STABILITY AND INSTABILITY RESULTS OF THE WAVE EQUATION ...
SIAM J. CONTROL OPTIM. Vol. 45, No. 5, pp. 1561–1585
c 2006 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. 35L05, 93D15 DOI. 10.1137/060648891
1. Introduction. We investigate the effect of time delay in boundary or internal stabilization of the wave equation in domains of Rn . 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 Ω ⊂ Rn be an open bounded set with a boundary Γ of class C 2 . We assume that Γ is divided into two parts ΓD and ΓN , i.e., Γ = ΓD ∪ ΓN , with ΓD ∩ ΓN = ∅ and ΓD = ∅. In this domain Ω, we consider the initial boundary value problem (1.1) (1.2) (1.3) (1.4) (1.5)
utt (x, t) − Δu(x, t) = 0 in Ω × (0, +∞), u(x, t) = 0 on ΓD × (0, +∞), ∂u (x, t) = −μ1 ut (x, t) − μ2 ut (x, t − τ ) on ΓN × (0, +∞), ∂ν u(x, 0) = u0 (x) and ut (x, 0) = u1 (x) in Ω, ut (x, t − τ ) = f0 (x, t − τ ) in ΓN × (0, τ ),
where ν(x) denotes the outer unit normal vector to the point x ∈ Γ and ∂u ∂ν is the normal derivative. Moreover, τ > 0 is the time delay, μ1 and μ2 are positive real numbers, and the initial data (u0 , u1 , f0 ) belong to a suitable space. We are interested in giving an exponential stability result for such a problem. ∗ Received by the editors January 3, 2006; accepted for publication (in revised form) May 24, 2006; published electronically November 14, 2006. http://www.siam.org/journals/sicon/45-5/64889.html † Universit´ e de Valenciennes et du Hainaut Cambr´esis, MACS, Institut des Sciences et Techniques de Valenciennes, 59313 Valenciennes Cedex 9, France ([email protected]). ‡ Dipartimento di Matematica Pura e Applicata, Universit` a di L’Aquila, Via Vetoio, Loc. Coppito, 67010 L’Aquila, Italy ([email protected]).
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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 ∈ Rn . We assume that there exists a scalar function v ∈ C 2 (Ω) such that (i) v is strictly convex in Ω; that is, there exists α > 0 such that (1.6)
D2 (v)(x)ξ, ξ ≥ 2α|ξ|2
∀x ∈ Ω, ∀ξ ∈ Rn ,
where D2 (v) denotes the Hessian matrix of v; (ii) the vector field H := ∇v verifies (1.7)
H(x) · ν(x) ≤ 0
∀x ∈ ΓD .
For the above assumptions see [14], where some observability estimates for secondorder hyperbolic equations are given. It is well known that if μ2 = 0, that is, in absence of delay, the energy of problem (1.1)–(1.5) is exponentially decaying to zero. See for instance Chen [2, 3], Lagnese [11, 12], Lasiecka and Triggiani [13], Komornik and Zuazua [10], and Komornik [8, 9]. On the contrary, if μ1 = 0, that is, if we have only the delay part in the boundary condition on ΓN , 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 μ1 and μ2 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 μ2 < μ1 . In their paper it is also shown that if μ2 > μ1 , the system is unstable and if μ1 = μ2 , some instabilities may occur. Here, in agreement with [17] and assuming that (1.8)
μ2 < μ1 ,
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 μ1 = μ2 , we further show that there exists a sequence of arbitrary small (and large) delays such that instabilities occur. In the case μ2 > μ1 , 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 ξ 1 {u2t (x, t) + |∇u(x, t)|2 }dx + u2 (x, t − τ ρ)dρdΓ, (1.9) E(t) := 2 Ω 2 ΓN 0 t where ξ is a positive constant verifying (1.10)
τ μ2 < ξ < τ (2μ1 − μ2 ).
Clearly this energy is larger than the standard energy 1 {u2 (x, t) + |∇u(x, t)|2 }dx 2 Ω t and contains an additional term that comes from the delay term.
STABILIZATION OF THE WAVE EQUATION WITH DELAY
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Theorem 1.1. Assume that (1.8) holds. There exist positive constants C1 , C2 such that, for any solution of problem (1.1)–(1.5), (1.11)
E(t) ≤ C1 E(0)e−C2 t
∀t ≥ 0.
Theorem 1.2. 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 (1.12) utt (x, t) − Δu(x, t) + a(x)[μ1 ut (x, t) + μ2 ut (x, t − τ )] = 0 in Ω × (0, +∞), (1.13) u(x, t) = 0 on ΓD × (0, +∞), ∂u (1.14) (x, t) = 0 on ΓN × (0, +∞), ∂ν (1.15) u(x, 0) = u0 (x) and ut (x, 0) = u1 (x) in Ω, (1.16) ut (x, t − τ ) = g0 (x, t − τ ) in Ω × (0, τ ), where a ∈ L∞ (Ω) is a function such that (1.17)
a(x) ≥ 0
a. e.
in
Ω,
and (1.18)
a(x) > a0 > 0
a. e.
in ω,
where ω ⊂ Ω is an open neighborhood of ΓN . Exponential stability results for the above problem in the case of μ2 = 0, 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 μ2 < μ1 , 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 μ2 ≥ μ1 , 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 (1.12)– (1.16) as 1 1 ξ 2 2 (1.19) F(t) := {u (x, t) + |∇u(x, t)| }dx + a(x) u2t (x, t − τ ρ)dρdx, 2 Ω t 2 Ω 0 where ξ is a positive constant verifying (1.10). 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 C1 , C2 such that, for any solution of problem (1.12)–(1.16), (1.20)
F(t) ≤ C1 F(0)e−C2 t
∀t ≥ 0.
Theorem 1.4. If (1.8) does not hold, there exist a sequence of arbitrary small (or large) delays, and solutions of problem (1.12)–(1.16) corresponding to these delays, such that their standard energy does not tend to 0.
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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 2 using semigroup theory. In subsection 2.1 we study the well-posedness of problem (1.1)–(1.5), while in subsection 2.2 we concentrate on problem (1.12)–(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. 2. Well-posedness of the problems. In this section we will give well-posedness results for problem (1.1)–(1.5) and problem (1.12)–(1.16) using semigroup theory. 2.1. Boundary feedback. Let us set (2.1)
z(x, ρ, t) = ut (x, t − τ ρ),
x ∈ ΓN , ρ ∈ (0, 1), t > 0.
Then, problem (1.1)–(1.5) is equivalent to
(2.7)
utt (x, t) − Δu(x, t) = 0 in Ω × (0, +∞), τ zt (x, ρ, t) + zρ (x, ρ, t) = 0 in ΓN × (0, 1) × (0, +∞), u(x, t) = 0 on ΓD × (0, +∞), ∂u (x, t) = −μ1 ut (x, t) − μ2 z(x, 1, t) on ΓN × (0, +∞), ∂ν z(x, 0, t) = ut (x, t) on ΓN × (0, ∞), u(x, 0) = u0 (x) and ut (x, 0) = u1 (x) in Ω,
(2.8)
z(x, ρ, 0) = f0 (x, −ρτ )
(2.2) (2.3) (2.4) (2.5) (2.6)
in
ΓN × (0, 1).
If we denote T
U := (u, ut , z) , then T T U := (ut , utt , zt ) = ut , Δu, −τ −1 zρ . Therefore, problem (2.2)–(2.8) can be rewritten as (2.9)
U = AU, T U (0) = (u0 , u1 , f0 (·, − · τ )) ,
where the operator A is defined by ⎞ ⎞ ⎛ u v ⎠, A ⎝ v ⎠ := ⎝ Δu z −τ −1 zρ ⎛
STABILIZATION OF THE WAVE EQUATION WITH DELAY
1565
with domain (2.10)
D(A) := (u, v, z)T ∈ E(Δ, L2 (Ω)) ∩ HΓ1D (Ω) × H 1 (Ω) × L2 (ΓN ; H 1 (0, 1)) : ∂u = −μ1 v − μ2 z(·, 1) on ΓN ; v = z(·, 0) on ΓN , ∂ν where, as usual, HΓ1D (Ω) = { u ∈ H 1 (Ω) : u = 0
on ΓD },
and E(Δ, L2 (Ω)) = {u ∈ H 1 (Ω) : Δu ∈ L2 (Ω)}. −1/2 (ΓN ) and the next Recall that for a function u ∈ E(Δ, L2 (Ω)), ∂u ∂ν belongs to H Green formula is valid (see section 1.5 of [7])
∂u ;w ∇u∇wdx = − Δuwdx + ∀w ∈ HΓ1D (Ω), (2.11) ∂ν Ω Ω ΓN
where ·; · ΓN means the duality pairing between H −1/2 (ΓN ) and H 1/2 (ΓN ). 2 Note further that for (u, v, z)T ∈ D(A), ∂u ∂ν belongs to L (ΓN ) since z(·, 1) is in 2 L (ΓN ). Denote by H the Hilbert space H := HΓ1D (Ω) × L2 (Ω) × L2 (ΓN × (0, 1)).
(2.12) Assuming that
μ2 ≤ μ1 ,
(2.13)
we will show that A generates a C0 semigroup on H. Let ξ be a positive real number such that τ μ2 ≤ ξ ≤ τ (2μ1 − μ2 ).
