proposition 4. below). Then, by (3.11) and the boundary condition (1.3), we have T (3.1) E() c {u t (x, t)+u t (x, t τ)}dγdt + c u H 1/+ε ( (,T )) for a suitable positive constant c. From (3.9) we have that (3.13) E N () c u t (x, s)dsdγ. τ By a change of variable in (3.13) we obtain, for T τ, T (3.14) E N () c u t (x, t τ)dγdt.

12 157 SERGE NICAISE AND CRISTINA PIGNOTTI Denote T := max{τ,t }. Then, from (3.1) and (3.14), for any T>T we have (3.15) E() = E() + E N () T c {u t (x, t)+u t (x, t τ)}dγdt + c u H 1/+ε ( (,T )) for a suitable positive constant c depending on T. In order to obtain (3.1) we need to absorb the lower order term u H 1/+ε ( (,T )). To do this, we argue by contradiction. Suppose that (3.1) is not true. Then, there exists a sequence {u n } n of solutions of problem (1.1) (1.5) such that, denoting by E n () the energy E related to u n at the time, T (3.16) E n () >n {u nt(x, t)+u nt(x, t τ)}dγdt. From (3.15), we have { T } (3.17) E n () c {u nt(x, t)+u nt(x, t τ)}dγdt + u n H 1/+ε ( (,T )). Then, from (3.16) and (3.17), T n {u nt(x, t)+u nt(x, t τ)}dγdt { T } <c {u nt(x, t)+u nt(x, t τ)}dγdt + u n H 1/+ε ( (,T )), that is, T (3.18) (n c) {u nt(x, t)+u nt(x, t τ)}dγdt<c u n H 1/+ε ( (,T )). Renormalizing, we obtain a sequence {w n } n of solutions of problem (1.1) (1.5) verifying (3.19) w n H 1/+ε ( (,T )) =1, and (3.) T {w nt(x, t)+w nt(x, t τ)}dγdt < c n c. From (3.19), (3.), and (3.17), it follows that the sequence {w n } n is bounded in H 1 ( (,T)). Since H 1 ( (,T)) is compactly embedded in H 1/+ε ( (,T)), there exists a subsequence which, for simplicity of notation, we still denote by {w n } n such that Then, from (3.19), w n w strongly in H 1/+ε ( (,T)). (3.1) w H 1/+ε ( (,T )) =1.

13 STABILIZATION OF THE WAVE EQUATION WITH DELAY 1573 Moreover, by (3.), T {w t (x, t)+w t (x, t τ)}dγdt =. Therefore, we have that w t = on (,T) and w ν = on (,T). Putting v := w t,vsolves in a distributional sense v Δv = in (,T), with v v = on Γ (,T), ν = on (,T). Therefore, from Holmgren s uniqueness theorem (see [15, Chap. I, Thm. 8., p. 9]) v. This implies that Thus, w verifies w(x, t) =w(x). Δw = in, w = on Γ D, w ν = on, and so w. This is in contradiction with (3.1). Then, the observability inequality (3.1) is proved. From (3.1) easily follows the stability result. Proof of Theorem 1.1. From (3.1), we have T (3.) E(T ) E() C {u t (x, t)+u t (x, t τ)}dγdt. By (3.) and the observability estimate (3.1), we obtain T E(T ) E() C {u t (x, t)+u t (x, t τ)}dγdt C C 1 (E() E(T )); then E(T ) CE(), with C <1. This easily implies the stability estimate (1.11), since our system (1.1) (1.5) is invariant by translation and the energy E is decreasing.

