Asymptotic stability of wave equations coupled by velocities

This paper is devoted to study the asymptotic stability of wave equations with constant coefficients coupled by velocities. By using Riesz basis approach, multiplier method and frequency domain approach respectively, we find the sufficient and necessary condition, that the coefficients satisfy, leading to the exponential stability of the system. In addition, we give the optimal decay rate in one dimensional case.


Introduction and Main Results
In this paper, we consider the long time behavior of the solution to a system of wave equations with constant coefficients coupled by velocities. In particular, we want to study what kind of conditions, that the coefficients satisfy, lead to the exponential stability of the system.
In the case of scalar wave equation, there are numerous results on asymptotic stability or stabilization with internal or boundary damping. Cox and Zuazua [7] studied, by Fourier analysis, the energy decay of in (0, 1) × (0, +∞), u(0, t) = u(1, t) = 0 in (0, +∞), (u, u t )(0) = (u 0 , u 1 ) in (0, 1) with indefinite damping. As a corollary, the exponential decay of the energy and its optimal rate were shown in [7] when a > 0 in (0, 1). Using Multiplier method, Alabau-Boussouira [3,4] and Alabau et al. [2] proved the indirect stabilization of wave systems coupled by displacements, for instance, in Ω × (0, +∞), That is, the damping is acted only in one equation and the total energy of the whole system decays polynomially due to the coupling effect. Using the criteria of polynomial decay in [16], Liu and Rao proved in [17], by frequency domain approach, the polynomial stability of a partially damped wave system with weak coupling by displacements and multiple propagation speeds. Liu and Rao [18] also proved the indirect stabilization, by Riesz basis approach, with optimal polynomial decay rate for a wave system which is coupled by displacements in one dimensional case.
Recently, Alabau-Boussouira, Wang and Yu [5] obtained the indirect stabilization, combining multiplier method and weighted energy techniques, for the following damped wave system with variable coefficients coupled by velocities in Ω × (0, +∞), The decay speed is shown to change corresponding to the various properties of the nonlinear damping ρ(x, u t ), especially, if ρ(x, u t ) ≡ αu t (α > 0), b(x) ≡ b > 0, the total energy decays exponentially. This phenomenon indicates that the velocity coupling has different impact compared to the displacement coupling. Being regarded as perturbation, the coupling through displacements is compact, while the coupling through velocities is bounded. On the other hand, the controllability (asymptotic stability) of coupled wave systems with general coefficients is closely related to the synchronization (asymptotic synchronization), according to the pioneer results on synchronization by Li and Rao [14,15] (see also [10]). For the above reasons, we focus on the question: What kind of coefficients can lead to the exponential stability of the general coupled wave system by velocities?
Suppose that Ω is an bounded open set in R d with C 2 boundary Γ = ∂Ω. Consider the following general wave system coupled by velocities where α, β, γ, η ∈ R are constants. and ω > 0 such that where the (total) energy is defined by and ω is the corresponding decay rate.
The task of this paper is to find the conditions that the coefficients matrix α β γ η should satisfy such that System (1.1) is exponential stable. Our main result is the following theorem: i.e., the two eigenvalues of the coefficient matrix α β γ η both have positive real part.
(2.1) Let U = (u, y, v, z) and let U 0 = (u 0 , u 1 , v 0 , v 1 ). System (1.1) can be rewritten as the Cauchy Problem in abstract form: Obviously, A can be regarded as a bounded perturbation to the standard wave operator, namely, By classical Hille-Yoshida Theorem and perturbation theory (see [20, page 76, Theorem For the sake of convenience of statement, we use real Schur decomposition for the coefficient matrix so that we can reduce the original general problem into the same problem in two canonical forms. Moreover, (1.4) is equivalent to a > 0 in Case i) and to a > 0 and c > 0 in Case ii).
Lemma 2.2. Let P ∈ R 2×2 be a real orthogonal matrix and let (u, v) be solution of (1.1), whereB = αβ γη = P T BP and the energy of (2.4) is equivalent to that of (1.1).
3 Proof of Proposition 2.2

Riesz basis approach
In this subsection, we adopt Riesz basis approach to prove Proposition 2.2 in one dimensional case, that is Ω = (0, π) ⊂ R. The key ingredient is to prove the Riesz basis property and to analyze the spectrum of A. Additionally, we obtain the optimal and explicit decay rate in this situation, see Corollary 3.1.
In Case i), System (1.1) can be rewritten as the Cauchy Problem (2.2) in one dimensional Let λ be the eigenvalue of A and E = (u, y, v, z) ∈ D(A) be its eigenvector: Canceling v, we obtain a forth order ODE of u: whose general solution is given by 1, · · · , 4) are four zeroes of the algebraic equation and A i (i = 1, · · · , 4) are constants to be determined. Then it follows that where the eigenvalue λ satisfies the characteristic equation Consequently, there are two classes of eigenvalues: yielding that X n is a decreasing function of n ∈ Z + . It follows that The corresponding eigenvectors are If a ≤ 0, there exists an eigenvalue with nonnegative real part according to (3.4)-(3.5).
When a > 0 and c > 0, we would like to prove that the eigenvectors as well as root-vectors form a Riesz basis of H. For this purpose, we discuss the different situations that multiple eigenvalues may appear with various values of a and c.
• Case 3.1. b = 0. System (1.1) is decoupled. The dimension of eigenspace of λ ± 1,n is two. The eigenvectors are which forms a Riesz basis of H. For every U 0 ∈ H given by (3.8) with (3.9), the solution of (2.2) is still (3.10).
From the above proof of Proposition 2.2, we get the corollary concerning the decay rate.