(2.14)
Note that, from (2.13), such a constant ξ exists. Let us define on the Hilbert space H the inner product (2.15) ⎞ ⎛ u ⎞ ⎛ u 1 ˜ ⎝ v ⎠ , ⎝ v˜ ⎠ := {∇u(x)∇˜ u(x)+v(x)˜ v (x)}dx+ξ z(x, ρ)˜ z (x, ρ)dρdΓ. Ω ΓN 0 z z˜ H Theorem 2.1. For any initial datum U0 ∈ H there exists a unique solution U ∈ C([0, +∞), H) of problem (2.9). Moreover, if U0 ∈ D(A), then U ∈ C([0, +∞), D(A)) ∩ C 1 ([0, +∞), H). Proof. Take U = (u, v, z)T ∈ D(A). Then ⎞ ⎛ ⎞ ⎛ v u ⎠,⎝ v ⎠ (AU, U ) = ⎝ Δu −1 z −τ zρ H −1 = {∇v(x)∇u(x) + v(x)Δu(x)}dx − ξτ Ω
ΓN
1
zρ (x, ρ)z(x, ρ)dρdΓ. 0
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SERGE NICAISE AND CRISTINA PIGNOTTI
So, by Green’s formula, (2.16)
(AU, U ) = ΓN
∂u (x)v(x)dΓ − ξτ −1 ∂ν
1
zρ (x, ρ)z(x, ρ)dρdΓ. ΓN
0
Integrating by parts in ρ, we get
1
1
zρ (x, ρ)z(x, ρ)dρdΓ = − ΓN
0
ΓN
zρ (x, ρ)z(x, ρ)dρdΓ+
0
{z 2 (x, 1)−z 2 (x, 0)}dΓ,
ΓN
that is
(2.17)
1
zρ (x, ρ)z(x, ρ)dρdΓ = ΓN
0
1 2
{z 2 (x, 1) − z 2 (x, 0)}dΓ. ΓN
Therefore, from (2.16) and (2.17), (AU, U ) = ΓN
=−
ξτ −1 ∂u (x)v(x)dΓ − ∂ν 2
{z 2 (x, 1) − z 2 (x, 0)}dΓ ΓN
(μ1 v(x) + μ2 z(x, 1))v(x)dΓ − ΓN
ξτ −1 2
{z 2 (x, 1) − z 2 (x, 0)}dΓ ΓN
ξτ −1 = −μ1 v 2 (x)dΓ − μ2 z(x, 1)v(x)dΓ − 2 ΓN ΓN
z 2 (x, 1)dΓ + ΓN
ξτ −1 2
v 2 (x)dΓ, ΓN
from which follows, using the Cauchy–Schwarz inequality, (2.18) ξτ −1 μ2 μ2 ξτ −1 2 − (AU, U ) ≤ −μ1 + v (x)dΓ + z 2 (x, 1)dΓ. + 2 2 2 2 ΓN ΓN Now, observe that from (2.14), −μ1 +
ξτ −1 μ2 + ≤ 0, 2 2
μ2 ξτ −1 − ≤ 0. 2 2
Then, (AU, U ) ≤ 0, which means that the operator A is dissipative. Now, we will show that λI −A is surjective for a fixed λ > 0. Given (f, g, h)T ∈ H, we seek a U = (u, v, z)T ∈ D(A) solution of ⎛ ⎞ ⎛ ⎞ u f (λI − A) ⎝ v ⎠ = ⎝ g ⎠ , z h that is, verifying (2.19)
⎧ ⎨ λu − v = f, λv − Δu = g, ⎩ λz + τ −1 zρ = h.
Suppose that we have found u with the appropriate regularity. Then, (2.20)
v := λu − f
STABILIZATION OF THE WAVE EQUATION WITH DELAY
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and we can determine z. Indeed, by (2.10), (2.21)
for x ∈ ΓN ,
z(x, 0) = v(x)
and, from (2.19), (2.22)
λz(x, ρ) + τ −1 zρ (x, ρ) = h(x, ρ)
for x ∈ ΓN , ρ ∈ (0, 1).
Then, by (2.21) and (2.22), we obtain z(x, ρ) = v(x)e−λρτ + τ e−λρτ
ρ
h(x, σ)eλστ dσ. 0
So, from (2.20), −λρτ
(2.23) z(x, ρ) = λu(x)e
−λρτ
−f (x)e
−λρτ
ρ
h(x, σ)eλστ dσ
+τ e
on ΓN ×(0, 1),
0
and, in particular, z(x, 1) = λu(x)e−λτ + z0 (x),
(2.24)
x ∈ ΓN ,
with z0 ∈ L2 (ΓN ) defined by z0 (x) = −f (x)e−λτ + τ e−λτ
(2.25)
1
h(x, σ)eλστ dσ,
x ∈ ΓN .
0
By (2.20) and (2.19), the function u verifies λ(λu − f ) − Δu = g, that is, λ2 u − Δu = g + λf.
(2.26)
Problem (2.26) can be reformulated as (2.27) (λ2 u − Δu)wdx = (g + λf )wdx ∀w ∈ HΓ1D (Ω). Ω
Ω
Integrating by parts, (λ2 u − Δu)wdx = (λ2 uw + ∇u∇w)dx − Ω
Ω
ΓN
(λ2 uw + ∇u∇w)dx +
= Ω
(μ1 vw + μ2 z(x, 1))wdΓ ΓN
{μ1 (λu − f )w + μ2 (λue−λτ + z0 )w}dΓ,
(λ uw + ∇u∇w)dx + 2
=
∂u wdΓ ∂ν
Ω
ΓN
where we have used (2.20) and (2.24). Therefore, (2.27) can be rewritten as (λ2 uw + ∇u∇w)dx + (μ1 + μ2 e−λτ )λuwdΓ Ω ΓN (2.28) = (g + λf )wdx + μ1 f wdΓ − μ2 z0 wdΓ ∀w ∈ HΓ1D (Ω). Ω
ΓN
ΓN
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SERGE NICAISE AND CRISTINA PIGNOTTI
As the left-hand side of (2.28) is coercive on HΓ1D (Ω), the Lax–Milgram lemma guarantees the existence and uniqueness of a solution u ∈ HΓ1D (Ω) of (2.28). If we consider w ∈ D(Ω) in (2.28), we have that u solves in D (Ω) λ2 u − Δu = g + λf,
(2.29)
and thus u ∈ E(Δ, L2 (Ω)). Using Green’s formula (2.11) in (2.28) and using (2.29), we obtain
∂u −λτ (μ1 + μ2 e )λuwdΓ + = μ1 f wdΓ − μ2 z0 wdΓ, ;w ∂ν ΓN ΓN ΓN ΓN from which follows (2.30)
∂u + (μ1 + μ2 e−λτ )λu = μ1 f − μ2 z0 ∂ν
on ΓN .
Therefore, from (2.30), ∂u = −μ1 v − μ2 z(·, 1) ∂ν
on ΓN ,
where we have used (2.20) and (2.24). So, we have found (u, v, z)T ∈ D(A), which verifies (2.19). Now, the well-posedness result follows from the Hille–Yosida theorem. 2.2. Internal feedback. Setting (2.31)
z(x, ρ, t) = ut (x, t − τ ρ),
x ∈ Ω, ρ ∈ (0, 1), t > 0,
problem (1.12)–(1.16) is equivalent to (2.32) (2.33) (2.34) (2.35) (2.36) (2.37) (2.38)
utt − Δu + a(x)[μ1 ut (x, t) + μ2 ut (x, t − τ )] = 0 in Ω × (0, +∞), τ zt (x, ρ, t) + zρ (x, ρ, t) = 0 in Ω × (0, 1) × (0, +∞), u(x, t) = 0 on ΓD × (0, +∞), ∂u (x, t) = 0 on ΓN × (0, +∞), ∂ν z(x, 0, t) = ut (x, t) on ΓN × (0, +∞), u(x, 0) = u0 (x) and ut (x, 0) = u1 (x) in z(x, ρ, 0) = g0 (x, −ρτ ) in Ω × (0, 1).
Ω,
If we denote by T
U := (u, ut , z) , then T T U := (ut , utt , zt ) = ut , Δu − a(μ1 ut + μ2 z(·, 1, ·)), −τ −1 zρ . Therefore, problem (2.32)–(2.38) can be rewritten as U = A0 U, (2.39) T U (0) = (u0 , u1 , g0 (·, − · τ )) ,
STABILIZATION OF THE WAVE EQUATION WITH DELAY
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where the operator A0 is defined by ⎛ ⎞ ⎛ ⎞ u v A0 ⎝ v ⎠ := ⎝ Δu − aμ1 v − aμ2 z(·, 1) ⎠ , z −τ −1 zρ with domain (2.40)
D(A0 ) :=
(u, v, z)T ∈ H 2 (Ω) ∩ HΓ1D (Ω) × H 1 (Ω) × L2 (Ω; H 1 (0, 1)) : ∂u = 0 on ΓN ; v = z(·, 0) in Ω . ∂ν
Denote by H0 the Hilbert space H0 := HΓ1D (Ω) × L2 (Ω) × L2 (Ω × (0, 1)),
(2.41)
equipped with the inner product (2.42) ⎞ ⎛ u ⎞ ⎛ u 1 ˜ ⎝ v ⎠ , ⎝ v˜ ⎠ := {∇u(x)∇˜ u(x)+v(x)˜ v (x)}dx+ξ z(x, ρ)˜ z (x, ρ)dρdx, Ω Ω 0 z z˜ H0 where ξ is a fixed positive number satisfying (2.14). Arguing analogously to the previous case, we can show that the operator A0 generates a C0 semigroup on H0 . Consequently we have the following well-posedness result. Theorem 2.2. For any initial datum U0 ∈ H0 there exists a unique solution U ∈ C([0, +∞), H0 ) of problem (2.39). Moreover, if U0 ∈ D(A0 ), then U ∈ C([0, +∞), D(A0 )) ∩ C 1 ([0, +∞), H0 ). 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 (1.10). 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 {u2t (x, t) + u2t (x, t − τ )}dΓ. ΓN
Proof. Differentiating (1.9), we obtain E (t) = {ut utt + ∇u∇ut }dx + ξ Ω
ΓN
1
ut (x, t − τ ρ)utt (x, t − τ ρ)dρdΓ, 0
and then, applying Green’s formula, 1 ∂u (3.2) E (t) = ut dΓ + ξ ut (x, t − τ ρ)utt (x, t − τ ρ)dρdΓ. ∂ν ΓN ΓN 0 Now, observe that ut (x, t − τ ρ) = −τ −1 uρ (x, t − τ ρ)
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SERGE NICAISE AND CRISTINA PIGNOTTI
and utt (x, t − τ ρ) = τ −2 uρρ (x, t − τ ρ). Therefore, (3.3)
1
ut (x, t− τ ρ)utt (x, t− τ ρ)dρdΓ = −τ ΓN
−3
uρ (x, t− τ ρ)uρρ (x, t− τ ρ)dρdΓ.
