Yan CUI Zhiqiang WANG

Abstract In this paper, the authors study the asymptotic stability of two wave equations coupled by velocities of anti-symmetric type via only one damping. They adopt the frequency domain method to prove that the system with smooth initial data is logarithmically stable, provided that the coupling domain and the damping domain intersect each other.Moreover, they show, by an example, that this geometric assumption of the intersection is necessary for 1-D case.

Keywords Wave equations, Coupled by velocities, Logarithmic stability

1 Introduction and Main Results

Let Ω ⊂Rnbe a bounded domain with smooth boundary∂Ω. We are interested in the asymptotic stability of the following system of two wave equations with Dirichlet boundary condition:

Here the coefficients of elliptic operatorgjk(·)∈C1(Ω;R) satisfy

and

for some constanta>0.

We assume that the coupling coefficientα∈L∞(Ω;R) and the damping coefficientβ∈L∞(Ω;R) are both nonnegative, and furthermore

It is classical to consider system (1.1) as the following Cauchy problem in space H ≜H10(Ω)×L2(Ω)×H10(Ω)×L2(Ω):

withU=(y,u,z,v) and the linear operator A:D(A)⊂H →H is defined as

It is easy to know from the theory of linear operator semigroup(see [21]) that system(1.5)has a unique solutionU(t) = etAU0inC0([0,+∞),H). Then, we can define the total energy of system (1.1) by

which implies immediately the equivalence

Obviously, the total energy is non-increasing:

We are interested in the following questions:

• Under what conditions onαandβ, system (1.1) is asymptotically stable?

• If system(1.1)is stable, what is the decay rate of the total energy E(y,z)(t)ast→+∞?

More precisely, the main result that we obtain is the following theorem.

Throughout this paper,we useC=C(Ω,(gjk)n×n,α,β)to denote generic positive constants which may vary from line to line unless otherwise stated.

Theorem 1.1Assume that(1.2)-(1.4)hold. Assume furthermore that there exist a constant δ >0and a nonempty open subset ωδ⊂ωα∩ωβ⊂Ωsuch that

Then, there exists a constant C >0, such that for any initial data(y0,y1,z0,z1) ∈D(A), the energy of solution to(1.1)satisfies

Moreover, system(1.1)is strongly stable inH, i.e., for any initial data(y0,y1,z0,z1)∈H,

Remark 1.1In Theorem 1.1, if we assume instead that the damping and coupling coeffi-cientsα,βare both continuous on Ω, then the assumption (1.9) can be simply replaced by the following geometric condition

Independently,based on frequency domain method and multiplier method,Kassem-Mortada-Toufayli-Wehbe[15]proved strong stability (1.11)when two waves propagate at different speed under the assumption (1.12). One can also refer to [10, Theorem 2.1] for indirect stability results of other coupled wave system by displacements under the same geometric conditions.

Remark 1.2We provide an example in Section 4 to show that the geometric assumptionωα∩ωβ/= ∅is necessary in general, which is different from the situation with coupling by displacements. One can refer to the open problem raised in[10,Remark 2.2]. As a supplement,we also refer readers to [13, section 5.2.1.3], some numerical examples have been provided to show that for some initial data, system (1.1) seems also strongly stable whenωα∩ωβ=∅.

Remark 1.3The result on logarithmical stability in Theorem 1.1 is sharp. Indeed,ifα≡0,system(1.1)is decoupled into a dissipative system for(y,yt)which is only logarithmically stable(see [18]) and a conservative one for (z,zt). Hence one can not expect a faster decay rate than the logarithmical one for the coupled system (1.1) no matter what the couplingαis.

Remark 1.4In the setting of Theorem 1.1, similar stability results still hold for system(1.1)with other types of boundary conditions,for instance,Robin conditions or mixed Dirichlet-Neumann conditions(see [10]). However,there are no such stability results for the system with Neumann conditions, since all the constant states are equilibrium of the system and will stay at the equilibrium all the time.

In order to prove the logarithmic stability of system (1.1) with regular initial data in Theorem 1.1, we adopt the frequency domain approach to prove certain spectral estimates of the infinitesimal generator A of the solution semigroup. One can refer to [6, 18] for the case of single wave equation and [10] for the case of wave systems.

Let us denote the real part and the imaginary part ofγ∈C by ℜγand ℑγ,respectively. We denote also the resolvent set and spectrum of the operator A byρ(A) andSp(A), respectively.