Multiplier method
In this subsection, we apply the multiplier method to establish the decay estimates of the total energy. The key ingredient is to use an integral inequality (see Lemma A.2) which leads to the exponential decay of the energy. In this subsection, the initial data are assumed to be all real functions.
In Case i), System (1.1) is reduced to in Ω × (0, +∞), Using the multiplier u t to u-equation and v t to v-equation, we obtain Integrating by parts and using the definition of the total energy E(t), we get Using the multiplier u to u-equation and v to v-equation, we obtain for 0 ≤ S ≤ T that and after integration by parts, Thanks to Cauchy Inequality and Poincaré Inequality, we get for every ε > 0 Choosing ε > 0 small we get Thanks to Lemma A.2, we conclude that E(t) decays to 0 exponentially.

In Case ii), System (1.1) is reduced to
It is easy to see that if a ≤ 0, then the energy of u does not decay; while if c ≤ 0, by taking u ≡ 0, then the energy of v does not decay. Hence, the total energy E(t) of (3.25) does not decay to 0 if a ≤ 0 or c ≤ 0. It remains to prove that if a > 0 and c > 0, E(t) decays to 0 exponentially.
For κ > 0, we set the equivalent energy Using the multiplier κu t to u-equation, v t to v-equation, Choosing κ > 0 suitably large, namely, b 2 − 4κac < 0 then there exists δ > 0 such that Using the multiplier κu to u-equation, v t to v-equation, Thanks to Cauchy Inequality and Poincaré Inequality, we get for every ε > 0 Similarly as as in Case i), we can prove by (3.27) and choosing ε > 0 small the integral inequality for E κ : Then it follows by Lemma A.2 E κ (t) (or equivalently, E(t)) decays to 0 exponentially. This ends the proof of Proposition 2.2.

Frequency Domain Approach
In this subsection, we use the frequency domain approach to prove Proposition 2.2. More precisely, we want to get the exponential stability of the semigroup through the uniform estimate of the resolvent on the imaginary axis by Lemma A.3 [11,22] (see also [19]). We start, by contradiction arguments, to assume that (A.1) does not hold, i.e., there exists ξ ∈ R and U = (u, y, v, z) ∈ D(A) with U H = 1 such that (iξ − A)U = 0, namely, By definition (2.1), we easily calculate Then it follows that y = z = 0 in L 2 and thus The theory of elliptic equation with Dirichlet boundary condition implies that u = v = 0 ∈ H 2 H 1 0 . Consequently we have U = 0 ∈ H which contradicts with U H = 1. Therefore (A.1) is satisfied for a > 0.
Next we continue to prove that (A.2) holds for a > 0. Otherwise, thanks to the continuity of the resolvent R(iξ, A) with respect to ξ ∈ R, (A.2) is not valid at ∞, i.e., there exists {U n = (u n , y n , v n , z n )} n∈Z + ⊂ D(A) with U n H = 1 and {ξ n } n∈Z + ⊂ R with |ξ n | → +∞, (iξ n − A)U n H → 0 as n → +∞, namely, Similarly as (3.28), we have On the other hand, Then z n → 0 ∈ L 2 , y n → 0 ∈ L 2 , and thus Thanks to Nirenberg Inequality [1, page 135, Theorem 5.2], we get ||∇u n || L 2 ≤ C||∆u n || In Case (ii), it is easy to see, as in Section 3.2, that if a ≤ 0 or c ≤ 0, system (3.25) is not asymptotically stable. It remains to prove that (3.25) is exponentially stable if a > 0 and c > 0.
Theorem 4.3. If we consider a wave system with multiple propagation speeds in Ω × (0, +∞), v tt − c 2 ∆v + γu t + ηv t = 0 in Ω × (0, +∞), in Ω with c = 1, we can adopt the multiplier method or the frequency domain approach to prove the exponential stability of the above system under the assumption Remark 4.1 (Comparison of Three Approaches). The advantage of Riesz basis approach is to give the explicit expression of the solution so that the relation between the exponential stability of the system and the coefficients can be relatively easy to discover. In addition, the optimal decay rate can be obtained through very careful analysis of both the spectrum and the eigenvectors. However, the Riesz basis properties of the eigenvectors are hard to check in higher dimensional case. The multiplier method is rather simple to apply without restrains in space dimension and works for variable coefficients case (even for nonlinear problems), but it requires strong geometric assumptions. The frequency domain approach is also applicable for all space dimension and for the variable coefficients case without much information of the eigenvalues and eigenvectors. Nevertheless, the optimal decay rate can not be obtained in general by multiplier method or by frequency domain approach.
A Appendix