14 1574 SERGE NICAISE AND CRISTINA PIGNOTTI Remark 3.3. Analogous arguments apply if we have more than one delay term in the boundary feedback, that is, if condition (1.3) is substituted by ν (x, t) = μ u t (x, t) k μ i u t (x, t τ i ) i=1 on (, + ), with μ,μ i,τ i,i=1,...,k, positive parameters. In this case, the right energy for our problem is E(t) := 1 {u t (x, t)+ u(x, t) }dx + k i=1 ξ i with suitable positive constants ξ i,i=1,...,k. Indeed, if 1 u t (x, t ρτ i )dρdγ, choosing ξ i such that k μ > μ i, i=1 μ i <ξ i τ 1 i,i=1,...,k, and k i=1 ξ i τ 1 i < μ k μ i, we can prove that the energy is exponentially decaying to zero. Remark 3.4. If Γ D, the strong solution of (1.1) (1.5) has a singular behavior along Γ D since u(t) E(Δ,L ()) with ν L ( ) and u =on Γ D (see, for instance, [7] for two-dimensional domains). Therefore the results from [14] cannot be invoked. For the standard feedback law (i.e., the case μ = ), the multiplier method has been used as an alternative in [1] to obtain stability results under strong geometrical conditions. Unfortunately that approach cannot be directly applied here because for μ >, we have only ν L ( ), which forbids the use of the multiplier identity. 4. Internal stability result. In this section, under assumption (1.8) we want to prove exponential stability for problem (1.1) (1.16). We first show that the energy F, defined by (1.19) and (1.1), is decreasing. Proposition 4.1. For any regular solution of problem (1.1) (1.16), the energy is decreasing and there exists a positive constant C such that (4.1) F (t) C a(x){u t (x, t)+u t (x, t τ)}dx. Proof. Differentiating (1.19), we obtain F (t) = {u t u tt + u u t }dx + ξ a(x) and then, applying Green s formula, (4.) F (t) = u t (u tt Δu)dx + ξ a(x) where we have used the boundary conditions (1.13) and (1.14). 1 1 i=1 u t (x, t τρ)u tt (x, t τρ)dρdx, u t (x, t τρ)u tt (x, t τρ)dρdx,

15 (4.3) STABILIZATION OF THE WAVE EQUATION WITH DELAY 1575 As in the proof of Proposition 3.1, we can compute 1 a(x) u t (x, t τρ)u tt (x, t τρ)dρdx 1 = τ 3 a(x) u ρ (x, t τ)u ρρ (x, t τρ)dρdx = τ 3 = τ 1 a(x){u ρ(x, t) u ρ(x, t τ)}dx a(x){u t (x, t) u t (x, t τ)}dγ. Now, using (4.3) and (1.1) in identity (4.), we have F (t) = μ 1 a(x)u t (x, t)dx μ a(x)u t (x, t)u t (x, t τ)dx (4.4) + ξτ 1 a(x)u t (x, t)dx ξτ 1 a(x)u t (x, t τ)dx. From (4.4), applying the Cauchy Schwarz inequality and recalling (1.1), we obtain estimate (4.1). We can write F(t) =E(t)+E (t), where E(t) is the standard energy for the wave equation defined in (3.7) and where (4.5) E (t) := ξ 1 a(x) u t (x, t τρ)dρdx. With a change of variable, we can rewrite (4.6) E (t) = ξ t a(x) u t (x, s)dsdx. τ t τ Let w be the solution of the homogeneous problem for the wave equation with mixed Dirichlet Neumann boundary condition, (4.7) (4.8) (4.9) (4.1) w tt (x, t) Δw(x, t) = in (, + ), w(x, t) = on Γ D (, + ), w ν (x, t) = on (, + ), w(x, ) = w (x) and w t (x, ) = w 1 (x) in. Denote by E w (t) the standard energy for the wave equation corresponding to w, that is, (4.11) E w (t) = 1 {wt (x, t)+ w(x, t) }dx. Note that E w (t) is constant. We can give an observability inequality for problem (4.7) (4.1).

16 1576 SERGE NICAISE AND CRISTINA PIGNOTTI Proposition 4.. There exists a time T such that for all times T>T, there exists a positive constant C 1 (depending on T ) for which T (4.1) E w () C 1 wt (x, t)dxdt for any regular solution w of problem (4.7) (4.1). Proof. Inequality (4.1) easily follows from some estimates of [14] and standard arguments with multipliers. We give the proof for the reader s convenience. Let ω,ω 1 be open neighborhoods of such that (4.13) ω ω ω 1. Let ϕ be a smooth function such that (4.14) ϕ(x) 1, ϕ on \ ω, ϕ 1 on ω 1. Then, the function ϕw verifies (ϕw) tt Δ(ϕw) =F (w), where F (w) = wδϕ + ϕ w, with the same boundary conditions as w. Therefore, we can apply to ϕw the result of Proposition 4..1 in [14]. Let us recall some notation from [14]. Without loss of generality, we can suppose that the function v satisfying assumptions (1.6) and (1.7) is nonnegative on. Denote ( ) 1/ maxx v(x) (4.15) T =, α with α as in (1.6). Define the function φ : R R by ( (4.16) φ(x, t) :=v(x) c t T ), where T > T is fixed and the constant c is chosen as follows. From (4.15), there exists a constant δ> such that αt > 4 max x For fixed δ, there is c such that (4.17) c T > 4 max x Note that ω v(x)+4δ, v(x)+4δ. c (,α). (4.18) φ(x, ) < δ and φ(x, T ) < δ uniformly in. Set BTw Γ (,T ) = 1 T ( ) w e γφ H ν dγdt Γ (4.19) D ν + 1 T e γφ H ν(wt T w )dγdt, where T w denotes the tangential gradient of w.

