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Mathematical Problems in Engineering

Volume 2014 (2014), Article ID 176583, 5 pages

http://dx.doi.org/10.1155/2014/176583

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

^{1}College of Computer Science, Beijing University of Technology, Beijing 100124, China^{2}State Engineering Laboratory of Information System Classified Protection Key Technologies, Beijing University of Technology, Beijing 100124, China

Received 24 April 2014; Accepted 7 June 2014; Published 29 June 2014

Academic Editor: Yoshinori Hayafuji

Copyright © 2014 Jian-Sheng Tian 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 and nonlinear term. By the Riemannian geometry methods and a suitable assumption of nonlinearity and the time-varying term, we obtain the uniform decay of the energy of the system.

#### 1. Introduction

There are many 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 time-varying 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 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: and there exists a positive constant such that and satisfies where is a positive constant and .

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

Define the energy of the system (2) by

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 (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 , such that
*

*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 (8).*

*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 [13] in Lemma 2.1.*

*Lemma 2. 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 need several lemmas further. Define
Then, we have
*

*Lemma 3. Let be the solution of system (2). Then there exists a constant such that
where . The assertion (22) implies that is decreasing.*

* Proof. *Differentiating (7), we obtain
Then the inequality (22) holds true.

*3. Proofs of Theorem 1*

*3. Proofs of Theorem 1*

*From Proposition 2.1 in [13], we have the following identities.*

*Lemma 4. Suppose that solves equation , and that is a vector field defined on . Then, for ,
*

Moreover, assume that . 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 for which
where .*

* Proof. *We let be a positive constant satisfying
Set
Substituting the identity (25) into the identity (24), we obtain
where

Decompose as
Since , we obtain ; that is,
Similarly, we have
Using the formulas (32) and (33) in the formula (30) on the portion , with (12), we obtain
From (12), we have

Substituting the formulas (34) and (35) into the formula (29), with (27), we obtain

It follows from (22) that

Substituting the formulas (22) and (37) into the formula (36), the inequality (26) holds.

*Proof of Theorem 1. *Since is decreasing, with (4) and (26), for sufficiently large , we have
Note that is decreasing; the estimate (15) holds.

*4. Application of the System (2)*

*4. 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.*

*Conflict of Interests*

*Conflict of Interests*

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

*References*

*References*

- J. E. Lagnese, “Note on boundary stabilization of wave equations,”
*SIAM Journal on Control and Optimization*, vol. 26, no. 5, pp. 1250–1256, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. L. Russell, “Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions,”
*SIAM Review*, vol. 20, no. 4, pp. 639–739, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Triggiani, “Wave equation on a bounded domain with boundary dissipation: an operator approach,”
*Journal of Mathematical Analysis and Applications*, vol. 137, no. 2, pp. 438–461, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. You, “Energy decay and exact controllability for the Petrovsky equation in a bounded domain,”
*Advances in Applied Mathematics*, vol. 11, no. 3, pp. 372–388, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Aassila, M. M. Cavalcanti, and V. N. D. Cavalcanti, “Existence and uniform decay of the wave equation with nonlinear boundary damping and boundary memory source term,”
*Calculus of Variations and Partial Differential Equations*, vol. 15, no. 2, pp. 155–180, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. M. Cavalcanti, V. N. Domingos Cavalcanti, and P. Martinez, “Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term,”
*Journal of Differential Equations*, vol. 203, no. 1, pp. 119–158, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Conrad and B. Rao, “Decay of solutions of the wave equation in a star-shaped domain with nonlinear boundary feedback,”
*Asymptotic Analysis*, vol. 7, no. 3, pp. 159–177, 1993. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Komornik, “On the nonlinear boundary stabilization of the wave equation,”
*Chinese Annals of Mathematics B*, vol. 14, no. 2, pp. 153–164, 1993. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Komornik,
*Exact Controllability and Stabilization: The Multiplier Method*, John Wiley & Sons, Chichester, UK, 1994. View at MathSciNet - V. Komornik and E. Zuazua, “A direct method for the boundary stabilization of the wave equation,”
*Journal de Mathématiques Pures et Appliquées*, vol. 69, no. 1, pp. 33–54, 1990. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. Lasiecka and D. Tataru, “Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping,”
*Differential and Integral Equations*, vol. 6, no. 3, pp. 507–533, 1993. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Zuazua, “Uniform stabilization of the wave equation by nonlinear boundary feedback,”
*SIAM Journal on Control and Optimization*, vol. 28, no. 2, pp. 466–477, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P.-F. Yao,
*Modeling and Control in Vibrational and Structural Dynamics: A Differential Geometric Approach*, Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series, CRC Press, Boca Raton, Fla, USA, 2011. View at Publisher · View at Google Scholar · View at 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 · View at Google Scholar · View at Zentralblatt MATH · View at 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 · View at Google Scholar · View at Zentralblatt MATH · View at 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 · View at Google Scholar · View at MathSciNet - Z. H. Ning, C. X. Shen, X. Zhao, H. Li, C. Lin, and Y. M. Zhang, “Nonlinear boundary stabilization of the wave equations with variable coefficients and time dependent delay,”
*Applied Mathematics and Computation*, vol. 232, pp. 511–520, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - 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 · View at Zentralblatt MATH · View at MathSciNet - B. Gong and X. Zhao, “Boundary stabilization of a semilinear wave equation with variable coefficients under the time-varying and nonlinear feedback,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 728760, 6 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - H. Li, C. S. Lin, S. P. Wang, and Y. M. Zhang, “Stabilization of the wave equation with boundary time-varying delay,”
*Advances in Mathematical Physics*, vol. 2014, Article ID 735341, 6 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - B.-Z. Guo and Z.-C. Shao, “On exponential stability of a semilinear wave equation with variable coefficients under the nonlinear boundary feedback,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 71, no. 12, pp. 5961–5978, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. H. Ning, C. X. Shen, and X. P. Zhao, “stabilization of the wave equation with variable coefficients and a internal memory type,”
*Nonlinear Analysis: Real World Applications*. In press. - M. M. Cavalcanti, V. N. D. Cavalcanti, and I. Lasiecka, “Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping—source interaction,”
*Journal of Differential Equations*, vol. 236, no. 2, pp. 407–459, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet

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