Theorem 1.2Suppose that the assumptions of Theorem 1.1 hold. Then there exists a constant C >0such that

and the following estimate holds

Obviously, the energy decay given by (1.11) implies directly the factρ(A)⊂{γ∈C|ℜγ <0} and in particular, the originO∈ρ(A). Sinceρ(A) is an open set, then the corollary follows from (1.13), upon choosingClarge enough, as a byproduct of Theorem 1.2.

Corollary 1.1Suppose that the assumptions of Theorem 1.1 hold. Then there exists a constant C >0such that

Remark 1.5The proof of Theorem 1.2 is based on global Carleman estimates (see [11]),which is quite elementary and applied to address many stabilization problems for the system with lower order terms. Moreover,it can be used to obtain explicit bounds on some estimates of decay rate or constant costs in terms of the coefficients. Roughly speaking, (1.14)is equivalent to an observable estimate with constant cost like eC|γ|for coupled elliptic system, which seems quite natural to adopt global Carleman estimates to obtain these types of estimates(see Lemma 3.1 in section 3.1 for more details).

Remark 1.6We should point out that we can not directly adopt the approach in this paper to obtain the logarithmic stability of system (1.1) when two waves have different propagating speed. Roughly speaking, one key step in the proof of important Lemma 3.1 is using an easy fact that∂sp·[∂ssq+∂j(gjk∂kq)]+∂sq·[∂ssp+∂j(gjk∂kp)] =∂s[∂sp∂sq+p∂j(gjk∂k∂sq)]+∂j(gjk∂sq∂kp)−∂j(gjkp∂k∂sq), which can be used to give an estimate thatL2norm of the coupling term with force terms can control theH1energy. However, this fact is invalid for the case of two waves with different propagating speed.

1.1 Previous results

There are a lot of results about asymptotic stability or stabilization of wave equations.Among them, Rauch-Taylor [24] and Bardos-Lebeau-Rauch [5] pointed out that, the single damped wave equation is exponentially stable if and (almost) only if the geometric control condition(GCC for short)is satisfied: There existsT >0 such that every geodesic flow touches the support set of damping term beforeT. If the damping acts on a small open set but the GCC is not satisfied,Lebeau[18]and Burq[6]proved that the wave equation is logarithmically stable for regular initial data. There are also many results about polynomial stability of a single wave equation with special condition on the damping domain (see [4, 7, 17, 23]). Recently, Jin proved in [14] that the damped wave equation on hyperbolic surface with constant curvature is exponentially stable even if the damping domain is arbitrarily small.

As for the case of coupled wave equations or other reversible equations, indirect stability is an important issue both in mathematical theory and in engineering application. Indeed, it arises whenever it is impossible or too expensive to damp all the components of the state,and it is hence important to study stabilization properties of coupled systems with a reduced number of feedbacks. For finite dimensional systems, it is fully understood thanks to the Kalman rank condition. While in the case of coupled partial differential equations,the situation is much more complicated. It depends not only on the algebraic structure of coupling but also the geometric properties of the damping and coupling domain.

Alabau-Boussouira first studied indirect stability of a weekly coupled wave system where the coupling is through the displacements. In [1], the author adopted multiplier method to obtain polynomial stability for wave system with anti-symmetric type coupling under stronger geometric conditions for both the coupling and damping terms. Moreover,she proved that this result was sharp for coupling with displacement. In [2], the polynomial stability results for coupled systems under an abstract framework(including wave-wave system, wave-plate system etc.) were obtained under the conditions that both coupling and damping are localized and satisfy the piecewise multipliers geometric conditions(PMGC for short,see[19]). For 1-D case,a sharp decay rate of polynomial stability was obtained by Riesz basis method in [20]. In [10],Fu adopted global Carleman estimates and frequency domain method to prove that the system with coupling by displacement of symmetric type is logarithmically stable with the assumption that coupling domain intersects the damping domain.

