Stabilization of the Wave Equation with Boundary Time-Varying Delay

We study the stabilization of the wave equation with variable coefficients in a bounded domain and a time-varying delay term in the time-varying, weakly nonlinear boundary feedbacks. By the Riemannian geometry methods and a suitable assumption of nonlinearity, we obtain the uniform decay of the energy of the closed loop system.

The authors in [11] considered the system (2) with constant coefficients operator and dissipative boundary conditions of time dependent delay and proved the exponential decay of the energy by combining the multiplier method with the use of suitable integral inequalities.Different from this paper,  1 is assumed to be linearly bounded and  is assumed to be a constant function in the paper [11].
Based on [11], the purpose of this paper is to solve the stability of the system (2) with variable coefficients and timevarying, weakly nonlinear terms.To obtain our stabilization result, we assume that where  2 is defined in (8).Define the energy of the system (2) by where  is a positive constant satisfying 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 (17) was introduced by [12] as a checkable assumption for the exact controllability of the wave equation with variable coefficients.Assumption A is also useful for the controllability and the stabilization of the quasilinear wave equation [15].For the examples of the condition, see [12,23].
Based on Assumption (17), Assumption A was given by [22] to study the stabilization of the wave equation with variable coefficients and boundary condition of memory type.The authors in [22] also constructed some examples of the condition based on the assumption that () = () or () is a perturbation of a symmetric positive definite matrix . Define To obtain the stabilization of the system (2), we assume that the system (2) is well posed such that The main result of this paper is the following.
Theorem 1.Let Assumption A hold true.Then, there exists a constant , such that Remark 2. If  = 1 and () satisfies where  0 and  1 are positive constants, then it follows from (13) that there exist constants  0 > 0 and 0 <  0 < 1 such that Then, the decay of the energy () is exponential.Methods in [21,22] are useful for Theorem 1.

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 (15).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 [12], Lemma 3.
Lemma 3. 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).

Proofs of Theorem 1
From Proposition 2.1 in [12], we have the following identities.