Boundary Stabilization of a Semilinear Wave Equation with Variable Coefficients under the Time-Varying and Nonlinear Feedback

Abstract

We study the boundary stabilization of a semilinear wave equation with variable coefficients under the time-varying and nonlinear feedback. By the Riemannian geometry methods, we obtain the stability results of the system under suitable assumptions of the bound of the time-varying term and the nonlinearity of the nonlinear term.

1. Introduction

Many results concerning the boundary stabilization of classical wave equations are available in literatures. See [1–6] for linear cases and [7–14] for nonlinear ones. The stability of a nondissipative system described by partial differential equations (PDEs) has attracted much attention. Reference [15] developed the exponential stability for an abstract nondissipative linear system, and in [16], the Riesz basis property was developed for a beam equation with nondissipativity.

In [17], the following semilinear wave equation was considered: and the well-posedness and uniform decay of the energy of the system (1) was also established with linearly bounded in [17].

Based on [17], we study the system (1) with time-varying and nonlinear feedback: The decay rate of the energy (when goes to infinity) of the wave equation with time-varying feedback was established under the assumption is decreasing [18–20] or has an upper bound [21].

In this paper, we consider the decay rate of the energy under suitable assumptions of the bound of the time-varying term and the nonlinearity of the nonlinear term .

2. Some Notation

Let be a bounded domain in    with smooth boundary . It is assumed that consists of two parts and with , .

Let be symmetric, positively definite matrices for each , and are smooth functions on . As in [22], we define as a Riemannian metric on and consider the couple as a Riemannian manifold with an inner product:

Denote by , , , and the Levi-Civita connection, the gradient operator, the divergence operator, and the Beltrami-Laplace operator in terms of the Riemannian metric , respectively. It can be easily shown that, under the Euclidean coordinate, where is the gradient of in the standard metric and .

Let be a vector field on . Then for each , the covariant differential of determines a bilinear form on : where stands for the covariant derivative of the vector field with respect to .

3. The Main Results

We consider the semilinear wave equation with variable coefficients under the time-varying and nonlinear boundary feedback: where are continuous nonlinear functions and is the outside unit normal vector of the Riemannian manifold for each . Different from [18–21], in this paper, we consider a general ; that is, satisfies where is a positive and nondecreasing function satisfying

Let be a positive and nondecreasing function with 0 as the limit. Then satisfies (9). There are many examples of such as and .

The main assumptions are listed as follows.

Assumption A. derives from a potential : and satisfies where are positive constants, and the parameter satisfies
Being different from [17], we assume the nonlinear term has no growth restriction near zero as in [23, 24].

Assumption B. is a nondecreasing function satisfying

Assumption C. There exists a vector field on such that where and where is a constant. Moreover we assume that

Condition (14) as a checkable assumption is very useful to study the control and stabilization of the wave equation with variable coefficients and the quasilinear wave equation [22, 25]. For the examples of the condition, see [22, 26].

Based on condition (14), Assumption C was given by [17] to study the stabilization of the wave equation with variable coefficients and nonlinear boundary condition. Being different from [17], the lower bound of was relaxed on from a positive constant to zero.

To facilitate the writing, we denote the volume element of by and denote the volume element of by . Define the energy of the system (7) by

As in [23, 24], we let be a concave increasing function such that With (18), the stabilization of the wave equation with variable coefficients and time dependent delay was studied by [27].

The main result of this paper is as follows.

Theorem 1. Let Assumptions A–C hold true. Assume that where is defined in (15).
(a) If the function in (7) satisfies then there exist constants such that
(b) If the functions in (7) satisfy where is a positive constant, then there exist constants such that
(c) If the function in (8) is a constant function; that is, then there exist constants such that

4. Well Posedness of the System

Define By a similar proof as Lemma  7.1 in [17], we have the following result.

Theorem 2. Let Assumptions A-B hold true. For any initial data , system (7) admits a unique weak solution u such that .

To prove Theorem 1, we still need several lemmas further. Define Then, we have

The following lemma shows the energy of the system (7) is decreasing.

Lemma 3. Suppose that Assumptions A-B hold true. Let be the solution of the system (7). Then The assertion (29) implies that is decreasing.

Proof. Differentiating (17), we obtain Then the inequality (29) follows directly from (30) integrating from to .

5. Proofs of Theorem 1

Lemma 4. Let be the solution of the equation and that is a vector field defined on . Then for
Moreover, assume that . Then

Proof. Note that The equality (31) and the equality (32) follow from Proposition  2.1 in [22].

Lemma 5. Suppose that all assumptions in Theorem 1 hold true. Let solve the system (7). Then there exist positive constants for which where .

Proof. From (15), we choose a positive constant satisfying Set We substitute the formula (32) into the formula (31), and we have where
Decompose as where stands by the value of the terms on the right side of (38) integrating on .
Similar to [5, 22], we deal with as follows.
Since , we have ; that is, Similarly, we obtain Using the equality (40) and (41) in the equality (38) on the portion , with (16) we obtain
Let be a vector field on such that Set ; it follows from (31) that Then we obtain that With (16) and (45), we have
Note that Substituting the formulas (42) and (46) into the formula (37), with (19) and (35), we obtain
Since from (48), we have
Since is decreasing, we deduce that Substituting the formulas (51) into the formula (50), for sufficiently large , we have The inequality (34) holds.

Proof of Theorem 1. (a) From (8), (13), (20), (29), and (34), for we deduce that Note that is decreasing, and the estimate (21) holds.
(b) From (8), (13), (22), (29), and (34), for we deduce that Note that is decreasing, and the estimate (23) holds.
(c) From (8), (13), (24), (29), and (34), for we deduce that Note that is decreasing, and the estimate (25) holds.

Conflict of Interests

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

Acknowledgments

The authors would like to express their deep thanks to Zhen-Hu Ning for bringing conditions (8) and (16) to us as the requirements for the stability results in Theorem 1. The research is supported by the Open Fund of Trust Computing Laboratory of Beijing University of Technology Start-up Fund of Beijing University of Technology under Grants (007000543113522).