0
0
ΓN
Integrating by parts in ρ, we obtain 1 uρ (x, t − τ ρ)uρρ (x, t − τ ρ)dρdΓ = − ΓN
1
uρ (x, t − τ ρ)uρρ (x, t − τ ρ)dρdΓ
0
1
0
ΓN
{u2ρ (x, t − τ ) − u2ρ (x, t)}dΓ,
+ ΓN
that is, (3.4)
1
uρ (x, t − τ ρ)uρρ (x, t − τ ρ)dρdΓ =
ΓN
0
1 2
{u2ρ (x, t − τ ) − u2ρ (x, t)}dΓ. ΓN
Then, from (3.3) and (3.4), (3.5)
1
ut (x, t − τ ρ)utt (x, t − τ ρ)dρdΓ = −τ ΓN
0
=
τ −3 2
−3
uρ (x, t − τ )uρρ (x, t − τ ρ)dρdΓ ΓN
{u2ρ (x, t) − u2ρ (x, t − τ )}dΓ = ΓN
1
τ −1 2
0
{u2t (x, t) − u2t (x, t − τ )}dΓ. ΓN
Using (3.2), (3.5), and the boundary condition (1.3) on ΓN , we have E (t) = −μ1 u2t (x, t)dΓ − μ2 ut (x, t)ut (x, t − τ )dΓ (3.6)
ΓN −1
ξτ + 2
ΓN
u2t (x, t)dΓ ΓN
−1
ξτ − 2
u2t (x, t − τ )dΓ. ΓN
From (3.6), applying the Cauchy–Schwarz inequality we obtain μ2 μ2 ξτ −1 ξτ −1 2 E (t) ≤ −μ1 + ut (x, t)dΓ + u2t (x, t − τ )dΓ, + − 2 2 2 2 ΓN ΓN which implies E (t) ≤ −C
{u2t (x, t) + u2t (x, t − τ )}dΓ, ΓN
with
C = min
μ1 −
μ2 ξτ −1 μ2 ξτ −1 , − . − + 2 2 2 2
Since ξ is chosen satisfying assumption (1.10), the constant C is positive.
STABILIZATION OF THE WAVE EQUATION WITH DELAY
1571
We can write E(t) = E(t) + EN (t), where E(t) is the standard energy for the wave equation 1 (3.7) E(t) := {u2 (x, t) + |∇u(x, t)|2 }dx, 2 Ω t and (3.8)
EN (t) :=
ξ 2
1
u2t (x, t − τ ρ)dρdΓ. 0
ΓN
With a change of variable we can rewrite t ξ (3.9) EN (t) = u2 (x, s)dsdΓ. 2τ ΓN t−τ t We can now give a boundary observability inequality which we will use to prove the exponential decay of the energy E(t). Proposition 3.2. There exists a time T > 0 such that for all times T > T , there exists a positive constant C0 (depending on T ) for which (3.10)
T
E(0) ≤ C0
{u2t (x, t) + u2t (x, t − τ )}dΓdt 0
ΓN
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 T0 , and any ε > 0, we have (3.11)
T
∂u 2
E(0) ≤ c 0
ΓN
∂ν
+ u2t dΓdt + cuH 1/2+ε (Ω×(0,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 2 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 4.2 below). Then, by (3.11) and the boundary condition (1.3), we have (3.12)
T
E(0) ≤ c
{u2t (x, t) + u2t (x, t − τ )}dΓdt + cuH 1/2+ε (Ω×(0,T )) 0
ΓN
for a suitable positive constant c. From (3.9) we have that
(3.13)
EN (0) ≤ c
0
−τ
ΓN
u2t (x, s)dsdΓ.
By a change of variable in (3.13) we obtain, for T ≥ τ, (3.14)
T
EN (0) ≤ c
u2t (x, t − τ )dΓdt. 0
ΓN
1572
SERGE NICAISE AND CRISTINA PIGNOTTI
Denote T := max{τ, T0 }. Then, from (3.12) and (3.14), for any T > T we have
(3.15)
E(0) = E(0) + EN (0) T ≤c {u2t (x, t) + u2t (x, t − τ )}dΓdt + cuH 1/2+ε (Ω×(0,T )) 0
ΓN
for a suitable positive constant c depending on T. In order to obtain (3.10) we need to absorb the lower order term uH 1/2+ε (Ω×(0,T )) . To do this, we argue by contradiction. Suppose that (3.10) is not true. Then, there exists a sequence {un }n of solutions of problem (1.1)–(1.5) such that, denoting by E n (0) the energy E related to un at the time 0, T n {u2nt (x, t) + u2nt (x, t − τ )}dΓdt. (3.16) E (0) > n 0
ΓN
From (3.15), we have
T
(3.17) E n (0) ≤ c
{u2nt (x, t) + u2nt (x, t − τ )}dΓdt + un H 1/2+ε (Ω×(0,T )) 0
.
ΓN
Then, from (3.16) and (3.17), T n {u2nt (x, t) + u2nt (x, t − τ )}dΓdt 0 ΓN T 2 2 T0 , there exists a positive constant C1 (depending on T ) for which T (4.12) Ew (0) ≤ C1 wt2 (x, t)dxdt ω
0
for any regular solution w of problem (4.7)–(4.10). Proof. Inequality (4.12) easily follows from some estimates of [14] and standard arguments with multipliers. We give the proof for the reader’s convenience. Let ω0 , ω1 be open neighborhoods of ΓN such that ω ⊃⊃ ω0 ⊃⊃ ω1 ⊃ ΓN .
(4.13)
Let ϕ be a smooth function such that (4.14)
0 ≤ ϕ(x) ≤ 1,
ϕ≡0
on Ω \ ω0 ,
ϕ≡1
on ω1 .
Then, the function ϕw verifies (ϕw)tt − Δ(ϕw) = F (w), where F (w) = wΔϕ + 2∇ϕ∇w, with the same boundary conditions as w. Therefore, we can apply to ϕw the result of Proposition 4.2.1 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 Ω. Denote 1/2 maxx∈Ω v(x) (4.15) T0 = 2 , α with α as in (1.6). Define the function φ : Ω × R → R by 2 T , (4.16) φ(x, t) := v(x) − c∗ t − 2 where T > T0 is fixed and the constant c∗ is chosen as follows. From (4.15), there exists a constant δ > 0 such that αT 2 > 4 max v(x) + 4δ. x∈Ω
For fixed δ, there is c∗ such that (4.17)
c∗ T 2 > 4 max v(x) + 4δ, x∈Ω
c∗ ∈ (0, α).
Note that (4.18) Set
φ(x, 0) < −δ
and φ(x, T ) < −δ
Ω.
2 1 T ∂w eγφ H · ν dΓdt 2 0 ΓD ∂ν 1 T + eγφ H · ν(wt2 − |∇T w|2 )dΓdt, 2 0 ΓN
BT w|Γ×(0,T ) = (4.19)
uniformly in
where ∇T w denotes the tangential gradient of w.
1577
STABILIZATION OF THE WAVE EQUATION WITH DELAY
Then, from Proposition 4.2.1 of [14], using (4.18), and recalling that Ew (t) is constant, we have T T γφ 2 e |∇(ϕw)| dxdt + ϕ2 wt2 dxdt BT w|Γ×(0,T ) ≤ c 0 0 Ω Ω T
ϕ2 w2 dxdt + e−γδ Ew (0)
+
(4.20)
T
0
≤c
Ω
T
e |∇w| dxdt + γφ
0
2
ω0
T
wt2 dxdt 0
ω −γδ
2
+
w dxdt + e 0
Ew (0)
Ω
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 0 ≤ ψ(x) ≤ 1,
(4.21)
ψ≡0
on Ω \ ω,
ψ≡1
on ω0 .
Integrating by parts, T T (wtt − Δw)ψweγφ dxdt = ψwwt eγφ dx 0
Ω
T
Ω
T
∇w∇(ψweγφ )dxdt −
+ 0
Ω
(4.22)
T
T
wt (ψweγφ )t dxdt 0
−
ψwwt eγφ dx
=
Ω T
Ω
ψwt (wt eγφ + weγφ γφt )dxdt 0
0
Ω T
ψe |∇w| dxdt + γφ
+ 0
0
w∇w∇(ψeγφ )dxdt.
2
0
Ω
Ω
Then, from (4.22), recalling that w satisfies (4.7), we have T T γφ 2 ψe |∇w| dxdt = (ψwt2 eγφ + wwt ψeγφ γφt )dxdt 0
(4.23)
Ω
−
T
Ω
T
−2
ψwwt eγφ dx Ω
0
0
0
T
−
ψw∇w∇( ψ)eγφ dxdt
Ω
ψw∇w∇eγφ dxdt. 0
Ω
Since Ew (t) is constant, by using the Cauchy–Schwarz inequality and Poincar´e’s theorem, we can estimate T ψwwt eγφ dx ≤ ce−δγ Ew (0), Ω
0
and so, from (4.23), we obtain T 1 T γφ 2 −δγ ψe |∇w| dxdt ≤ ce Ew (0) + ψeγφ |∇w|2 dxdt 2 0 Ω 0 Ω T T (4.24) 2 2 +c wt dxdt + w dxdt 0
ω
0
Ω
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SERGE NICAISE AND CRISTINA PIGNOTTI
for a suitable positive constant c. By (4.24) we deduce T T T γφ 2 2 2 −δγ e |∇w| dxdt ≤ c wt dxdt + w dxdt + e Ew (0) , 0
ω0
ω
0
Ω
0
which, used in (4.20), gives (4.25)
T
BT w|Γ×(0,T ) ≤ c
T
wt2 dxdt + ω
0
0
w2 dxdt + e−δγ Ew (0).
Ω
Then, from (4.25) and Theorem 3.4 of [14] (Carleman estimate (3.14)), taking γ sufficiently large, we obtain
T
Ew (0) ≤ c
wt2 (x, t)dxdt + cw2L2 (Ω×(0,T )) . ω
0
Now, estimate (4.12) 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 C0 (depending on T ) for which
T
F(0) ≤ C0
(4.26)
a(x){u2t (x, t) + u2t (x, t − τ )}dxdt 0
Ω
for any regular solution u of problem (1.12)–(1.16). Proof. Following Zuazua [18], we can decompose the solution u of problem (1.12)– (1.16) as u = w + w, ˜ where w solves (4.7)–(4.9) with initial condition w(x, 0) = u0 (x),
wt (x, 0) = u1 (x)
in Ω,
and w ˜ verifies (4.27) (4.28) (4.29) (4.30)
˜ = −a(x)[μ1 ut (x, t) + μ2 ut (x, t − τ )] w ˜tt − Δw w(x, ˜ t) = 0 on ΓD × (0, +∞), ∂w ˜ (x, t) = 0 on ΓN × (0, +∞), ∂ν w(x, ˜ 0) = 0 and w ˜t (x, 0) = 0 in Ω.
in
Ω × (0, +∞),
Then, from (4.5) and (4.11), (4.31)
F(0) = E(0) + E 0 (0) = Ew (0) +
ξ 2
Ω
1
u2t (x, −ρτ )dρdx.
