Stabilization of hyperbolic equations with mixed boundary conditions

This paper is devoted to study decay properties of solutions to hyperbolic equations in a bounded domain with two types of dissipative mechanisms, i.e. either with a small boundary or an internal damping. Both of the equations are equipped with the mixed boundary conditions. When the Geometric Control Condition on the dissipative region is not satisfied, we show that sufficiently smooth solutions to the equations decay logarithmically, under sharp regularity assumptions on the coefficients, the damping and the boundary of the domain involved in the equations. Our decay results rely on an analysis of the size of resolvent operators for hyperbolic equations on the imaginary axis. To derive this kind of resolvent estimates, we employ global Carleman estimates for elliptic equations with mixed boundary conditions.


1.
Motivation and formulation of the problem. In this paper, we will consider the longtime behavior of solutions to damped hyperbolic equations in a bounded domain. When the energy of the solution tends to zero as time tends to infinity, we will analyze what the explicit decay rate of the solution is. In the literature, most of the works are concerned with three types of decay rates for the solutions of evolution equations, i.e. exponential decay, polynomial decay and logarithmic decay. Consider the following abstract Cauchy problem on a Hilbert space X: dx(t) dt = Ax(t), t ≥ 0, where A generates a C 0 semigroup T (t) = e At on X. The solution of (1.1) is then given by x(t) = T (t)x 0 . Assume that ilR ⊂ ρ(A), where ρ(A) is the resolvent set of A. We recall the following characterization on decay rates: (i) The solution of (1.1) decays exponentially if and only if sup{||(A − iβI) −1 || L(X) | β ∈ lR} < ∞ (see [13,22]); (ii) The solution of (1.1) decays polynomially if and only if sup β −l ||(A − iβI) −1 || L(X) ≤ L for some l, L > 0 and all |β| ≥ 1 (see [5,18]); (iii) The solution of (1.1) decays logarithmically if and only if there exists a constant C > 0 such that ∀β ∈ lR, ||(A − iβI) −1 || L(X) ≤ Ce C|β| (see [9,16,17]). We refer to [3,8] and 762 XIAOYU FU references therein for the recent progress on characterization of decay rates for the solutions of abstract evolution equations. The aim of this paper is to derive the explicit decay rates of the solutions to hyperbolic equations with a small boundary or internal damping term. This work was motivated by the above mentioned references, especially reference [9] for logarithmic stabilization of wave equations with Zaremba boundary conditions. More precisely, our problem can be formulated as follows.
There are numerous studies on the decay properties of hyperbolic equations. For exponential decay of the solutions of wave equations, Bardos-Lebeau-Rauch [2] show that the energy of the solutions decays exponentially if and only if the effective damping domain satisfy the Geometric Control Conditions. We refer to [15,24,23,27] and references therein for more exponential decay results for the wave equations with Dirichlet or Neumann boundary conditions; and [6,7,20] for polynomial decay for the wave equations with Dirichlet boundary condition under special geometries. When the Geometric Control Condition is not satisfied, the logarithmic stabilization of the following wave equation was first considered by Lebeau [16]: (u(x, 0), u t (x, 0)) = (u 0 , u 1 ) ∈ H 1 0 (M ) × L 2 (M ) where (M, g) be a Riemannian, compact manifold with compact boundary ∂M , ∆ g is the "Laplacian" associated to the metrix g = (a jk ) 1≤j,k≤n ∈ C ∞ (M ; lR n×n ), the damping term a(x) ∈ C ∞ (Ω) is a non-negative bounded function supported in ω. Later, Lebeau-Robbiano [17] obtained a similar boundary stabilization result with a Neumann dissipative term ∂ ν u + a(x)u t = 0 on ∂M . Bellassoued [4] established the logarithmic decay results for both the boundary stabilization of the elastic wave equation (∂ 2 t −∆ e )u = 0 with Neumann dissipative term supported on the boundary and the internal