Boundary Stabilization of the Wave Equation with Time-Varying and Nonlinear Feedback

There aremany results concerning the boundary stabilization of classical wave equations. See [1–6] for linear cases and [7– 12] for nonlinear ones.The stability of the wave equation with variable coefficients has attracted much attention. See [13– 23], and many others. In [20], by the methods in [11, 24], the authors study the stability of the wave equation with nonlinear term and time-varying term. However, under the condition the nonlinear term has upper bound and the timevarying term has lower bound, the stability of the wave equation was not studied in [20]. In this paper, our purpose is to study the stability of thewave equation under the condition the nonlinear term has upper bound and the time-varying term has lower bound. LetΩ be a bounded domain inR with smooth boundary Γ. It is assumed that Γ consists of two parts Γ 1 and Γ 2 (Γ =


Introduction
There are many results concerning the boundary stabilization of classical wave equations.See [1][2][3][4][5][6] for linear cases and [7][8][9][10][11][12] for nonlinear ones.The stability of the wave equation with variable coefficients has attracted much attention.See [13][14][15][16][17][18][19][20][21][22][23], and many others.In [20], by the methods in [11,24], the authors study the stability of the wave equation with nonlinear term and time-varying term.However, under the condition the nonlinear term has upper bound and the timevarying term has lower bound, the stability of the wave equation was not studied in [20].In this paper, our purpose is to study the stability of the wave equation under the condition the nonlinear term has upper bound and the time-varying term has lower bound.
Let Ω be a bounded domain in R  with smooth boundary Γ.It is assumed that Γ consists of two parts Γ 1 and Γ where div is the divergence operator of the standard metric of R  ; () = (  ()) is symmetric, positively definite matrices for each  ∈ R  and   () are smooth functions on R  .
We consider the stabilization of the wave equations with variable coefficients and time-varying delay in the dissipative boundary feedback: 1 ∈ (R) and there exists a positive constant  1 such that and () ∈ ([0, +∞)) satisfies lim where  0 is a positive constant and () = max 0≤≤ ().
where ⟨⋅, ⋅⟩ denotes the standard metric of the Euclidean space R  and ]() is the outside unit normal vector for each  ∈ Γ.Moreover, the initial data ( 0 ,  1 ) belongs to a suitable space.
Define the energy of the system (2) by We define as a Riemannian metric on R  and consider the couple (R  , ) as a Riemannian manifold with an inner product: Let   denote the Levi-Civita connection of the metric .For the variable coefficients, the main assumptions are as follows.
Assumption A. There exists a vector field  on Ω and a constant  0 > 0 such that Moreover we assume that sup where  is a positive constant.
Assumption (10) was introduced by [13] as a checkable assumption for the exact controllability of the wave equation with variable coefficients.For examples on the condition, see [13,14].
Based on Assumption (10), Assumption A was given by [19] to study the stabilization of the wave equation with variable coefficients and boundary condition of memory type. Define To obtain the stabilization of the system (2), we assume the system (2) is well-posed such that The main result of this paper is stated as follows.
Theorem 1.Let Assumption A holds true.Then there exist positive constants ,  2 , such that

Basic Inequality of the System
In this section we work on Ω with two metrics at the same time, the standard dot metric ⟨⋅, ⋅⟩ and the Riemannian metric  = ⟨⋅, ⋅⟩  given by (8).
If  ∈  1 (R  ), we define the gradient ∇   of  in the Riemannian metric , via the Riesz representation theorem, by where  is any vector field on (R  , ).The following lemma provides further relations between the two metrics; see [13] in Lemma 2.1.
Lemma 2. Let  = ( 1 , . . .,   ) be the natural coordinate system in R  .Let , ℎ be functions and let H,  be vector fields.Then where ∇ is the gradient of  in the standard metric; (c) where the matrix () is given in formula (1).
To prove Theorem 1, we still need several lemmas further.Define Then, we have Lemma 3. Let () be the solution of system (2).Then there exists a constant  1 such that where  ≥ 0. The assertion (22) implies that () is decreasing.

Proofs of Theorem 1
From Proposition 2.1 in [13], we have the following identities.
Then, for  ≥ 0, Moreover, assume that  ∈  1 (Ω).Then Lemma 5. Suppose that all assumptions in Theorem 1 hold true.Let  be the solution of the system (2).Then there exist positive constants ,  0 for which where  ≥  0 .

Application of the System (2)
Nonlinear feedback describes a property of a physical system; that is, the response by the physical system to an applied force is nonlinear in its effect.One of the applications of the system (2) is in sound waves, where the system (2) describes the reflection of sound in heterogeneous materials at surfaces of some materials with nonlinearity of interest in engineering practice.Theorem 1 indicates that the energy of the sound waves with the reflection of sound at surfaces in heterogeneous materials at surfaces of some materials with nonlinearity is uniform decay under a suitable assumption of the nonlinearity.