a(x) 0
If we take T > T := max{T0 , τ }, from (4.31) with a change of variable we obtain F(0) ≤ Ew (0) + c
Ω
T
u2t (x, t − τ )dtdx,
a(x) 0
STABILIZATION OF THE WAVE EQUATION WITH DELAY
1579
and then, from (4.12),
F(0) ≤ c (4.32)
Ω
a(x)
≤c
T
0
{wt2 (x, t) + u2t (x, t − τ )}dtdx T
{w ˜t2 (x, t) + u2t (x, t) + u2t (x, t − τ )}dtdx
a(x) 0
Ω
for a suitable positive constant c. Therefore, from standard energy estimates for w, ˜ we obtain T F(0) ≤ C0 a(x){u2t (x, t) + u2t (x, t − τ )}dxdt. 0
Ω
Now, using estimate (4.26), 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 (1.12) is replaced with k utt (x, t) − Δu(x, t) + a(x) μ0 ut (x, t) + μi ut (x, t − τi ) = 0
in Ω × (0, +∞),
i=1
with μ0 , μi , τi , i = 1, . . . , k, positive parameters. In this case, if μ0 >
k
μi ,
i=1
the right energy to consider, in order to prove exponential decay, is E(t) :=
1 2
{u2t (x, t) + |∇u(x, t)|2 }dx + Ω
k ξi i=1
2
Ω
1
u2t (x, t − ρτi )dρdx,
a(x) 0
with constants ξi , i = 1, . . . , k, chosen as in Remark 3.3. Remark 4.5. In the case ΓD ∩ΓN = ∅, since for internal feedbacks we have ∂u ∂ν = 0 on ΓN , 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 μ2 ≥ μ1 . 5.1. Boundary feedback. In this subsection we consider problem (1.1)–(1.5) with boundary feedback, and we prove Theorem 1.2. Let us consider the spectral problem for the system ⎧ ⎪ ⎨ utt (x, t) − Δu(x, t) = 0 in Ω × (0, +∞), u(x, t) = 0 on ΓD × (0, +∞), (5.1) ⎪ ⎩ ∂u (x, t) = −μ1 ut (x, t) − μ2 ut (x, t − τ ) on ΓN × (0, +∞). ∂ν We seek a solution of (5.1) in the form u(x, t) = eλt ϕ(x),
λ ∈ C.
1580
SERGE NICAISE AND CRISTINA PIGNOTTI
Then, ϕ has to be a solution of the eigenvalue problem ⎧ 2 ⎪ ⎨ −Δϕ + λ ϕ = 0 in Ω, ϕ = 0 on ΓD , (5.2) ⎪ ⎩ ∂ϕ = −(μ1 + μ2 e−λτ )λϕ on ΓN , ∂ν which can be reformulated, in a variational form, as 2 −λτ ∇ϕ∇vdx + λ ϕvdx + (μ1 + μ2 e )λ (5.3) Ω
Ω
ϕvdΓ = 0 ∀v ∈ HΓ1D (Ω).
ΓN
We want to find a solution for λ := ib, with b ∈ R. For this choice of λ, problem (5.3) can be rewritten as 2 −ibτ (5.4) ∇ϕ∇vdx − b ϕvdx + (μ1 + μ2 e )ib ϕvdΓ = 0 ∀v ∈ HΓ1D (Ω). Ω
Ω
ΓN
Assume that cos(bτ ) = −
(5.5)
μ1 . μ2
Note that, since we are considering the case μ2 ≥ μ1 , there exist b, τ such that (5.5) holds. Then, we choose ! (5.6) μ2 sin(bτ ) = μ22 − μ21 . Under these assumptions, (5.4) becomes ! 2 2 2 (5.7) ∇ϕ∇vdx − b ϕvdx + b μ2 − μ1 Ω
Ω
ϕvdΓ = 0 ∀v ∈ HΓ1D (Ω).
ΓN
In particular, for v = ϕ, (5.7) gives ! 2 2 2 2 2 (5.8) |∇ϕ| dx − b |ϕ| dx + b μ2 − μ1 Ω
Ω
|ϕ|2 dΓ = 0.
ΓN
Without loss of generality, we can assume |ϕ|2 dx = 1, (5.9) ϕ22 := Ω
and then the identity (5.8) can be rewritten as ! (5.10) b2 − b μ22 − μ21 q0 (ϕ) − q1 (ϕ) = 0, where (5.11)
|ϕ|2 dΓ,
q0 (ϕ) :=
|∇ϕ|2 dx.
q1 (ϕ) :=
ΓN
Ω
Now we distinguish two cases. Case (a): μ1 = μ2 . In this case, under our assumptions, (5.10) becomes (5.12)
b2 = q1 (ϕ).
STABILIZATION OF THE WAVE EQUATION WITH DELAY
1581
Define b2 :=
(5.13)
min
w∈H 1 (Ω) ΓD w2 =1
q1 (w).
If ϕ verifies q1 (ϕ) =
min
q1 (w),
w∈H 1 (Ω) ΓD w2 =1
then it easy to see that ϕ is a solution of (5.4) with b as in (5.13). Then ϕ verifies (5.2), and so u(x, t) := eibt ϕ(x)
(5.14)
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)|2 + |ut (x, t)|2 )dx = 2b2 > 0 ∀t ≥ 0. Ω
Note that, from our assumptions (λ = ib, cos(bτ ) = −1, sin(bτ ) = 0), problem (5.2) becomes the classical eigenvalue problem for the Laplace operator with a mixed Dirichlet–Neumann boundary condition. So, we can take a sequence {bn }n of positive real numbers defined by b2n = Λ2n ,
n ∈ N,
where Λ2n , n ∈ N, are the eigenvalues for the Laplace operator. Then, putting l ∈ N,
bn τ = (2l + 1)π, we obtain a sequence of delays τn,l =
(2l + 1)π , bn
l, n ∈ N,
which become arbitrarily small (or large) for suitable choices of the indices n, l ∈ N. Therefore, in the case μ1 = μ2 , we have found a set of time delays for which problem (1.1)–(1.5) is not asymptotically stable. Case (b): μ2 > μ1 . In this case, from (5.10) we have ! ! 1 b= μ22 − μ21 q0 (ϕ) ± (μ22 − μ21 )q02 (ϕ) + 4q1 (ϕ) . 2 Define (5.15)
b :=
1 2
! min 1
w∈H
(Ω) ΓD w2 =1
μ22 − μ21 q0 (w) +
!
(μ22 − μ21 )q02 (w) + 4q1 (w) .
We now prove that if the minimum in the right–hand side of (5.15) is attained at ϕ, that is,
1582 (5.16)!
SERGE NICAISE AND CRISTINA PIGNOTTI
μ22 − μ21 q0 (ϕ) + :=
! (μ22 − μ21 )q02 (ϕ) + 4q1 (ϕ) ! ! 2 − μ2 q (w) + 2 − μ2 )q 2 (w) + 4q (w) , min μ (μ 1 2 1 0 2 1 0 1
w∈H
(Ω) ΓD w2 =1
then ϕ is a solution of (5.7) with b as in (5.15). To show this, take for ε ∈ R, (5.17)
with v ∈ HΓ1D (Ω) such that
w = ϕ + εv,
ϕvdx = 0. Ω
Then, w22 = ϕ22 + ε2 v22 = 1 + ε2 v22 . If we denote (5.18) g(ε) :=
1 1 + ε2 v22
!
μ22 − μ21 q0 (ϕ + εv) +
!
(μ22 − μ21 )q02 (ϕ + εv) + 4q1 (ϕ + εv) ,
then, by definition (5.16), g(ε) ≥ g(0) =
! ! μ22 − μ21 q0 (ϕ) + (μ22 − μ21 )q02 (ϕ) + 4q1 (ϕ) .
So, we have that " dg(ε) "" = 0, dε "ε=0 which, after an easy computation, gives ∇ϕ∇vdx + b
(5.19) Ω
! μ22
−
μ21
ϕvdΓ = 0. ΓN
Since any function v˜ ∈ HΓ1D (Ω) can be decomposed as v˜ = γϕ + v,
γ ∈ R, v ∈ HΓ1D (Ω) with
ϕvdx = 0, Ω
from (5.19) and (5.8) we obtain that ϕ satisfies (5.7) with b defined in (5.15). So, for such positive b, μ1 bτ = arccos − + 2lπ, μ2
l ∈ N,
defines a sequence of time delays for which problem (1.1)–(1.5) is not asymptotically stable. The above examples prove Theorem 1.2.
STABILIZATION OF THE WAVE EQUATION WITH DELAY
1583
5.2. Internal feedback. In this subsection we will give instability examples for problem (1.12)–(1.16) with internal feedback, proving Theorem 1.4. Let us consider the spectral problem for the system (5.20) ⎧ ⎪ utt (x, t) − Δu(x, t) + a(x)[μ1 ut (x, t) + μ2 ut (x, t − τ )] = 0 ⎨ u(x, t) = 0 on ΓD × (0, +∞), ⎪ ⎩ ∂u (x, t) = 0 on ΓN × (0, +∞). ∂ν
in Ω × (0, +∞),
We restrict our analysis to the case a(x) ≡ 1 in Ω. We seek a solution of (5.20) in the form u(x, t) = eλt ϕ(x),
λ ∈ C.
Then, ϕ has to solve the eigenvalue problem ⎧ 2 −λτ )λ]ϕ ⎪ ⎨ Δϕ = [λ + (μ1 + μ2 e ϕ = 0 on Γ , D (5.21) ⎪ ⎩ ∂ϕ = 0 on ΓN . ∂ν
in Ω,
Let us consider the standard problem for the Laplace operator with a mixed Dirichlet–Neumann boundary condition ⎧ 2 ⎪ ⎨ Δϕ = −μ ϕ in Ω, ϕ = 0 on ΓD , (5.22) ⎪ ⎩ ∂ϕ = 0 on ΓN . ∂ν We want to show that for any Λ2 eigenvalue of problem (5.22), there exists a λ ∈ C solution of the equation (5.23)
λ2 + (μ1 + μ2 e−λτ )λ = −Λ2 .
We seek a solution λ = α + iβ, α, β ∈ R, with (5.24)
βτ = (2l + 1)π,
l ∈ N.
Under this assumption, (5.23) becomes 2 α + β 2 = Λ2 , (5.25) μ2 e−ατ = 2α + μ1 . Now we distinguish two cases. Case (a): μ1 = μ2 . In this case, from (5.25) we have α = 0,
β 2 = Λ2 .
Therefore, for any Λ2n eigenvalue of problem (5.22), if βn ∈ R verifies βn2 = Λ2n , then for λ = iβn problem (5.21) admits a nonzero solution.