The above results concern only the weakly coupled system. In[3], Alabau-Wang-Yu studied the indirect stability for wave equations coupled by velocities with a general nonlinear damping.By multiplier method, they obtained various types of stability results, including exponential stability, under strong geometric condition on the coupling and damping domains. They also pointed out, for the first time, that it is more efficient to transfer the energy in case the of coupling by velocities compared to the case with coupling by displacements. For 1-D case with constant coefficients, the sharp decay rate was explicitly given in [8]. In [16], Klein computed the best exponent for the stabilization of wave equations on compact manifolds.The coefficient he obtained is therefore the solution of some ODE system of matrices. Kassem-Mortada-Toufayli-Wehbe [15] studied a system of two wave equations coupled by velocities with only one localized damping, the waves propagate at different speed and the positivity and smallness assumptions of the continuous coupling coefficient can be removed. They obtained a strong stability result with the assumption that coupling domain intersects the damping domain. Moreover, assuming that coupling and damping coefficient belong toW1,∞(Ω) and the intersection of coupling domain and damping domain holds PMGC, based on frequency domain method and a multiplier method, they established an exponential energy decay when the waves propagate at the same speed and a polynomial energy decay when the waves propagate at the different speed. Recently, the exponential energy decay result has been generalized by Gerbi-Kassem-Mortada-Wehbe in [13] to the case that the intersection of the coupling and the damping domain satisfies GCC.

To the authors’ knowledge, most known indirect stability results are obtained under the geometric conditions that the damping domain intersects the coupling domain. Indeed, this guarantees effectively the energy transmission in higher space dimension. It is remarkable that Alabau-Boussouira and L´eauteau [2] proved an indirect stability result in 1-D case where the damping domain and the coupling domain are two intervals which do not intersect.

1.2 Main contribution and ideas

As already pointed out in [3], the energy transition is more efficient through the first order coupling (by velocities) compared to zero order coupling (by displacements). This is natural since the first order coupling effect can be seen as a bounded perturbation to the system while the zero order coupling is a compact one. Nevertheless,one can not expect a faster decay(than logarithmical one) of the whole system (1.1) even if a first order coupling appears, because the energy of the single wave equation with damping localized in small domain only decays logarithmically. In this sense, the stability results are sharp.

Not surprisingly,the indirect stability result is obtained by assuming essentially the damping and coupling domain intersect. However, we give an example to show that this geometric condition is necessary in general for the wave system coupled by velocities. This is quite different from the system coupled by displacements (see [2]).

As for the proof of the main theorems, we adopt the frequency domain approach to reduce the stability problem to an estimate on resolvent which can be obtained by global Carleman estimates of an elliptic equation as in [10]. Different from the system in [10], there are no zero order terms explicitly in system (1.1). In order to derive theL2energy of the solution, we then need to make fully use of the coupling structure together with Poincar´e inequality under homogeneous Dirichlet boundary condition.

1.3 Organization of the paper

This paper is organized as follows. In Section 2, we recall some basic facts about frequency domain method and global Carlemann estimates for an elliptic equation. In Section 3, we give the proofs of Theorem 1.1 and Theorem 1.2 as well as the technical Lemma 3.1,which is crucial to the proof of Theorem 1.2. Finally in Section 4, we give an example of system (1.1) withωα∩ωβ=∅, which is indeed unstable.

2 Preliminaries

In this section,we briefly recall the frequency domain method and global Carleman estimates for an elliptic equation.

2.1 Frequency domain method

Thanks to classical semigroup theory, A generates aC0-semigroup operator{etA}t≥0on H.It is well-known that the logarithmic stability of system (1.1) can be obtained by a resolvent estimate (see [6, 18]). More precisely, we have the following lemma.

Lemma 2.1(see [6, Theorem 3])LetAbe defined by(1.6). If

then, there exists C >0such that for any U0∈D(A2)≜{U∈H|AU∈D(A)},

Obviously,(1.14)in Theorem 1.2 implies the assumption(2.1)in Lemma 2.1. Once Theorem 1.2 is proved,the logarithmical decay estimate (1.10)in Theorem 1.1 can be easily obtained by Calderon-Lions interpolation theorem and (2.2).

2.2 Global Carleman estimates

To obtain resolvent estimate (1.14), we need to introduce the global Carlemann estimates for elliptic equations (see [9, 10, 12]).

Letω0be an open set such thatω0⊂⊂ωδ⊂ωα∩ωβ.There existssuch that

Next, we introduce some weight functions

with

whereb>0 andλ,µ,s∈R are all constants.

Let us consider a single elliptic equation

wheregjk(·)∈C1(Ω;R)satisfy (1.2)—(1.3). Therefore for everyf∈L2((−b,b)×Ω),the elliptic system(2.6)has a unique solutionw∈H10((−b,b)×Ω). Therefore we have the following global Carleman estimates of solution.