stabilization of the elastic wave equation (∂ 2 t − ∆ e + a(x)u)u = 0 with zero Dirichlet boundary condition, where ∆ e = µ∆ g + (µ + λ)∇ g (div ·) with real parameters λ, µ satisfying µ > 0, 2λ + µ > 0. Very recently, Cornilleau-Robbiano [9] considered the logarithmic stabilization of the wave equation (∂ 2 t − ∆)u = 0, i.e. the coefficients (a jk ) n×n = I, the identity matrix, with the Zaremba boundary conditions, i.e. u = 0 on Γ 0 and ∂ ν u + a(x)u t = 0 on Γ 1 . We remark that the logarithmic decay results obtained in [4,16,17] depend on C ∞ -regularity of a jk (·), a(·) and ∂M , while Cornilleau-Robbiano [9] weakened the regularity of the damping a(·) to C s for s > 1 2 , and kept the C ∞ -regularity on the boundary ∂M . Since the classical local Carleman estimate was employed in [9], it seems that the authors do need these strong regularity. Naturally, we want to know whether the logarithmic decay results still hold under some sharp regularity on the coefficients a jk (·), the damping a(·) and the boundary ∂M . In this respect, we refer to [1] for the Dirichlet boundary stabilization of wave equation and [10,11] for both the boundary stabilization and the internal stabilization of hyperbolic equations by imposing the Neumann boundary conditions. However, to the best of our knowledge, there are no references addressing the stabilization of hyperbolic equations with Neumann-Robin boundary conditions even for C ∞ -regularity of the coefficients, the damping and the boundary. Moreover, as we mentioned before, most of very interesting logarithmic decay results were given in [9,17] and the references therein for the hyperbolic equation under the regularity assumption that the coefficients a jk (·) and a(·) are C ∞ -smooth, and the damping at least is C s (s > 1 2 )-smooth. In this paper, we shall develop an approach based on two global Carleman estimates to prove the boundary stabilization of system (1.4) equipped with boundary condition (1.5), and the internal stabilization of system (1.6) equipped with boundary conditions (1.7), under sharp regularity of the coefficients a jk (·) ∈ C 1 (Ω), the damping a(·) ∈ L ∞ (∂Ω; lR + ), d(·) ∈ L ∞ (Ω; lR + ) and the boundary ∂Ω is C 2 -smooth.
2. Statement of the main results. To begin with, we assume that a(·) is a bounded real valued functions satisfying where a 0 ∈ lR. As to the hypersurface Γ of the ∂Ω, we have the following assumption: Also, we assume that d(·) is a bounded real valued functions satisfying Then, we define an unbounded operator A 1 : As for Neumann-Robin boundary condition on Γ 0 , we define It is easy to show that A j (j = 1, 2) generate C 0 -semigroups {e tAj } t∈lR + (j = 1, 2) on H. Therefore, system (1.4) is well-posed in H. For the equation (1.4) with u| Γ0 = 0 boundary condition, we define its energy by a jk u xj u x k + |u t | 2 dx.

STABILIZATION OF HYPERBOLIC EQUATIONS 765
For the equation (1.4) with the boundary condition n j,k=1 a jk u xj ν x k + p(x)u Γ0 = 0, we define its energy by Hence, the energy of every solution to (1.4) with the boundary condition (1.5) is non-increasing. Following the well-known unique continuation property for solutions of the wave equation, it is easy to show that there are no nonzero solutions of (1.4) which conserve energy. Hence, by using LaSalle's invariance principle ([12, p. 18]), we conclude that the energy of every solution of (1.4) tends to zero as t tends to infinity, without any geometric conditions on the domain Ω.
For system (1.6) equipped with boundary conditions (1.7), we define an unbounded operator B : Also, it is easy to check that B generate C 0 -semigroups {e tB } t∈lR + on H. Thus, system (1.6) is well-posed in H and the following energy of every solution of (1.6) with the boundary condition (1.7) is non-increasing as well: Throughout this paper, we will use C = C Ω, Γ * 1 , n j,k=1 ||a jk || C 1 (Ω) , s 0 and ||a jk || C 1 (Ω) , s 0 to denote generic positive constants which may vary from line to line (unless otherwise stated). Our main results can be stated as follows.