1584
SERGE NICAISE AND CRISTINA PIGNOTTI
Take βn positive. From our assumption (5.24), τn,l =
(2l + 1)π , βn
n, l ∈ 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 (1.12)–(1.16) admits solutions in the form u(x, t) = eiβt ϕ(x), whose energy is constant and strictly positive. So, system (1.12)–(1.16) is not asymptotically stable. Case (b): μ2 > μ1 . For a fixed α > 0, from the second equation of (5.25), we obtain μ2 1 , (5.26) τ (α) = ln α μ1 + 2α and so, in order to have τ (α) > 0, we consider 0
c 2006 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. 35L05, 93D15 DOI. 10.1137/060648891
1. Introduction. We investigate the effect of time delay in boundary or internal stabilization of the wave equation in domains of Rn . 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 Ω ⊂ Rn be an open bounded set with a boundary Γ of class C 2 . We assume that Γ is divided into two parts ΓD and ΓN , i.e., Γ = ΓD ∪ ΓN , with ΓD ∩ ΓN = ∅ and ΓD = ∅. In this domain Ω, we consider the initial boundary value problem (1.1) (1.2) (1.3) (1.4) (1.5)
utt (x, t) − Δu(x, t) = 0 in Ω × (0, +∞), u(x, t) = 0 on ΓD × (0, +∞), ∂u (x, t) = −μ1 ut (x, t) − μ2 ut (x, t − τ ) on ΓN × (0, +∞), ∂ν u(x, 0) = u0 (x) and ut (x, 0) = u1 (x) in Ω, ut (x, t − τ ) = f0 (x, t − τ ) in ΓN × (0, τ ),
where ν(x) denotes the outer unit normal vector to the point x ∈ Γ and ∂u ∂ν is the normal derivative. Moreover, τ > 0 is the time delay, μ1 and μ2 are positive real numbers, and the initial data (u0 , u1 , f0 ) belong to a suitable space. We are interested in giving an exponential stability result for such a problem. ∗ Received by the editors January 3, 2006; accepted for publication (in revised form) May 24, 2006; published electronically November 14, 2006. http://www.siam.org/journals/sicon/45-5/64889.html † Universit´ e de Valenciennes et du Hainaut Cambr´esis, MACS, Institut des Sciences et Techniques de Valenciennes, 59313 Valenciennes Cedex 9, France ([email protected]). ‡ Dipartimento di Matematica Pura e Applicata, Universit` a di L’Aquila, Via Vetoio, Loc. Coppito, 67010 L’Aquila, Italy ([email protected]).
1561
1562
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 ∈ Rn . We assume that there exists a scalar function v ∈ C 2 (Ω) such that (i) v is strictly convex in Ω; that is, there exists α > 0 such that (1.6)
D2 (v)(x)ξ, ξ ≥ 2α|ξ|2
∀x ∈ Ω, ∀ξ ∈ Rn ,
where D2 (v) denotes the Hessian matrix of v; (ii) the vector field H := ∇v verifies (1.7)
H(x) · ν(x) ≤ 0
∀x ∈ ΓD .
For the above assumptions see [14], where some observability estimates for secondorder hyperbolic equations are given. It is well known that if μ2 = 0, that is, in absence of delay, the energy of problem (1.1)–(1.5) is exponentially decaying to zero. See for instance Chen [2, 3], Lagnese [11, 12], Lasiecka and Triggiani [13], Komornik and Zuazua [10], and Komornik [8, 9]. On the contrary, if μ1 = 0, that is, if we have only the delay part in the boundary condition on ΓN , 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 μ1 and μ2 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 μ2 < μ1 . In their paper it is also shown that if μ2 > μ1 , the system is unstable and if μ1 = μ2 , some instabilities may occur. Here, in agreement with [17] and assuming that (1.8)
μ2 < μ1 ,
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 μ1 = μ2 , we further show that there exists a sequence of arbitrary small (and large) delays such that instabilities occur. In the case μ2 > μ1 , 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 ξ 1 {u2t (x, t) + |∇u(x, t)|2 }dx + u2 (x, t − τ ρ)dρdΓ, (1.9) E(t) := 2 Ω 2 ΓN 0 t where ξ is a positive constant verifying (1.10)
τ μ2 < ξ < τ (2μ1 − μ2 ).
Clearly this energy is larger than the standard energy 1 {u2 (x, t) + |∇u(x, t)|2 }dx 2 Ω t and contains an additional term that comes from the delay term.
STABILIZATION OF THE WAVE EQUATION WITH DELAY
1563
Theorem 1.1. Assume that (1.8) holds. There exist positive constants C1 , C2 such that, for any solution of problem (1.1)–(1.5), (1.11)
E(t) ≤ C1 E(0)e−C2 t
∀t ≥ 0.
Theorem 1.2. 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 (1.12) utt (x, t) − Δu(x, t) + a(x)[μ1 ut (x, t) + μ2 ut (x, t − τ )] = 0 in Ω × (0, +∞), (1.13) u(x, t) = 0 on ΓD × (0, +∞), ∂u (1.14) (x, t) = 0 on ΓN × (0, +∞), ∂ν (1.15) u(x, 0) = u0 (x) and ut (x, 0) = u1 (x) in Ω, (1.16) ut (x, t − τ ) = g0 (x, t − τ ) in Ω × (0, τ ), where a ∈ L∞ (Ω) is a function such that (1.17)
a(x) ≥ 0
a. e.
in
Ω,
and (1.18)
a(x) > a0 > 0
a. e.
in ω,
where ω ⊂ Ω is an open neighborhood of ΓN . Exponential stability results for the above problem in the case of μ2 = 0, 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 μ2 < μ1 , 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 μ2 ≥ μ1 , 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 (1.12)– (1.16) as 1 1 ξ 2 2 (1.19) F(t) := {u (x, t) + |∇u(x, t)| }dx + a(x) u2t (x, t − τ ρ)dρdx, 2 Ω t 2 Ω 0 where ξ is a positive constant verifying (1.10). 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 C1 , C2 such that, for any solution of problem (1.12)–(1.16), (1.20)
F(t) ≤ C1 F(0)e−C2 t
∀t ≥ 0.
Theorem 1.4. If (1.8) does not hold, there exist a sequence of arbitrary small (or large) delays, and solutions of problem (1.12)–(1.16) corresponding to these delays, such that their standard energy does not tend to 0.
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 2 using semigroup theory. In subsection 2.1 we study the well-posedness of problem (1.1)–(1.5), while in subsection 2.2 we concentrate on problem (1.12)–(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. 2. Well-posedness of the problems. In this section we will give well-posedness results for problem (1.1)–(1.5) and problem (1.12)–(1.16) using semigroup theory. 2.1. Boundary feedback. Let us set (2.1)
z(x, ρ, t) = ut (x, t − τ ρ),
x ∈ ΓN , ρ ∈ (0, 1), t > 0.
Then, problem (1.1)–(1.5) is equivalent to
(2.7)
utt (x, t) − Δu(x, t) = 0 in Ω × (0, +∞), τ zt (x, ρ, t) + zρ (x, ρ, t) = 0 in ΓN × (0, 1) × (0, +∞), u(x, t) = 0 on ΓD × (0, +∞), ∂u (x, t) = −μ1 ut (x, t) − μ2 z(x, 1, t) on ΓN × (0, +∞), ∂ν z(x, 0, t) = ut (x, t) on ΓN × (0, ∞), u(x, 0) = u0 (x) and ut (x, 0) = u1 (x) in Ω,
(2.8)
z(x, ρ, 0) = f0 (x, −ρτ )
(2.2) (2.3) (2.4) (2.5) (2.6)
in
ΓN × (0, 1).
If we denote T
U := (u, ut , z) , then T T U := (ut , utt , zt ) = ut , Δu, −τ −1 zρ . Therefore, problem (2.2)–(2.8) can be rewritten as (2.9)
U = AU, T U (0) = (u0 , u1 , f0 (·, − · τ )) ,
where the operator A is defined by ⎞ ⎞ ⎛ u v ⎠, A ⎝ v ⎠ := ⎝ Δu z −τ −1 zρ ⎛
STABILIZATION OF THE WAVE EQUATION WITH DELAY
1565
with domain (2.10)
D(A) := (u, v, z)T ∈ E(Δ, L2 (Ω)) ∩ HΓ1D (Ω) × H 1 (Ω) × L2 (ΓN ; H 1 (0, 1)) : ∂u = −μ1 v − μ2 z(·, 1) on ΓN ; v = z(·, 0) on ΓN , ∂ν where, as usual, HΓ1D (Ω) = { u ∈ H 1 (Ω) : u = 0
on ΓD },
and E(Δ, L2 (Ω)) = {u ∈ H 1 (Ω) : Δu ∈ L2 (Ω)}. −1/2 (ΓN ) and the next Recall that for a function u ∈ E(Δ, L2 (Ω)), ∂u ∂ν belongs to H Green formula is valid (see section 1.5 of [7])
∂u ;w ∇u∇wdx = − Δuwdx + ∀w ∈ HΓ1D (Ω), (2.11) ∂ν Ω Ω ΓN
where ·; · ΓN means the duality pairing between H −1/2 (ΓN ) and H 1/2 (ΓN ). 2 Note further that for (u, v, z)T ∈ D(A), ∂u ∂ν belongs to L (ΓN ) since z(·, 1) is in 2 L (ΓN ). Denote by H the Hilbert space H := HΓ1D (Ω) × L2 (Ω) × L2 (ΓN × (0, 1)).
(2.12) Assuming that
μ2 ≤ μ1 ,
(2.13)
we will show that A generates a C0 semigroup on H. Let ξ be a positive real number such that τ μ2 ≤ ξ ≤ τ (2μ1 − μ2 ).