17 STABILIZATION OF THE WAVE EQUATION WITH DELAY 1577 Then, from Proposition 4..1 of [14], using (4.18), and recalling that E w (t) is constant, we have { T T BTw Γ (,T ) c e γφ (ϕw) dxdt + ϕ wt dxdt T } + ϕ w dxdt + e γδ E w () (4.) { T T c e γφ w dxdt + wt dxdt ω ω T } + w dxdt + e γδ E w () for a suitable positive constant c, where the parameter γ can be chosen sufficiently large in order to have the desired inequality. Now, consider another smooth cut-off function ψ such that (4.1) ψ(x) 1, ψ on \ ω, ψ 1 on ω. Integrating by parts, T [ ] T (w tt Δw)ψwe γφ dxdt = ψww t e γφ dx T T + w (ψwe γφ )dxdt w t (ψwe γφ ) t dxdt (4.) [ ] T T = ψww t e γφ dx ψw t (w t e γφ + we γφ γφ t )dxdt T T + ψe γφ w dxdt + w w (ψe γφ )dxdt. Then, from (4.), recalling that w satisfies (4.7), we have T T ψe γφ w dxdt = (ψwt e γφ + ww t ψe γφ γφ t )dxdt [ ] T T (4.3) ψww t e γφ dx ψw w ( ψ)e γφ dxdt T ψw w e γφ dxdt. Since E w (t) is constant, by using the Cauchy Schwarz inequality and Poincaré s theorem, we can estimate [ ] T ψww t e γφ dx ce δγ E w (), and so, from (4.3), we obtain T ψe γφ w dxdt ce δγ E w () + 1 T ψe γφ w dxdt (4.4) { T T } +c wt dxdt + w dxdt ω

18 1578 SERGE NICAISE AND CRISTINA PIGNOTTI for a suitable positive constant c. By (4.4) we deduce T { T T } e γφ w dxdt c wt dxdt + w dxdt + e δγ E w (), ω ω which, used in (4.), gives { T (4.5) BTw Γ (,T ) c ω T } wt dxdt + w dxdt + e δγ E w (). Then, from (4.5) and Theorem 3.4 of [14] (Carleman estimate (3.14)), taking γ sufficiently large, we obtain T E w () c wt (x, t)dxdt + c w L ( (,T )). ω Now, estimate (4.1) follows from compactness-uniqueness arguments. Proposition 4.3. There exists a time T such that for all times T > T, there exists a positive constant C (depending on T ) for which T (4.6) F() C a(x){u t (x, t)+u t (x, t τ)}dxdt for any regular solution u of problem (1.1) (1.16). Proof. Following Zuazua [18], we can decompose the solution u of problem (1.1) (1.16) as u = w + w, where w solves (4.7) (4.9) with initial condition and w verifies w(x, ) = u (x), w t (x, ) = u 1 (x) in, (4.7) (4.8) (4.9) (4.3) w tt Δ w = a(x)[μ 1 u t (x, t)+μ u t (x, t τ)] in (, + ), w(x, t) = on Γ D (, + ), w ν (x, t) = on (, + ), w(x, ) = and w t (x, ) = in. Then, from (4.5) and (4.11), (4.31) F() = E() + E () = E w () + ξ a(x) 1 u t (x, ρτ)dρdx. If we take T>T := max{t,τ}, from (4.31) with a change of variable we obtain T F() E w () + c a(x) u t (x, t τ)dtdx,

19 and then, from (4.1), (4.3) STABILIZATION OF THE WAVE EQUATION WITH DELAY 1579 F() c c a(x) a(x) T T {w t (x, t)+u t (x, t τ)}dtdx { w t (x, t)+u t (x, t)+u t (x, t τ)}dtdx for a suitable positive constant c. Therefore, from standard energy estimates for w, we obtain T F() C a(x){u t (x, t)+u t (x, t τ)}dxdt. Now, using estimate (4.6), as in the case of boundary feedback we obtain the exponential stability result of Theorem 1.3. Remark 4.4. Analogous arguments apply if we have more than one delay term in the internal feedback, that is, if (1.1) is replaced with [ u tt (x, t) Δu(x, t)+a(x) μ u t (x, t)+ with μ,μ i,τ i,i=1,...,k, positive parameters. In this case, if μ > k i=1 ] μ i u t (x, t τ i ) = in (, + ), k μ i, the right energy to consider, in order to prove exponential decay, is E(t) := 1 i=1 {u t (x, t)+ u(x, t) }dx + k i=1 ξ i a(x) 1 u t (x, t ρτ i )dρdx, with constants ξ i,i=1,...,k, chosen as in Remark 3.3. Remark 4.5. In the case Γ D, since for internal feedbacks we have ν = on, we can use the multiplier identity from [1] and then obtain stability results under the same geometrical conditions as those from [1]. 5. Some instability examples. In this section we will give some instability examples for the case μ μ Boundary feedback. In this subsection we consider problem (1.1) (1.5) with boundary feedback, and we prove Theorem 1.. Let us consider the spectral problem for the system (5.1) u tt (x, t) Δu(x, t) = in (, + ), u(x, t) = onγ D (, + ), ν (x, t) = μ 1u t (x, t) μ u t (x, t τ) We seek a solution of (5.1) in the form u(x, t) =e λt ϕ(x), λ C. on (, + ).

