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.