(2.14)
Note that, from (2.13), such a constant ξ exists. Let us define on the Hilbert space H the inner product (2.15) ⎞ ⎛ u ⎞ ⎛ u 1 ˜ ⎝ v ⎠ , ⎝ v˜ ⎠ := {∇u(x)∇˜ u(x)+v(x)˜ v (x)}dx+ξ z(x, ρ)˜ z (x, ρ)dρdΓ. Ω ΓN 0 z z˜ H Theorem 2.1. For any initial datum U0 ∈ H there exists a unique solution U ∈ C([0, +∞), H) of problem (2.9). Moreover, if U0 ∈ D(A), then U ∈ C([0, +∞), D(A)) ∩ C 1 ([0, +∞), H). Proof. Take U = (u, v, z)T ∈ D(A). Then ⎞ ⎛ ⎞ ⎛ v u ⎠,⎝ v ⎠ (AU, U ) = ⎝ Δu −1 z −τ zρ H −1 = {∇v(x)∇u(x) + v(x)Δu(x)}dx − ξτ Ω
ΓN
1
zρ (x, ρ)z(x, ρ)dρdΓ. 0
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SERGE NICAISE AND CRISTINA PIGNOTTI
So, by Green’s formula, (2.16)
(AU, U ) = ΓN
∂u (x)v(x)dΓ − ξτ −1 ∂ν
1
zρ (x, ρ)z(x, ρ)dρdΓ. ΓN
0
Integrating by parts in ρ, we get
1
1
zρ (x, ρ)z(x, ρ)dρdΓ = − ΓN
0
ΓN
zρ (x, ρ)z(x, ρ)dρdΓ+
0
{z 2 (x, 1)−z 2 (x, 0)}dΓ,
ΓN
that is
(2.17)
1
zρ (x, ρ)z(x, ρ)dρdΓ = ΓN
0
1 2
{z 2 (x, 1) − z 2 (x, 0)}dΓ. ΓN
Therefore, from (2.16) and (2.17), (AU, U ) = ΓN
=−
ξτ −1 ∂u (x)v(x)dΓ − ∂ν 2
{z 2 (x, 1) − z 2 (x, 0)}dΓ ΓN
(μ1 v(x) + μ2 z(x, 1))v(x)dΓ − ΓN
ξτ −1 2
{z 2 (x, 1) − z 2 (x, 0)}dΓ ΓN
ξτ −1 = −μ1 v 2 (x)dΓ − μ2 z(x, 1)v(x)dΓ − 2 ΓN ΓN
z 2 (x, 1)dΓ + ΓN
ξτ −1 2
v 2 (x)dΓ, ΓN
from which follows, using the Cauchy–Schwarz inequality, (2.18) ξτ −1 μ2 μ2 ξτ −1 2 − (AU, U ) ≤ −μ1 + v (x)dΓ + z 2 (x, 1)dΓ. + 2 2 2 2 ΓN ΓN Now, observe that from (2.14), −μ1 +
ξτ −1 μ2 + ≤ 0, 2 2
μ2 ξτ −1 − ≤ 0. 2 2
Then, (AU, U ) ≤ 0, which means that the operator A is dissipative. Now, we will show that λI −A is surjective for a fixed λ > 0. Given (f, g, h)T ∈ H, we seek a U = (u, v, z)T ∈ D(A) solution of ⎛ ⎞ ⎛ ⎞ u f (λI − A) ⎝ v ⎠ = ⎝ g ⎠ , z h that is, verifying (2.19)
⎧ ⎨ λu − v = f, λv − Δu = g, ⎩ λz + τ −1 zρ = h.
Suppose that we have found u with the appropriate regularity. Then, (2.20)
v := λu − f
STABILIZATION OF THE WAVE EQUATION WITH DELAY
1567
and we can determine z. Indeed, by (2.10), (2.21)
for x ∈ ΓN ,
z(x, 0) = v(x)
and, from (2.19), (2.22)
λz(x, ρ) + τ −1 zρ (x, ρ) = h(x, ρ)
for x ∈ ΓN , ρ ∈ (0, 1).
Then, by (2.21) and (2.22), we obtain z(x, ρ) = v(x)e−λρτ + τ e−λρτ
ρ
h(x, σ)eλστ dσ. 0
So, from (2.20), −λρτ
(2.23) z(x, ρ) = λu(x)e
−λρτ
−f (x)e
−λρτ
ρ
h(x, σ)eλστ dσ
+τ e
on ΓN ×(0, 1),
0
and, in particular, z(x, 1) = λu(x)e−λτ + z0 (x),
(2.24)
x ∈ ΓN ,
with z0 ∈ L2 (ΓN ) defined by z0 (x) = −f (x)e−λτ + τ e−λτ
(2.25)
1
h(x, σ)eλστ dσ,
x ∈ ΓN .
0
By (2.20) and (2.19), the function u verifies λ(λu − f ) − Δu = g, that is, λ2 u − Δu = g + λf.
(2.26)
Problem (2.26) can be reformulated as (2.27) (λ2 u − Δu)wdx = (g + λf )wdx ∀w ∈ HΓ1D (Ω). Ω
Ω
Integrating by parts, (λ2 u − Δu)wdx = (λ2 uw + ∇u∇w)dx − Ω
Ω
ΓN
(λ2 uw + ∇u∇w)dx +
= Ω
(μ1 vw + μ2 z(x, 1))wdΓ ΓN
{μ1 (λu − f )w + μ2 (λue−λτ + z0 )w}dΓ,
(λ uw + ∇u∇w)dx + 2
=
∂u wdΓ ∂ν
Ω
ΓN
where we have used (2.20) and (2.24). Therefore, (2.27) can be rewritten as (λ2 uw + ∇u∇w)dx + (μ1 + μ2 e−λτ )λuwdΓ Ω ΓN (2.28) = (g + λf )wdx + μ1 f wdΓ − μ2 z0 wdΓ ∀w ∈ HΓ1D (Ω). Ω
ΓN
ΓN
1568
SERGE NICAISE AND CRISTINA PIGNOTTI
As the left-hand side of (2.28) is coercive on HΓ1D (Ω), the Lax–Milgram lemma guarantees the existence and uniqueness of a solution u ∈ HΓ1D (Ω) of (2.28). If we consider w ∈ D(Ω) in (2.28), we have that u solves in D (Ω) λ2 u − Δu = g + λf,
(2.29)
and thus u ∈ E(Δ, L2 (Ω)). Using Green’s formula (2.11) in (2.28) and using (2.29), we obtain
∂u −λτ (μ1 + μ2 e )λuwdΓ + = μ1 f wdΓ − μ2 z0 wdΓ, ;w ∂ν ΓN ΓN ΓN ΓN from which follows (2.30)
∂u + (μ1 + μ2 e−λτ )λu = μ1 f − μ2 z0 ∂ν
on ΓN .
Therefore, from (2.30), ∂u = −μ1 v − μ2 z(·, 1) ∂ν
on ΓN ,
where we have used (2.20) and (2.24). So, we have found (u, v, z)T ∈ D(A), which verifies (2.19). Now, the well-posedness result follows from the Hille–Yosida theorem. 2.2. Internal feedback. Setting (2.31)
z(x, ρ, t) = ut (x, t − τ ρ),
x ∈ Ω, ρ ∈ (0, 1), t > 0,
problem (1.12)–(1.16) is equivalent to (2.32) (2.33) (2.34) (2.35) (2.36) (2.37) (2.38)
utt − Δu + a(x)[μ1 ut (x, t) + μ2 ut (x, t − τ )] = 0 in Ω × (0, +∞), τ zt (x, ρ, t) + zρ (x, ρ, t) = 0 in Ω × (0, 1) × (0, +∞), u(x, t) = 0 on ΓD × (0, +∞), ∂u (x, t) = 0 on ΓN × (0, +∞), ∂ν z(x, 0, t) = ut (x, t) on ΓN × (0, +∞), u(x, 0) = u0 (x) and ut (x, 0) = u1 (x) in z(x, ρ, 0) = g0 (x, −ρτ ) in Ω × (0, 1).
Ω,
If we denote by T
U := (u, ut , z) , then T T U := (ut , utt , zt ) = ut , Δu − a(μ1 ut + μ2 z(·, 1, ·)), −τ −1 zρ . Therefore, problem (2.32)–(2.38) can be rewritten as U = A0 U, (2.39) T U (0) = (u0 , u1 , g0 (·, − · τ )) ,
STABILIZATION OF THE WAVE EQUATION WITH DELAY
1569
where the operator A0 is defined by ⎛ ⎞ ⎛ ⎞ u v A0 ⎝ v ⎠ := ⎝ Δu − aμ1 v − aμ2 z(·, 1) ⎠ , z −τ −1 zρ with domain (2.40)
D(A0 ) :=
(u, v, z)T ∈ H 2 (Ω) ∩ HΓ1D (Ω) × H 1 (Ω) × L2 (Ω; H 1 (0, 1)) : ∂u = 0 on ΓN ; v = z(·, 0) in Ω . ∂ν
Denote by H0 the Hilbert space H0 := HΓ1D (Ω) × L2 (Ω) × L2 (Ω × (0, 1)),
(2.41)
equipped with the inner product (2.42) ⎞ ⎛ u ⎞ ⎛ u 1 ˜ ⎝ v ⎠ , ⎝ v˜ ⎠ := {∇u(x)∇˜ u(x)+v(x)˜ v (x)}dx+ξ z(x, ρ)˜ z (x, ρ)dρdx, Ω Ω 0 z z˜ H0 where ξ is a fixed positive number satisfying (2.14). Arguing analogously to the previous case, we can show that the operator A0 generates a C0 semigroup on H0 . Consequently we have the following well-posedness result. Theorem 2.2. For any initial datum U0 ∈ H0 there exists a unique solution U ∈ C([0, +∞), H0 ) of problem (2.39). Moreover, if U0 ∈ D(A0 ), then U ∈ C([0, +∞), D(A0 )) ∩ C 1 ([0, +∞), H0 ). 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 (1.10). 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 {u2t (x, t) + u2t (x, t − τ )}dΓ. ΓN
Proof. Differentiating (1.9), we obtain E (t) = {ut utt + ∇u∇ut }dx + ξ Ω
ΓN
1
ut (x, t − τ ρ)utt (x, t − τ ρ)dρdΓ, 0
and then, applying Green’s formula, 1 ∂u (3.2) E (t) = ut dΓ + ξ ut (x, t − τ ρ)utt (x, t − τ ρ)dρdΓ. ∂ν ΓN ΓN 0 Now, observe that ut (x, t − τ ρ) = −τ −1 uρ (x, t − τ ρ)
1570
SERGE NICAISE AND CRISTINA PIGNOTTI
and utt (x, t − τ ρ) = τ −2 uρρ (x, t − τ ρ). Therefore, (3.3)
1
ut (x, t− τ ρ)utt (x, t− τ ρ)dρdΓ = −τ ΓN
−3
uρ (x, t− τ ρ)uρρ (x, t− τ ρ)dρdΓ.
0
0
ΓN
Integrating by parts in ρ, we obtain 1 uρ (x, t − τ ρ)uρρ (x, t − τ ρ)dρdΓ = − ΓN
1
uρ (x, t − τ ρ)uρρ (x, t − τ ρ)dρdΓ
0
1
0
ΓN
{u2ρ (x, t − τ ) − u2ρ (x, t)}dΓ,
+ ΓN
that is, (3.4)
1
uρ (x, t − τ ρ)uρρ (x, t − τ ρ)dρdΓ =
ΓN
0
1 2
{u2ρ (x, t − τ ) − u2ρ (x, t)}dΓ. ΓN
Then, from (3.3) and (3.4), (3.5)
1
ut (x, t − τ ρ)utt (x, t − τ ρ)dρdΓ = −τ ΓN
0
=
τ −3 2
−3
uρ (x, t − τ )uρρ (x, t − τ ρ)dρdΓ ΓN
{u2ρ (x, t) − u2ρ (x, t − τ )}dΓ = ΓN
1
τ −1 2
0
{u2t (x, t) − u2t (x, t − τ )}dΓ. ΓN
Using (3.2), (3.5), and the boundary condition (1.3) on ΓN , we have E (t) = −μ1 u2t (x, t)dΓ − μ2 ut (x, t)ut (x, t − τ )dΓ (3.6)
ΓN −1
ξτ + 2
ΓN
u2t (x, t)dΓ ΓN
−1
ξτ − 2
u2t (x, t − τ )dΓ. ΓN
From (3.6), applying the Cauchy–Schwarz inequality we obtain μ2 μ2 ξτ −1 ξτ −1 2 E (t) ≤ −μ1 + ut (x, t)dΓ + u2t (x, t − τ )dΓ, + − 2 2 2 2 ΓN ΓN which implies E (t) ≤ −C
{u2t (x, t) + u2t (x, t − τ )}dΓ, ΓN
with
C = min
μ1 −
μ2 ξτ −1 μ2 ξτ −1 , − . − + 2 2 2 2
Since ξ is chosen satisfying assumption (1.10), the constant C is positive.
