Dynamic Behavior of a One-Dimensional Wave Equation with Memory and Viscous Damping

Abstract

We study the dynamic behavior of a one-dimensional wave equation with both exponential polynomial kernel memory and viscous damping under the Dirichlet boundary condition. By introducing some new variables, the time-variant system is changed into a time-invariant one. The detailed spectral analysis is presented. It is shown that all eigenvalues of the system approach a line that is parallel to the imaginary axis. The residual and continuous spectral sets are shown to be empty. The main result is the spectrum-determined growth condition that is one of the most difficult problems for infinite-dimensional systems. Consequently, an exponential stability is concluded.

1. Introduction

It is known that viscoelastic materials have been widely used in mechanics, chemical engineering, architecture, traffic, information, and so on. More and more researchers have paid close attention to the dynamic behavior and control of vibration for elastic structures with viscoelasticity over the past several decades. The most widely used models for viscoelasticity are the Boltzmann model and Kelvin-Voigt model. For instance, the results concerning the exponential asymptotic stability of a linear hyperbolic integrodifferential equation in Hilbert space are established in [1], which is an abstract version of the equation of motion for dynamic linear viscoelastic solids. In [2, 3], the exponential stabilities of a vibrating beam with one segment made of viscoelastic material of a Kelvin-Voigt type and a vibrating string with Boltzmann damping are proved under certain hypotheses of the smoothness and structural condition of the coefficients of the system. In [4], the global existence and exponential decay of solutions of a nonlinear unidimensional wave equation with a viscoelastic boundary condition are analyzed under some assumptions. Spectral analyses of a wave equation with internal Kelvin-Voigt damping and Boltzmann damping are considered in [5, 6], respectively. The Riesz basis property of the generalized eigenfunctions of a one-dimensional hyperbolic system which is a heat equation incorporating the effect of thermomechanical coupling and the effect of inertia is studied in [7]. In [8], a detailed spectral analysis for a heat equation system which is derived from a thermoelastic equation with memory type is presented and the spectrum-determined growth condition and strong exponential stability are then concluded. Similar studies from different aspects for elastic structures with viscoelasticity can also be found in [9–14] and the references therein.

In this paper, we are interested in the following one-dimensional wave equation with viscoelastic damping under the Dirichlet boundary condition: where and are positive constants and the kernel is taken as the finite sum of exponential polynomials: and we assume that, for simplicity,

Our interest is to investigate the dynamic behavior of (1)-(2), particularly the large time behavior. The asymptotic distribution of eigenvalues of the system is discussed in detail. It is shown that the system operator is of compact resolvent and hence its spectrum consists of isolated eigenvalues of finite algebraic multiplicity only. All eigenvalues approach a line that is parallel to the imaginary axis. The main result is the spectrum-determined growth condition which is one of the most difficult problems for infinite-dimensional systems. Consequently, a strongly exponential stability is concluded.

The paper is organized as follows. In Section 2, we introduce some new variables so that the system (1) with kernel (2) is reduced to be a time-invariant one. Section 3 is devoted to the detailed spectral analysis of the newly formulated system. The main result is the spectrum-determined growth condition that is presented in Section 4; finally, a strongly exponential stability is obtained.

2. System Operator Setup

Introduce Then satisfies Thus, we can rewrite the systems (1) and (2) as Obviously, the system (6) is a time-invariant system. The energy function of (6) is given by Motivated by the energy function, it is natural to consider the system (6) in the following Hilbert space equipped with the inner product, , Now, define the system operator by Then (6) can be formulated as an abstract evolution equation in as where is the state variable and is the initial value.

Lemma 1. Let be defined by (10). Then exists and hence , the resolvent set of . Moreover, generates a -semigroup on .

Proof. Let . Solve for ; that is, to give Set , together with the boundary condition to obtain where Collecting (13), (15), and (16), we get the unique solution to (12). Hence, and exists, or .
Now, we show that can generate a -semigroup on . Actually, for any , we have where we have used the fact . Let Then we have And, hence, . Therefore, by the Lumer-Phillips theorem [15, p.14, Theorem 4.3], generates a -semigroup of contractions on ; thus is an infinitesimal generator of a -semigroup by the perturbation theory of semigroup [15, p.76, Theorem 1.1].

3. Spectral Analysis of System Operator

In this section, we investigate the distribution of spectrum of in the complex plane. Firstly, we consider the eigenvalue problem.

Suppose that for and ; that is,

Proposition 2. Let be defined as in (10). Then , are eigenvalues of , which are corresponding to eigenfunctions , respectively, where is a constant function whose element is the th element of the canonical basis of . Moreover, each of these eigenvalues is algebraically simple.

Proof. We only give the proof for because other cases can be treated similarly. Let and the eigenvalue problem becomes From the third equation of (22), it has . This in turn, together with the fourth equation of (22), yields and, by (3), we arrive at By the first and the second equations of (22), these yield Therefore, is an eigenfunction of corresponding to . Further computation of , where , yields We claim that (26) has no solution since must satisfy which is impossible. Hence is algebraically simple. The proof is complete.

When , it follows from (21) that And satisfies

Lemma 3. Let be defined by (10); then for any and , .