Theorem 2.1(see [10])Let b∈(1,2]and θ,φ∈C2([−b,b]×Ω;R)be defined by(2.4)-(2.5). Then there exists µ0>0, such that for any µ≥µ0, there exist C=C(µ)>0and λ0=λ0(µ)such that for any f∈L2((−b,b)×Ω), the solution w to system(2.6)satisfies

for all λ≥λ0(µ).

3 Proofs of Main Theorems

In this section, we give the proofs of the main theorems,i.e., Theorem 1.1 and Theorem 1.2.First in Subsection 3.1,we prove Theorem 1.2,particularly the resolvent estimate(3.16), based on some interpolation estimates on elliptic equations. Then by Theorem 1.2 and Calderon-Lions interpolation inequality, we conclude Theorem 1.1 in Subsection 3.2. Finally in Subsection 3.3,we prove Lemma 3.1 concerning an interpolation inequality of coupled elliptic equations,which is crucial to the proof of Theorem 1.2.

3.1 Proof of Theorem 1.2

LetF=(f0,f1,g0,g1)∈H andU0=(y0,y1,z0,z1)∈D(A) be such that

whereγ∈C and A is given by (1.6). Then (3.1) is equivalent to

or furthermore

In order to prove (1.14), it suffices to prove that there exists a constantC >0, such that

For this purpose, we set

Thenpandqsatisfy the following coupled elliptic align

Note that there are no boundary conditions ons=±2 in the above system (3.5). We have the following lemma on interpolation estimate, while its proof is left in Subsection 3.3.

Lemma 3.1Under the assumption of Theorem 1.1, there exists a constant C >0such that for any λ>0big enough, the solution(p,q)to(3.5)with form(3.4)satisfies

where

On the other hand, by (3.4), we have

for some constantC >0. Combining (3.6) and (3.8), we get

Next, we turn to estimate ‖y0‖L2(ωδ). Letζ∈C20(Ω;R) be a cutofffunction such that

Multiplyingy-equation in (3.2) by 2ζy0and integrating by parts on Ω yield that

Similarly, multiplyingz-equation in (3.2) by 2ζz0and integrating by parts on Ω yield that

Note that

Adding (3.11) to (3.12) and taking the imaginary part result in

Then it follows by Cauchy inequality and the definition ofβ,F0,F1that

Thus by (3.9), we get

By the definition of OC, we takeC >0 large enough such that

for allγ∈OC. Note also the fact thatβ(x)≥δa.e. inωδ. Then it follows from (3.13)—(3.14)that

for someC >0 large enough. Combining (3.9) and (3.15) gives

Sincey1=f0+γy0,z1=g0+γz0, we have also

Hence the desired estimate (3.3) indeed holds for allγ∈OC.

Consequently, A −γIis a bijection from D(A) to H,which satisfies the resolvent estimate(1.14). We conclude the proof of Theorem 1.2.

3.2 Proof of Theorem 1.1

As a corollary of Theorem 1.2, there existsC >0 such that

Then by Lemma 2.1, we have forU0∈D(A2) that

On the other hand, the contraction of the semigroup etAimplies that

Note that D(A) is an interpolate space between D(A2) and H. Combining (3.17)—(3.18) and using Calderon-Lions interpolation theorem (see [25, p. 38, Example 1 and p. 44, Proposition 8]), we conclude for allU0∈D(A) that

which is equivalent to the logarithmical decay estimate (1.10).

Finally we conclude by (3.19) and density argument that system (1.1) is strongly stable,i.e., for allU0∈H,

The proof of Theorem 1.2 is complete.

3.3 Proof of Lemma 3.1

In this subsection,we give the proof of Lemma 3.1 which plays a key role in proving Theorem 1.2. The proof is divided into 6 steps. In this subsection, we denoteC >0 various constants independent ofλwhich can be different from one line to another.

Step 1We derive a weighted estimate (3.28) for (p,q), the solution of (3.5).

Note that there are no boundary conditions onp(±2,·),q(±2,·) in (3.5). Let us introduce a cutofffunctionφ=φ(s)∈C30((−b,b);R) (see for instance [22]) such that

where the constantsb0,bare given by

respectively, forµlarge enough which enables to apply Theorem 2.1. Obviously, ifµ >ln 2,then

Let

Then, we consider the elliptic equations thatp,qsatisfy in (−b,b)×Ω

where

By applying Theorem 2.1 to bothandthere existsµ0>ln 2 such that for anyµ≥µ0,there existC=C(µ)>0 andλ0=λ0(µ) such that for anyλ≥λ0(µ), we have

where

Step 2We estimate I0in (3.29) from below.