Following the analysis of resolvent estimate for the abstract evolution equations (see [6,Theorem 3] for example), the decay properties in Theorem 2.1 can be obtained via the size of the resolvent operator R(λ, the imaginary axis. Denote by ρ(A j ) and ρ(B) the resolvent sets of A j and B, respectively. We have the following result.
Theorem 2.2. Under the assumption of Theorem 2.1, it holds that ilR ⊂ ρ(A j ) (j = 1, 2) and ilR ⊂ ρ(B), and there exist two constants C > 0 and C * > 0 such that We have the following several remarks.
Remark 2.1. If we impose the boundary condition u = 0 on Γ 0 , this boundary condition together with the boundary condition on Γ 1 presented in (1.4) is the socalled Zaremba boundary conditions. In this respect, we refer to [9] for the related logarithmic decay of system (1.4). i.e. p(x) = 0 in (1.5), we can obtain the stabilization result for the hyperbolic equations with internal damping discussed in [11]. Similarly, if we impose zero Dirichlet boundary condition, i.e. u = 0 on ∂Ω, we can obtain the same stabilization result for the hyperbolic equations with internal damping discussed in [16].
The rest of this paper is organized as follows. In Section 3, we shall prove two global Carleman estimates for elliptic equations with mixed boundary conditions, via which, two interpolation inequalities for elliptic equations are presented in Section 4. Section 5 is devoted to the proof of our main results.
3. Global Carleman estimates for elliptic equations with mixed boundary conditions. In this section, we shall establish two global Carleman estimates for the elliptic equations with mixed boundary conditions. Denote Q = (−2, 2) × Ω, Σ 1 = (−2, 2) × Γ 1 , Σ 0 = (−2, 2) × Γ 0 . Let us consider the following elliptic equation: where z 0 ∈ L 2 (X), z 1 ∈ L 2 (Σ 1 ). As to the boundary Γ 0 , we impose any one of the following boundary conditions: Also, we will consider the global Carleman estimate for the following elliptic equation: where z 0 ∈ L 2 (X) and system (3.3) is equipped with the boundary conditions To present our global Carleman results for systems (3.1) and (3.3), we first recall the following known results.
Let ω 0 be an arbitrary fixed sub-domain of Ω such that ω 0 ⊂ ω * . Then there exists a function ψ 2 ∈ C 2 (Ω; lR) such that For any µ > ln 2, put It is easy to check that 1 < b 0 < b ≤ 2. For parameter λ, µ > 1, assume that there exists a function ψ * ∈ C 2 (Ω) (which will be given later), we define the weight functions θ and φ as follow: We have the following results.
Then, there is a constant µ 1 > 0 such that for all µ ≥ µ 1 , one can find two constants , and for all λ ≥ λ 1 , it holds that Remark 3.1. We refer to [26,Theorem 3.2] for more information on the pointwise weighted identity for a more general partial differential operator of second order than that appeared in Lemma 3.3. Such kind of weighted identities are quite useful for the study of control theory and inverse problems for various kinds of partial differential equations (see [26] and the references therein). Finally, we recall the following known identity.
1, · · · , n), and g = (g 1 , · · · , g n ) : lR s × lR n x → lR n be a vector field of class C 1 . Then for any z ∈ C 2 (lR s × lR n x ; l C), it holds (3.13) 3.1. Proof of Theorem 3.1. In this subsection, we will prove Theorem 3.1. Our proof is based on Lemma 3.3. The key point is to estimate the boundary terms in Lemma 3.3 with mixed boundary conditions. The proof is divided into several steps.