20 158 SERGE NICAISE AND CRISTINA PIGNOTTI Then, ϕ has to be a solution of the eigenvalue problem Δϕ + λ ϕ = in, (5.) ϕ = on Γ D, ϕ ν = (μ 1 + μ e λτ )λϕ on, which can be reformulated, in a variational form, as (5.3) ϕ vdx + λ ϕvdx +(μ 1 + μ e λτ )λ ϕvdγ = v HΓ 1 D (). We want to find a solution for λ := ib, with b R. For this choice of λ, problem (5.3) can be rewritten as (5.4) ϕ vdx b Assume that ϕvdx +(μ 1 + μ e ibτ )ib ϕvdγ = v H 1 Γ D (). (5.5) cos(bτ) = μ 1. μ Note that, since we are considering the case μ μ 1, there exist b, τ such that (5.5) holds. Then, we choose (5.6) μ sin(bτ) = μ μ 1. Under these assumptions, (5.4) becomes (5.7) ϕ vdx b ϕvdx + b μ μ 1 ϕvdγ = v HΓ 1 D (). In particular, for v = ϕ, (5.7) gives (5.8) ϕ dx b ϕ dx + b μ μ 1 ϕ dγ=. Without loss of generality, we can assume (5.9) ϕ := ϕ dx =1, and then the identity (5.8) can be rewritten as (5.1) b b μ μ 1 q (ϕ) q 1 (ϕ) =, where (5.11) q (ϕ) := ϕ dγ, q 1 (ϕ) := ϕ dx. Now we distinguish two cases. Case (a): μ 1 = μ. In this case, under our assumptions, (5.1) becomes (5.1) b = q 1 (ϕ).

21 STABILIZATION OF THE WAVE EQUATION WITH DELAY 1581 Define (5.13) b := min w H Γ 1 () D w =1 q 1 (w). If ϕ verifies q 1 (ϕ) = min w H Γ 1 () D w =1 q 1 (w), then it easy to see that ϕ is a solution of (5.4) with b as in (5.13). Then ϕ verifies (5.), and so (5.14) u(x, t) :=e ibt ϕ(x) is a solution of problem (5.1). Therefore, we have found a solution of our boundary problem, whose energy is constant. Indeed, an easy computation shows that, for the function u defined in (5.14), ( u(x, t) + u t (x, t) )dx =b > t. Note that, from our assumptions (λ = ib, cos(bτ) = 1, sin(bτ) = ), problem (5.) becomes the classical eigenvalue problem for the Laplace operator with a mixed Dirichlet Neumann boundary condition. So, we can take a sequence {b n } n of positive real numbers defined by b n =Λ n, n N, where Λ n,n N, are the eigenvalues for the Laplace operator. Then, putting we obtain a sequence of delays b n τ =(l +1)π, l N, (l +1)π τ n,l =, l,n N, b n which become arbitrarily small (or large) for suitable choices of the indices n, l N. Therefore, in the case μ 1 = μ, we have found a set of time delays for which problem (1.1) (1.5) is not asymptotically stable. Case (b): μ >μ 1. In this case, from (5.1) we have b = 1 ( ) μ μ 1 q (ϕ) ± (μ μ 1 )q (ϕ)+4q 1(ϕ). Define (5.15) b := 1 min w H Γ 1 () D w =1 ( ) μ μ 1 q (w)+ (μ μ 1 )q (w)+4q 1(w). We now prove that if the minimum in the right hand side of (5.15) is attained at ϕ, that is,

