Advances in Mathematical Physics

Volume 2014 (2014), Article ID 735341, 6 pages

http://dx.doi.org/10.1155/2014/735341

Research Article

Stabilization of the Wave Equation with Boundary Time-Varying Delay

Hao Li,¹ Changsong Lin,¹ Shupeng Wang,² and Yanmei Zhang³

¹School of Ocean Sciences, China University of Geosciences, Beijing 100083, China

²Institute of Information Engineering, Chinese Academy of Sciences, Beijing 100093, China

³School of Information Engineering, China University of Geosciences, Beijing 100083, China

Received 18 October 2013; Revised 10 January 2014; Accepted 10 January 2014; Published 26 February 2014

Academic Editor: Wen-Xiu Ma

Copyright © 2014 Hao Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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).

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