Proof. Firstly, we show that Re. When , (29) gives Multiply by , the conjugate of , and integrate over with respect to for the equation to give That is, Replace in the above formula to get Let the real and imaginary parts of the equation equal zero to obtain If , from the first equation of (34), then it has If , from the second equation of (34), then we have so
Secondly, we prove that . We use contradiction argument. Suppose that ; letting , from (34), we have From the second equation, we obtain or . If holds, it has ; then, from (28), we have . Hence, it has only . This is a contradiction. The proof is complete.

Now, we give the asymptotic fundamental solutions of (29) as .

Lemma 4. Suppose that and , , where stands for the point spectrum set of . Then and are linearly independent fundamental solutions of , and as has the following asymptotic fundamental solutions: where

Proof. The first claim is trivial. We only need to show that (40) is the asymptotic fundamental solution of (39). This can be done along the same way of Birkhoff [16] and Naimark [17]. Here we present briefly a simple calculation to (40). Let where and , , are some functions to be determined, and Substitute and into , respectively, to yield where Moreover, there exists some positive constant such that Thus, letting the coefficients of and be zero gives where is given by (41).
Now, use the condition , to get These are the expressions of (41). When is large enough, we can obtain the linearly independent asymptotic fundamental solutions of (39) given by (40) (see [16]): The proof is complete.

Assume that and , and let where , are given by (40). Then substitute the above into the boundary conditions of (29) to obtain where Hence (29) admits nontrivial solution if and only if and the eigenvalues of (29) are the zeros of (53). Notice that where and is given in Lemma 4.

Therefore produces which further leads to At last, the solutions of are Apply the Rouché theorem to (57) to give the solutions of (57) as follows:

We summarize the above analysis as the following Theorem.

Theorem 5. The eigenvalues of (29) have the following asymptotic expressions: especially, That is, is the asymptote of the eigenvalues given by (60). Furthermore, the corresponding eigenfunctions , have the asymptotic expressions Moreover, and are approximately normalized in in the sense that there exist positive constants and independent of such that, for ,

Proof. From above statement, (60) has been proved. We only need to show (62)-(63) to be valid. Since , , in view of (52) and (60), Lemma 4, and some facts in linear algebra, the eigenfunction corresponding to the eigenvalue is given by Owing to the fact of (60) that (62) is hence proved by taking Finally, These give (63). The proof is complete.

The following result is the consequence of Theorem 5 and Proposition 2.

Theorem 6. Let be defined as in (10). Then(i) has the eigenvalues  where have the asymptotic expressions (60),(ii)the eigenfunction corresponding to is for any ;(iii)the eigenfunctions corresponding to and are given by for , respectively, where and is given by

Concerning about , we have Theorem 7.

Theorem 7. Let be defined as in (10). Then and are empty sets, where and denote the set of residual and continuous spectrum of , respectively.

Proof. We need only to prove that , when . Letting , then . Solve for and ; that is, to get and so Set Then we can write (71) as where Hence (75) can be rewritten as the following first-order system of differential equations: Let Then it has where and is given by The general solution of (77) is then found to be Since , it has When , (83) and (84) reduce to the eigenvalue problem So, when , , if and only if , that is, which yields This is the characteristic determinant of , which has the asymptotic form given by (60).
Now, since , it has By the boundary condition , we have Hence is uniquely determined by (83) and . Once is known, and are also uniquely determined by (72). So exists and is bounded. Therefore . The proof is complete.

4. Spectrum-Determined Growth Condition and Exponential Stability

Now we are in a position to consider the main result of this part, the so-called spectrum-determined growth condition for system (11), which is one of the most hard problems for infinite-dimensional systems. Our proof is based on the following characterization condition (see [18, Corollary 3.40, p.144]).

Lemma 8. Let be a -semigroup on a Hilbert space with its generator . Let be the growth bound of and let be the spectral bound of . Then

We also need Lemma 1.2 of [19].

Lemma 9. Let where are polynomials of , are some complex numbers, and n is a positive integer. Then, for all outside those circles of radius that centered at the zeros of , one has for some constant that depends only on .

Lemma 10. Let where and are given by (76) and (81), respectively. Then, as indicated by (87), all eigenvalues claimed by (60) are zeros of . Moreover, for all that are outside those circles centered at with radius , one has for some constant that depends only on .

Proof. By (57), it has Then the desired result follows from Lemma 9. The proof is complete.

Theorem 11. Let be defined by (10). Then the spectrum-determined growth condition holds true for .

Proof. By Lemma 8, the proof will be accomplished if we can prove that, for any and with and , there exists a constant such that Let with and . For any , from the proof of Theorem 7, satisfies where , are given by (80).
Firstly, it is easy to see that there is a positive constant such that Secondly, for simplicity, let and , since So combing (99), there is a positive constant such that Thirdly, by Proposition 2 and Theorem 7, it has Define By Lemma 10, there exists a positive constant depending only on such that Hence, by (86), there exists a positive constant depending only on such that So, by the Cauchy-Schwartz inequality, it follows from (101) and the last equation of (98) that there is a positive constant such that Finally, from (98), there is a positive constant such that
This gives (97). The proof is complete.

The following result gives a strongly exponential stability for the system (1).

Theorem 12. is exponentially stable; that is, there exist constants such that or, equivalently,

Proof. By the spectrum-determined growth condition claimed by Theorem 11, the asymptote for eigenvalues of (61) and (109), , is exponentially stable if and only if From Proposition 2, Lemma 3, and Theorem 7, for arbitrary, . The proof is complete.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.