STABILIZATION OF THE WAVE EQUATION WITH DELAY
1571
We can write E(t) = E(t) + EN (t), where E(t) is the standard energy for the wave equation 1 (3.7) E(t) := {u2 (x, t) + |∇u(x, t)|2 }dx, 2 Ω t and (3.8)
EN (t) :=
ξ 2
1
u2t (x, t − τ ρ)dρdΓ. 0
ΓN
With a change of variable we can rewrite t ξ (3.9) EN (t) = u2 (x, s)dsdΓ. 2τ ΓN t−τ t We can now give a boundary observability inequality which we will use to prove the exponential decay of the energy E(t). Proposition 3.2. There exists a time T > 0 such that for all times T > T , there exists a positive constant C0 (depending on T ) for which (3.10)
T
E(0) ≤ C0
{u2t (x, t) + u2t (x, t − τ )}dΓdt 0
ΓN
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 T0 , and any ε > 0, we have (3.11)
T
∂u 2
E(0) ≤ c 0
ΓN
∂ν
+ u2t dΓdt + cuH 1/2+ε (Ω×(0,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 2 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 4.2 below). Then, by (3.11) and the boundary condition (1.3), we have (3.12)
T
E(0) ≤ c
{u2t (x, t) + u2t (x, t − τ )}dΓdt + cuH 1/2+ε (Ω×(0,T )) 0
ΓN
for a suitable positive constant c. From (3.9) we have that
(3.13)
EN (0) ≤ c
0
−τ
ΓN
u2t (x, s)dsdΓ.
By a change of variable in (3.13) we obtain, for T ≥ τ, (3.14)
T
EN (0) ≤ c
u2t (x, t − τ )dΓdt. 0
ΓN
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SERGE NICAISE AND CRISTINA PIGNOTTI
Denote T := max{τ, T0 }. Then, from (3.12) and (3.14), for any T > T we have
(3.15)
E(0) = E(0) + EN (0) T ≤c {u2t (x, t) + u2t (x, t − τ )}dΓdt + cuH 1/2+ε (Ω×(0,T )) 0
ΓN
for a suitable positive constant c depending on T. In order to obtain (3.10) we need to absorb the lower order term uH 1/2+ε (Ω×(0,T )) . To do this, we argue by contradiction. Suppose that (3.10) is not true. Then, there exists a sequence {un }n of solutions of problem (1.1)–(1.5) such that, denoting by E n (0) the energy E related to un at the time 0, T n {u2nt (x, t) + u2nt (x, t − τ )}dΓdt. (3.16) E (0) > n 0
ΓN
From (3.15), we have
T
(3.17) E n (0) ≤ c
{u2nt (x, t) + u2nt (x, t − τ )}dΓdt + un H 1/2+ε (Ω×(0,T )) 0
.
ΓN
Then, from (3.16) and (3.17), T n {u2nt (x, t) + u2nt (x, t − τ )}dΓdt 0 ΓN T 2 2 T0 , there exists a positive constant C1 (depending on T ) for which T (4.12) Ew (0) ≤ C1 wt2 (x, t)dxdt ω
0
for any regular solution w of problem (4.7)–(4.10). Proof. Inequality (4.12) easily follows from some estimates of [14] and standard arguments with multipliers. We give the proof for the reader’s convenience. Let ω0 , ω1 be open neighborhoods of ΓN such that ω ⊃⊃ ω0 ⊃⊃ ω1 ⊃ ΓN .
(4.13)
Let ϕ be a smooth function such that (4.14)
0 ≤ ϕ(x) ≤ 1,
ϕ≡0
on Ω \ ω0 ,
ϕ≡1
on ω1 .
Then, the function ϕw verifies (ϕw)tt − Δ(ϕw) = F (w), where F (w) = wΔϕ + 2∇ϕ∇w, with the same boundary conditions as w. Therefore, we can apply to ϕw the result of Proposition 4.2.1 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 Ω. Denote 1/2 maxx∈Ω v(x) (4.15) T0 = 2 , α with α as in (1.6). Define the function φ : Ω × R → R by 2 T , (4.16) φ(x, t) := v(x) − c∗ t − 2 where T > T0 is fixed and the constant c∗ is chosen as follows. From (4.15), there exists a constant δ > 0 such that αT 2 > 4 max v(x) + 4δ. x∈Ω
For fixed δ, there is c∗ such that (4.17)
c∗ T 2 > 4 max v(x) + 4δ, x∈Ω
c∗ ∈ (0, α).
Note that (4.18) Set
φ(x, 0) < −δ
and φ(x, T ) < −δ
Ω.
2 1 T ∂w eγφ H · ν dΓdt 2 0 ΓD ∂ν 1 T + eγφ H · ν(wt2 − |∇T w|2 )dΓdt, 2 0 ΓN
BT w|Γ×(0,T ) = (4.19)
uniformly in
where ∇T w denotes the tangential gradient of w.
1577
STABILIZATION OF THE WAVE EQUATION WITH DELAY
Then, from Proposition 4.2.1 of [14], using (4.18), and recalling that Ew (t) is constant, we have T T γφ 2 e |∇(ϕw)| dxdt + ϕ2 wt2 dxdt BT w|Γ×(0,T ) ≤ c 0 0 Ω Ω T
ϕ2 w2 dxdt + e−γδ Ew (0)
+
(4.20)
T
0
≤c
Ω
T
e |∇w| dxdt + γφ
0
2
ω0
T
wt2 dxdt 0
ω −γδ
2
+
w dxdt + e 0
Ew (0)
Ω
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 0 ≤ ψ(x) ≤ 1,
(4.21)
ψ≡0
on Ω \ ω,
ψ≡1
on ω0 .
Integrating by parts, T T (wtt − Δw)ψweγφ dxdt = ψwwt eγφ dx 0
Ω
T
Ω
T
∇w∇(ψweγφ )dxdt −
+ 0
Ω
(4.22)
T
T
wt (ψweγφ )t dxdt 0
−
ψwwt eγφ dx
=
Ω T
Ω
ψwt (wt eγφ + weγφ γφt )dxdt 0
0
Ω T
ψe |∇w| dxdt + γφ
+ 0
0
w∇w∇(ψeγφ )dxdt.
2
0
Ω
Ω
Then, from (4.22), recalling that w satisfies (4.7), we have T T γφ 2 ψe |∇w| dxdt = (ψwt2 eγφ + wwt ψeγφ γφt )dxdt 0
(4.23)
Ω
−
T
Ω
T
−2
ψwwt eγφ dx Ω
0
0
0
T
−
ψw∇w∇( ψ)eγφ dxdt
Ω
ψw∇w∇eγφ dxdt. 0
Ω
Since Ew (t) is constant, by using the Cauchy–Schwarz inequality and Poincar´e’s theorem, we can estimate T ψwwt eγφ dx ≤ ce−δγ Ew (0), Ω
0
and so, from (4.23), we obtain T 1 T γφ 2 −δγ ψe |∇w| dxdt ≤ ce Ew (0) + ψeγφ |∇w|2 dxdt 2 0 Ω 0 Ω T T (4.24) 2 2 +c wt dxdt + w dxdt 0
ω
0
Ω
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SERGE NICAISE AND CRISTINA PIGNOTTI
for a suitable positive constant c. By (4.24) we deduce T T T γφ 2 2 2 −δγ e |∇w| dxdt ≤ c wt dxdt + w dxdt + e Ew (0) , 0
ω0
ω
0
Ω
0
which, used in (4.20), gives (4.25)
T
BT w|Γ×(0,T ) ≤ c
T
wt2 dxdt + ω
0
0
w2 dxdt + e−δγ Ew (0).
Ω
Then, from (4.25) and Theorem 3.4 of [14] (Carleman estimate (3.14)), taking γ sufficiently large, we obtain
T
Ew (0) ≤ c
wt2 (x, t)dxdt + cw2L2 (Ω×(0,T )) . ω
0
Now, estimate (4.12) 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 C0 (depending on T ) for which
T
F(0) ≤ C0
(4.26)
a(x){u2t (x, t) + u2t (x, t − τ )}dxdt 0
Ω
for any regular solution u of problem (1.12)–(1.16). Proof. Following Zuazua [18], we can decompose the solution u of problem (1.12)– (1.16) as u = w + w, ˜ where w solves (4.7)–(4.9) with initial condition w(x, 0) = u0 (x),
wt (x, 0) = u1 (x)
in Ω,
and w ˜ verifies (4.27) (4.28) (4.29) (4.30)
˜ = −a(x)[μ1 ut (x, t) + μ2 ut (x, t − τ )] w ˜tt − Δw w(x, ˜ t) = 0 on ΓD × (0, +∞), ∂w ˜ (x, t) = 0 on ΓN × (0, +∞), ∂ν w(x, ˜ 0) = 0 and w ˜t (x, 0) = 0 in Ω.
in
Ω × (0, +∞),
Then, from (4.5) and (4.11), (4.31)
F(0) = E(0) + E 0 (0) = Ew (0) +
ξ 2
Ω
1
u2t (x, −ρτ )dρdx.