Step 1. Note that in system (4.8) we need to consider the problem with inhomogeneous Neumann-type boundary. As in [10,11], we shall choose two weight functions, one is positive while another one is negative with respect to the space domain Ω, such that some undesired boundary terms vanish on ∂Ω. More precisely, we choose other weight functionsθ andφ as follows where b is given by (3.7) and ψ * will be given later, it is easy to check that Now, we choose ψ * = ψ 1 , which is given by (3.5), and we apply Lemma 3.3 to (3.1). By (3.15), and noting that z(−b, x) = z(b, x) = 0, there is a constant µ 1 > 0 such that for all µ ≥ µ 1 , one can find a λ 1 = λ 1 (µ) so that for all λ ≥ λ 1 : whereṼ k has the same form with V k , only replace , A by˜ ,Ã.
3.2. Proof of Theorem 3.2. The proof of Theorem 3.2 is divided into several steps.
Proof. We borrow some ideas from [21]. Note however that there is no boundary condition for w at s = ±2. Therefore, we need to introduce a cut-off function where b and b 0 were given by (3.7), and it is easy to see that 1 < b 0 < b ≤ 2. Next, we put z = ϕw. (4.7) Then, noting that ϕ does not depend on x, by (4.1), it follows Now, by applying Theorem 3.1 to system (4.8), we have Denoting c 0 = 2 + e µ > 1, and note that b 0 ∈ (1, b). By using (4.6) and (4.10), note that z = ϕw, we get (4.11) By (4.11), one concludes that there exists an ε 1 > 0 such that for every ε ∈ (0, ε 1 ], it holds Proof of Theorem 2.2. First, fix F = (f 0 , f 1 ) ∈ H and U = (u 0 , u 1 ) ∈ D(A j ) (j = 1, 2). It is easy to see that the following equation It is easy check that v satisfies the following equation: a jk v xj ν x k − iav s = −af 0 e −βs on lR × Γ 1 .
Now, applying Theorem 4.1 to (5.3), and by (5.4), we have On the other hand, multiplying (5.1) by u 0 and integrating it on Ω, by (2.1)-(2.2), it follows that a jk u 0 xj ν x k u 0 dx.
(5.10) By (5.9)-(5.10), we know that A j −iβI is injective. Therefore A j −iβI is bi-injective from D(A j ) to H (j = 1, 2). Moreover, This gives the first result of Theorem 2.2. The analysis for the internal damping case can be derived in a similar way. Here, we only give some explanation on the relationship of the spectral equation with the elliptic equation we considered in (4.2). For fixed F = (f 0 , f 1 ) ∈ H and U = (u 0 , u 1 ) ∈ D(B). It is easy to see that the following equation (B − iβI)U = F is equivalent to By setting the same transform v = e −βs u 0 , it is easy check that v satisfies the following equation: a jk v xj ν x k + p(x)v = 0 on lR × G 1 . Then, by using Theorem 4.1 and the multiplier technique used in (5.6), a short calculation will yield the second result in Theorem 2.2 directly. Now, we will give a brief proof of the decay rates of hyperbolic equations.
Proof of Theorem 2.1. As we said before that once suitable resolvent estimates are established, the existing abstract semi-group results can be adopted to yield the desired energy decay rate. Therefore, based on Theorem 2.2, for any positive integer k, by using an abstract theorem in [6, Théorème 3], we conclude that there exists a constant C > 0 such that ||U || H , ∀ t ≥ 0, U ∈ H, k ≥ 2, k ∈ lN, j = 1, 2 (5.13) that is ||e tAj || L(D(A k j ),H) ≤ C ln(2 + t) k , ∀ t ≥ 0, k ≥ 2, k ∈ lN, j = 1, 2. (5.14) On the other hand, for any s ∈ [0, 1] and k ≥ 2, by definition D(A sk j ) is the interpolate space of order s between D(A 0 j ) = H and D(A k j ) (j = 1, 2). Note also that ||e tAj U || L(H,H) ≤ C Therefore, for any s ∈ [0, 1], by applying the Calderon-Lions interpolation theorem as explained in [6,25], we have that