Research Article | Open Access

# Stabilization of the Wave Equation with Boundary Time-Varying Delay

**Academic Editor:**Wen-Xiu Ma

#### 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).

#### References

- A. Benaissa and A. Guesmia, “Energy decay for wave equations of $\varphi $-Laplacian type with weakly nonlinear dissipation,”
*Electronic Journal of Differential Equations*, vol. 2008, no. 109, pp. 1–22, 2008. View at: Google Scholar | MathSciNet - G. Chen, “Control and stabilization for the wave equation in a bounded domain,”
*SIAM Journal on Control and Optimization*, vol. 17, no. 1, pp. 66–81, 1979. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. Chen, “Control and stabilization for the wave equation in a bounded domain, part II,”
*SIAM Journal on Control and Optimization*, vol. 19, no. 1, pp. 114–122, 1981. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. Haraux, “Two remarks on dissipative hyperbolic problems,” in
*Nonlinear Partial Differential Equations and Their Applications*, vol. 122 of*Research Notes in Mathematics*, pp. 161–179, Pitman, Boston, Mass, USA, 1985. View at: Google Scholar | Zentralblatt MATH | MathSciNet - I. Lasiecka and D. Tataru, “Uniform boundary stabilization of semilinear wave equations with nonlinear boundary dampin,”
*Differential Integral Equations*, vol. 6, no. 3, pp. 507–533, 1993. View at: Google Scholar | Zentralblatt MATH | MathSciNet - K. Liu, “Locally distributed control and damping for the conservative systems,”
*SIAM Journal on Control and Optimization*, vol. 35, no. 5, pp. 1574–1590, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - E. Zuazua, “Exponential decay for the semilinear wave equation with locally distributed damping,”
*Communications in Partial Differential Equations*, vol. 15, no. 2, pp. 205–235, 1990. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Feng and D. X. Feng, “Locally distributed control of wave equations with variable coefficients,”
*Science in China A*, vol. 44, no. 3, pp. 345–350, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Nicaise and C. Pignotti, “Interior feedback stabilization of wave equations with time dependent delay,”
*Electronic Journal of Differential Equations*, vol. 2011, no. 41, pp. 1–20, 2011. View at: Google Scholar | Zentralblatt MATH | MathSciNet - S. Nicaise and C. Pignotti, “Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks,”
*SIAM Journal on Control and Optimization*, vol. 45, no. 5, pp. 1561–1585, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Nicaise, C. Pignotti, and J. Valein, “Exponential stability of the wave equation with boundary time-varying delay,”
*Discrete and Continuous Dynamical Systems S*, vol. 4, no. 3, pp. 693–722, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - P.-F. Yao, “On the observability inequalities for exact controllability of wave equations with variable coefficients,”
*SIAM Journal on Control and Optimization*, vol. 37, no. 5, pp. 1568–1599, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - P.-F. Yao, “Observability inequalities for shallow shells,”
*SIAM Journal on Control and Optimization*, vol. 38, no. 6, pp. 1729–1756, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - P.-F. Yao, “Global smooth solutions for the quasilinear wave equation with boundary dissipation,”
*Journal of Differential Equations*, vol. 241, no. 1, pp. 62–93, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - P.-F. Yao, “Boundary controllability for the quasilinear wave equation,”
*Applied Mathematics and Optimization*, vol. 61, no. 2, pp. 191–233, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - P.-F. Yao, “Energy decay for the Cauchy problem of the linear wave equation of variable coefficients with dissipation,”
*Chinese Annals of Mathematics B*, vol. 31, no. 1, pp. 59–70, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z.-F. Zhang and P.-F. Yao, “Global smooth solutions of the quasi-linear wave equation with internal velocity feedback,”
*SIAM Journal on Control and Optimization*, vol. 47, no. 4, pp. 2044–2077, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - I. Lasiecka, R. Triggiani, and P.-F. Yao, “Inverse/observability estimates for second-order hyperbolic equations with variable coefficients,”
*Journal of Mathematical Analysis and Applications*, vol. 235, no. 1, pp. 13–57, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z.-H. Ning and Q.-X. Yan, “Stabilization of the wave equation with variable coefficients and a delay in dissipative boundary feedback,”
*Journal of Mathematical Analysis and Applications*, vol. 367, no. 1, pp. 167–173, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z.-H. Ning, C.-X. Shen, and X. P. Zhao, “Stabilization of the wave equation with variable coefficients and a delay in dissipative internal feedback,”
*Journal of Mathematical Analysis and Applications*, vol. 405, no. 1, pp. 148–155, 2013. View at: Publisher Site | Google Scholar | MathSciNet - Z. H. Ning, C. X. Shen, X. P. Zhao, H. Li, C. S. Lin, and Y. M. Zhang, “Nonlinear boundary stabilization of the wave 5
equations with variable coeffcients and time dependent delay,”
*Applied Mathematics and Computation*, vol. 232, no. 1, pp. 511–520, 2014. View at: Publisher Site | Google Scholar - S. Nicaise and C. Pignotti, “Stabilization of the wave equation with variable coefficients and boundary condition of memory type,”
*Asymptotic Analysis*, vol. 50, no. 1-2, pp. 31–67, 2006. View at: Google Scholar | Zentralblatt MATH | MathSciNet - P.-F. Yao,
*Modeling and Control in Vibrational and Structural Dynamics: A Differential Geometric Approach*, Chapman & Hall/CRC Applied Mathematics & Nonlinear Science, CRC Press, Boca Raton, Fla, USA, 2011. View at: Publisher Site | MathSciNet - R. Gulliver, I. Lasiecka, W. Littman, and R. Triggiani, “The case for differential geometry in the control of single and coupled PDEs: the structural acoustic chamber,” in
*Geometric Methods in Inverse Problems and PDE Control*, vol. 137 of*The IMA Volumes in Mathematics and Its Applications*, pp. 73–181, Springer, New York, NY, USA, 2004. View at: Publisher Site | Google Scholar | MathSciNet

#### Copyright

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.