Mathematical Problems in Engineering

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

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

Research Article

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

Jian-Sheng Tian,1,2 Wei Wang,1,2 Fei Xue,1,2 and Pei-Yong Cong1,2

¹College of Computer Science, Beijing University of Technology, Beijing 100124, China

²State 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

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)

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

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

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