22 158 SERGE NICAISE AND CRISTINA PIGNOTTI (5.16) μ μ 1 q (ϕ)+ := min w H 1 Γ D () (μ μ 1 )q (ϕ)+4q 1(ϕ) ( μ μ 1 q (w)+ w =1 ) (μ μ 1 )q (w)+4q 1(w), then ϕ is a solution of (5.7) with b as in (5.15). To show this, take for ε R, (5.17) w = ϕ + εv, with v H 1 Γ D () such that Then, If we denote (5.18) 1 g(ε) := 1+ε v then, by definition (5.16), g(ε) g() = w = ϕ + ε v =1+ε v. ϕvdx =. ( ) μ μ 1 q (ϕ + εv)+ (μ μ 1 )q (ϕ + εv)+4q 1(ϕ + εv), ( ) μ μ 1 q (ϕ)+ (μ μ 1 )q (ϕ)+4q 1(ϕ). So, we have that dg(ε) dε =, ε= which, after an easy computation, gives (5.19) ϕ vdx + b μ μ 1 ϕvdγ =. Since any function ṽ H 1 Γ D () can be decomposed as ṽ = γϕ + v, γ R, v H 1 Γ D () with ϕvdx =, from (5.19) and (5.8) we obtain that ϕ satisfies (5.7) with b defined in (5.15). So, for such positive b, ( bτ = arccos μ ) 1 +lπ, l N, μ defines a sequence of time delays for which problem (1.1) (1.5) is not asymptotically stable. The above examples prove Theorem 1..

23 STABILIZATION OF THE WAVE EQUATION WITH DELAY Internal feedback. In this subsection we will give instability examples for problem (1.1) (1.16) with internal feedback, proving Theorem 1.4. Let us consider the spectral problem for the system (5.) u tt (x, t) Δu(x, t)+a(x)[μ 1 u t (x, t)+μ u t (x, t τ)] = in (, + ), u(x, t) = onγ D (, + ), (x, t) = ν on (, + ). We restrict our analysis to the case a(x) 1in. We seek a solution of (5.) in the form u(x, t) =e λt ϕ(x), λ C. Then, ϕ has to solve the eigenvalue problem Δϕ =[λ +(μ 1 + μ e λτ )λ]ϕ in, (5.1) ϕ = on Γ D, ϕ ν = on. Let us consider the standard problem for the Laplace operator with a mixed Dirichlet Neumann boundary condition (5.) Δϕ = μ ϕ in, ϕ = on Γ D, ϕ ν = on. We want to show that for any Λ eigenvalue of problem (5.), there exists a λ C solution of the equation (5.3) λ +(μ 1 + μ e λτ )λ = Λ. We seek a solution λ = α + iβ, α, β R, with (5.4) βτ =(l +1)π, l N. Under this assumption, (5.3) becomes (5.5) { α + β =Λ, μ e ατ =α + μ 1. Now we distinguish two cases. Case (a): μ 1 = μ. In this case, from (5.5) we have α =, β =Λ. Therefore, for any Λ n eigenvalue of problem (5.), if β n R verifies β n =Λ n, then for λ = iβ n problem (5.1) admits a nonzero solution.

24 1584 SERGE NICAISE AND CRISTINA PIGNOTTI Take β n positive. From our assumption (5.4), (l +1)π τ n,l =, n,l N, β n is a set of time delays that become arbitrarily small (or large) for suitable choices of the indices n, l N. For such delays, problem (1.1) (1.16) admits solutions in the form u(x, t) =e iβt ϕ(x), whose energy is constant and strictly positive. So, system (1.1) (1.16) is not asymptotically stable. Case (b): μ >μ 1. For a fixed α>, from the second equation of (5.5), we obtain (5.6) τ(α) = 1 α ln ( μ μ 1 +α and so, in order to have τ(α) >, we consider <α< 1 (μ μ 1 ). From (5.4), the first equation of (5.5) becomes (5.7) α + (l +1) π τ =Λ, (α) where τ(α) is given by (5.6). Denoting we have while g(α) :=α + (l +1) π τ (α) α ), (, μ ) μ 1, τ(α) + and g(α) + as α +, τ(α) + and g(α) + as α 1 (μ μ 1 ). Since g is a continuous function of α, for any fixed Λ eigenvalue of problem (5.) there exists α ( <α<(μ μ 1 )/) such that (5.7) is verified. Therefore, for such α there exists a delay τ(α) (defined by (5.6)) such that a function of the form e α+iβ ϕ(x) solves problem (1.1) (1.16). Since α, the energy of such a solution is not decaying to zero. So, this solution is not asymptotically stable. Note that, for any Λ n eigenvalue of problem (5.) and for any l N, there exist α n,l and a delay τ n,l = τ(α n,l ) such that (5.5) is verified with β n,l = (l +1)π τ n,l.

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