Stabilization of the Wave Equation with Boundary Time-Varying Delay

Abstract

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.

1. Introduction

Let be a bounded domain in with smooth boundary . It is assumed that consists of two parts and with . Define where is the divergence operator of the standard metric of . is symmetric, positively definite matrices for each and are smooth functions on .

We consider the stabilization of the wave equations with variable coefficients and time-varying delay in the dissipative boundary feedback as follows: where satisfies where and and are constants. and there exist positive constants such that satisfies and satisfies where , , and are positive constants and .

is the conormal derivative where denotes the standard metric of the Euclidean space and is the outside unit normal vector for each . Moreover, , , , and the initial data belongs to a suitable space.

There is a specific example for . Let be a constant. If satisfies then Conditions (7) and (8) hold.

In absence of delay (), the problem (2) was studied by [1–8] and many others. The decay rate of the energy (when goes to infinity) depends on the function and the growth of .

The system (2) with constant coefficient (the case: is a constant matrix on ) was studied by [9–11] and many other authors. For the system (2) with variable coefficients, the main tools to cope with the system (2) are the differential geometrical methods which were introduced by [12] and have been applied in many papers. See [13–22] and references cited therein. For a survey on the differential geometric methods, see [23, 24].

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, 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 time-varying, weakly nonlinear terms. To obtain our stabilization result, we assume that where 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 and consider the couple 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 and a constant such that Moreover, we assume that 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 and satisfies where and are positive constants, then it follows from (13) that there exist constants and such that Then, the decay of the energy is exponential. Methods in [21, 22] are useful for Theorem 1.

2. Basic Inequality of the System

In this section, we work on with two metrics at the same time: the standard dot metric and the Riemannian metric given by (15).

If , we define the gradient of in the Riemannian metric , via the Riesz representation theorem, by where is any vector field on . The following lemma provides further relations between the two metrics; see [12], Lemma 3.

Lemma 3. Let be the natural coordinate system in . Let , be functions and let , be vector fields. Then,(a) (b)  where is the gradient of in the standard metric;(c)  where the matrix is given in formula (1).

To prove Theorem 1, we still further need several lemmas. Define Then, we have

Lemma 4. Suppose that condition (14) holds true. Let be the solution of system (2). Then, there exist constants , such that where . Assertion (31) implies that is decreasing.

Proof. Differentiating (13), we obtain
Applying Green’s formula and by integrating by parts with (3) and (8), we arrive at It follows from (3), (4), (12), and (14) that where satisfies Then, inequality (31) follows directly from (34) integrating from to .

3. Proofs of Theorem 1

From Proposition 2.1 in [12], we have the following identities.

Lemma 5. Suppose that solves equation and that is a vector field defined on . Then, for ,
Moreover, assume that . Then,

Lemma 6. Suppose that all assumptions in Theorem 1 hold true. Let be the solution of the system (2). Then, there exists a positive constant for which where .

Proof. Let be a positive constant satisfying Set Substituting identity (37) into identity (36), we have where
We decompose as Since , we obtain ; that is, Similarly, we have Using formulas (44) and (45) in formula (42) on the portion , with (19), we obtain From (19), we have
Substituting formulas (46) and (47) into formula (41), with (39), we obtain Let , and from (3), (7), (8), and (30), we have Since substituting formula (49) into formula (48), we obtain Inequality (38) holds.

Proof of Theorem 1. Since is decreasing, from (38), for sufficiently large , we have where is defined in (8). With (4)–(8) and (31), we deduce that Therefore, Note that is decreasing; estimate (22) holds.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is supported by the National Science Foundation of China (nos. 91328201 and 41130422) and the National Basic Research Program of China (no. 2011CB201103).