a(x) 0
If we take T > T := max{T0 , τ }, from (4.31) with a change of variable we obtain F(0) ≤ Ew (0) + c
Ω
T
u2t (x, t − τ )dtdx,
a(x) 0
STABILIZATION OF THE WAVE EQUATION WITH DELAY
1579
and then, from (4.12),
F(0) ≤ c (4.32)
Ω
a(x)
≤c
T
0
{wt2 (x, t) + u2t (x, t − τ )}dtdx T
{w ˜t2 (x, t) + u2t (x, t) + u2t (x, t − τ )}dtdx
a(x) 0
Ω
for a suitable positive constant c. Therefore, from standard energy estimates for w, ˜ we obtain T F(0) ≤ C0 a(x){u2t (x, t) + u2t (x, t − τ )}dxdt. 0
Ω
Now, using estimate (4.26), 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 (1.12) is replaced with k utt (x, t) − Δu(x, t) + a(x) μ0 ut (x, t) + μi ut (x, t − τi ) = 0
in Ω × (0, +∞),
i=1
with μ0 , μi , τi , i = 1, . . . , k, positive parameters. In this case, if μ0 >
k
μi ,
i=1
the right energy to consider, in order to prove exponential decay, is E(t) :=
1 2
{u2t (x, t) + |∇u(x, t)|2 }dx + Ω
k ξi i=1
2
Ω
1
u2t (x, t − ρτi )dρdx,
a(x) 0
with constants ξi , i = 1, . . . , k, chosen as in Remark 3.3. Remark 4.5. In the case ΓD ∩ΓN = ∅, since for internal feedbacks we have ∂u ∂ν = 0 on ΓN , 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 μ2 ≥ μ1 . 5.1. Boundary feedback. In this subsection we consider problem (1.1)–(1.5) with boundary feedback, and we prove Theorem 1.2. Let us consider the spectral problem for the system ⎧ ⎪ ⎨ utt (x, t) − Δu(x, t) = 0 in Ω × (0, +∞), u(x, t) = 0 on ΓD × (0, +∞), (5.1) ⎪ ⎩ ∂u (x, t) = −μ1 ut (x, t) − μ2 ut (x, t − τ ) on ΓN × (0, +∞). ∂ν We seek a solution of (5.1) in the form u(x, t) = eλt ϕ(x),
λ ∈ C.
1580
SERGE NICAISE AND CRISTINA PIGNOTTI
Then, ϕ has to be a solution of the eigenvalue problem ⎧ 2 ⎪ ⎨ −Δϕ + λ ϕ = 0 in Ω, ϕ = 0 on ΓD , (5.2) ⎪ ⎩ ∂ϕ = −(μ1 + μ2 e−λτ )λϕ on ΓN , ∂ν which can be reformulated, in a variational form, as 2 −λτ ∇ϕ∇vdx + λ ϕvdx + (μ1 + μ2 e )λ (5.3) Ω
Ω
ϕvdΓ = 0 ∀v ∈ HΓ1D (Ω).
ΓN
We want to find a solution for λ := ib, with b ∈ R. For this choice of λ, problem (5.3) can be rewritten as 2 −ibτ (5.4) ∇ϕ∇vdx − b ϕvdx + (μ1 + μ2 e )ib ϕvdΓ = 0 ∀v ∈ HΓ1D (Ω). Ω
Ω
ΓN
Assume that cos(bτ ) = −
(5.5)
μ1 . μ2
Note that, since we are considering the case μ2 ≥ μ1 , there exist b, τ such that (5.5) holds. Then, we choose ! (5.6) μ2 sin(bτ ) = μ22 − μ21 . Under these assumptions, (5.4) becomes ! 2 2 2 (5.7) ∇ϕ∇vdx − b ϕvdx + b μ2 − μ1 Ω
Ω
ϕvdΓ = 0 ∀v ∈ HΓ1D (Ω).
ΓN
In particular, for v = ϕ, (5.7) gives ! 2 2 2 2 2 (5.8) |∇ϕ| dx − b |ϕ| dx + b μ2 − μ1 Ω
Ω
|ϕ|2 dΓ = 0.
ΓN
Without loss of generality, we can assume |ϕ|2 dx = 1, (5.9) ϕ22 := Ω
and then the identity (5.8) can be rewritten as ! (5.10) b2 − b μ22 − μ21 q0 (ϕ) − q1 (ϕ) = 0, where (5.11)
|ϕ|2 dΓ,
q0 (ϕ) :=
|∇ϕ|2 dx.
q1 (ϕ) :=
ΓN
Ω
Now we distinguish two cases. Case (a): μ1 = μ2 . In this case, under our assumptions, (5.10) becomes (5.12)
b2 = q1 (ϕ).
STABILIZATION OF THE WAVE EQUATION WITH DELAY
1581
Define b2 :=
(5.13)
min
w∈H 1 (Ω) ΓD w2 =1
q1 (w).
If ϕ verifies q1 (ϕ) =
min
q1 (w),
w∈H 1 (Ω) ΓD w2 =1
then it easy to see that ϕ is a solution of (5.4) with b as in (5.13). Then ϕ verifies (5.2), and so u(x, t) := eibt ϕ(x)
(5.14)
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)|2 + |ut (x, t)|2 )dx = 2b2 > 0 ∀t ≥ 0. Ω
Note that, from our assumptions (λ = ib, cos(bτ ) = −1, sin(bτ ) = 0), problem (5.2) becomes the classical eigenvalue problem for the Laplace operator with a mixed Dirichlet–Neumann boundary condition. So, we can take a sequence {bn }n of positive real numbers defined by b2n = Λ2n ,
n ∈ N,
where Λ2n , n ∈ N, are the eigenvalues for the Laplace operator. Then, putting l ∈ N,
bn τ = (2l + 1)π, we obtain a sequence of delays τn,l =
(2l + 1)π , bn
l, n ∈ N,
which become arbitrarily small (or large) for suitable choices of the indices n, l ∈ N. Therefore, in the case μ1 = μ2 , we have found a set of time delays for which problem (1.1)–(1.5) is not asymptotically stable. Case (b): μ2 > μ1 . In this case, from (5.10) we have ! ! 1 b= μ22 − μ21 q0 (ϕ) ± (μ22 − μ21 )q02 (ϕ) + 4q1 (ϕ) . 2 Define (5.15)
b :=
1 2
! min 1
w∈H
(Ω) ΓD w2 =1
μ22 − μ21 q0 (w) +
!
(μ22 − μ21 )q02 (w) + 4q1 (w) .
We now prove that if the minimum in the right–hand side of (5.15) is attained at ϕ, that is,
1582 (5.16)!
SERGE NICAISE AND CRISTINA PIGNOTTI
μ22 − μ21 q0 (ϕ) + :=
! (μ22 − μ21 )q02 (ϕ) + 4q1 (ϕ) ! ! 2 − μ2 q (w) + 2 − μ2 )q 2 (w) + 4q (w) , min μ (μ 1 2 1 0 2 1 0 1
w∈H
(Ω) ΓD w2 =1
then ϕ is a solution of (5.7) with b as in (5.15). To show this, take for ε ∈ R, (5.17)
with v ∈ HΓ1D (Ω) such that
w = ϕ + εv,
ϕvdx = 0. Ω
Then, w22 = ϕ22 + ε2 v22 = 1 + ε2 v22 . If we denote (5.18) g(ε) :=
1 1 + ε2 v22
!
μ22 − μ21 q0 (ϕ + εv) +
!
(μ22 − μ21 )q02 (ϕ + εv) + 4q1 (ϕ + εv) ,
then, by definition (5.16), g(ε) ≥ g(0) =
! ! μ22 − μ21 q0 (ϕ) + (μ22 − μ21 )q02 (ϕ) + 4q1 (ϕ) .
So, we have that " dg(ε) "" = 0, dε "ε=0 which, after an easy computation, gives ∇ϕ∇vdx + b
(5.19) Ω
! μ22
−
μ21
ϕvdΓ = 0. ΓN
Since any function v˜ ∈ HΓ1D (Ω) can be decomposed as v˜ = γϕ + v,
γ ∈ R, v ∈ HΓ1D (Ω) with
ϕvdx = 0, Ω
from (5.19) and (5.8) we obtain that ϕ satisfies (5.7) with b defined in (5.15). So, for such positive b, μ1 bτ = arccos − + 2lπ, μ2
l ∈ N,
defines a sequence of time delays for which problem (1.1)–(1.5) is not asymptotically stable. The above examples prove Theorem 1.2.
STABILIZATION OF THE WAVE EQUATION WITH DELAY
1583
5.2. Internal feedback. In this subsection we will give instability examples for problem (1.12)–(1.16) with internal feedback, proving Theorem 1.4. Let us consider the spectral problem for the system (5.20) ⎧ ⎪ utt (x, t) − Δu(x, t) + a(x)[μ1 ut (x, t) + μ2 ut (x, t − τ )] = 0 ⎨ u(x, t) = 0 on ΓD × (0, +∞), ⎪ ⎩ ∂u (x, t) = 0 on ΓN × (0, +∞). ∂ν
in Ω × (0, +∞),
We restrict our analysis to the case a(x) ≡ 1 in Ω. We seek a solution of (5.20) in the form u(x, t) = eλt ϕ(x),
λ ∈ C.
Then, ϕ has to solve the eigenvalue problem ⎧ 2 −λτ )λ]ϕ ⎪ ⎨ Δϕ = [λ + (μ1 + μ2 e ϕ = 0 on Γ , D (5.21) ⎪ ⎩ ∂ϕ = 0 on ΓN . ∂ν
in Ω,
Let us consider the standard problem for the Laplace operator with a mixed Dirichlet–Neumann boundary condition ⎧ 2 ⎪ ⎨ Δϕ = −μ ϕ in Ω, ϕ = 0 on ΓD , (5.22) ⎪ ⎩ ∂ϕ = 0 on ΓN . ∂ν We want to show that for any Λ2 eigenvalue of problem (5.22), there exists a λ ∈ C solution of the equation (5.23)
λ2 + (μ1 + μ2 e−λτ )λ = −Λ2 .
We seek a solution λ = α + iβ, α, β ∈ R, with (5.24)
βτ = (2l + 1)π,
l ∈ N.
Under this assumption, (5.23) becomes 2 α + β 2 = Λ2 , (5.25) μ2 e−ατ = 2α + μ1 . Now we distinguish two cases. Case (a): μ1 = μ2 . In this case, from (5.25) we have α = 0,
β 2 = Λ2 .
Therefore, for any Λ2n eigenvalue of problem (5.22), if βn ∈ R verifies βn2 = Λ2n , then for λ = iβn problem (5.21) admits a nonzero solution.
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SERGE NICAISE AND CRISTINA PIGNOTTI
Take βn positive. From our assumption (5.24), τn,l =
(2l + 1)π , βn
n, l ∈ 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 (1.12)–(1.16) admits solutions in the form u(x, t) = eiβt ϕ(x), whose energy is constant and strictly positive. So, system (1.12)–(1.16) is not asymptotically stable. Case (b): μ2 > μ1 . For a fixed α > 0, from the second equation of (5.25), we obtain μ2 1 , (5.26) τ (α) = ln α μ1 + 2α and so, in order to have τ (α) > 0, we consider 0