By the choices ofθ,l,φin (2.4) andb,b0in (3.21), we have

Then

Step 3We estimate I1in (3.29) from above.

Using Cauchy inequality, we get easily

By the choices ofθ,l,φin (2.4) andb,b0in (3.21), we have Then

Obviously,

which can be absorbed by I0ifλis large enough.

Therefore,

Step 4We estimate the localized term I2in (3.29) from above.

We write it as

where

Denoteωk(k=1,2,3)some open subsets in Ω such thatω0⊂⊂ω1⊂⊂ω2⊂⊂ω3⊂⊂ωδ. Letηj∈C30(ωj;R) (j=1,2,3) be suitable cut-offfunctions such that

Moreover,we choose furtherη2such that

for some constantC >0. The existence ofη2is shown at the end of the proof.

Step 4.1We estimate a weighted energy fordxds.

By the definition ofη1, it suffices to estimatedxds. To do this,we multiplyequation in (3.24) by

It reduces equivalently to

Integrating (3.3) over (−b,b)×Ω, upon using (3.24), (1.2)—(1.3) and Cauchy inequality, we obtain

Step 4.2We estimate a weighted energy fordxds.

By definition ofη2, it suffices to find an up bound fordxds. We claim that there exists a constantC >0, such that

Indeed, in view of (3.4), we have

Notice that

then

Recalling (3.30)and (3.34), we can obtain from the choices ofφ,b,b0in(3.20)and(3.22), that

Combining (3.45) and (3.46) yields (3.44) immediately.

Step 4.3We estimatedxdsby using the coupling relation in (3.24).

Direct computations yield that

and

We integrate (3.47) on (−b,b)×Ω. Then from (3.48)—(3.49) and particularly the facts that==0 on the boundary of (−b,b)×Ω and ∇p(±b,·)=∇(±b,·)=0, we derive

Sinceα(x) ≥δ >0 onωδ, the first term on the right hand side of (3.50) can be absorbed by the left hand side. Moreover, by direct computations, we have the following two facts

and

By definitions ofη2,θ,φ, in particular (3.40), (3.42) andgjk(·)∈C1(Ω;R), one can obtain the following estimate Thus by above estimate (3.53) and Cauchy inequality, we get for allε>0 that

Therefore we obtain from (3.44) and by choosingε>0 small that

Step 4.4We estimatedxds.

Similarly to (3.43), we can derive

Step 4.5We summarize the estimate of I2from above.

Applying (3.43), (3.54)—(3.56) for I21,I23,I24in (3.38), we end up with

Similarly to the estimate of I11in (3.35), we have

Note thatdxdsis the same as I1and the estimate has been given by(3.37). Therefore, we have

Step 5We derive the estimate (3.6) based on Steps 1—4.

Lettingλbe large enough,from(3.28),(3.31),(3.37)and(3.58),we finally obtain the desired interpolation estimate (3.6).

Step 6We provide an example of the cut-offfunctionη2∈C30(ω2) such that (3.41)—(3.42)indeed hold.

Without loss of generality, we may assumeω1⊂B(0,r2)⊂B(0,r)⊂ω2. Set

It is easy to check thatη2∈C30(ω2) and

Then it follows the property (3.42).

This finally concludes the proof of Lemma 3.1.

4 An Example with ωα ∩ωβ = ∅

In this section, we show, by an example, that the geometric assumptionωα∩ωβ/= ∅is necessary in general for asymptotic stability of the coupled system (1.1).

More precisely, we consider the following 1-D coupled wave equations

where the coefficientsαandβare given by s

uch thatωα∩ωβ=∅.

Let the initial data be the following

and

Clearly,(y0,y1,z0,z1)belongs to(H2(0,2π)∩H10(0,2π))×H10(0,2π)×H2(0,2π)∩(H10(0,2π)×H10(0,2π)). It is not hard to check that the unique solution of system(4.1)—(4.3)can be explicitly given by

and

In contrast to (1.11) in Theorem 1.1, the energy of system (4.1) is conserved

therefore the system is not asymptotically stable. According to the above example,we conclude that the decay estimate (1.10) may not hold ifωα∩ωβ=∅.