Abstract and Applied Analysis

Volume 2014 (2014), Article ID 835765, 5 pages

http://dx.doi.org/10.1155/2014/835765

Research Article

Stability of a Class of Coupled Systems

Kun-Peng Jin1,2

¹School of Mathematical Sciences, Fudan University, Shanghai 200433, China

²Department of Mathematics and Physics, Shanghai Dian Ji University, Shanghai 201306, China

Received 9 March 2014; Accepted 4 April 2014; Published 16 April 2014

Academic Editor: Xinan Hao

Copyright © 2014 Kun-Peng Jin. 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 consider a class of coupled systems with damping terms. By using multiplier method and the estimation techniques of the energy, we show that even if the kernel function is nonincreasing and integrable without additional conditions, the energy of the system decays also to zero in a good rate.

1. Introduction

This work is motivated by the recent researches on the Cauchy problem for the coupled evolution equations with memory (e.g., Alabau-Boussouira et al. [1], Cannarsa and Sforza [2], Wan and Xiao [3], and Xiao and Liang [4]).

We study the following abstract Cauchy problem for coupled systems with damping terms:

where is a positive self-adjoint linear operator in a Hilbert space ; and are two nonnegative functions on and denote the memory kernel, which will be specified later. The problem arises in the theory of viscoelasticity.

We are concerned with the delay behavior of the energy of the systems. In the real world, for the viscoelastic material, the kernel function is almost all nonincreasing and nonnegative. Therefore, we are more interested in decay behavior when the kernel is nonnegative and nonincreasing. In this case, is a strongly positive definite kernel (as in [2, 5]). By using multiplier method and the estimation techniques of the energy, we show that even if the kernel function is nonincreasing and integrable without additional conditions, the energy of the system decays also to zero in a good rate.

Let us recall the following assumptions which were used in related literature:

(I ₁) is a positive self-adjoint linear operator in , satisfying

 for a constant .(I ₂) is a nonincreasing and integrable function such that

 where .

A pair of functions is called a (classical) solution of (1)–(4) on , if

satisfying (1)–(4) for .

We define the energy of a solution () of (1)–(4) as

About the information on , see Xiao and Liang's monograph [6].

Theorem 1. Let hold. Then, for and , (1)–(4) have a unique solution on and

Proof. The existence and uniqueness of solution can be obtained by the standard operator theory. Here, we omit it.

Multiplying (1) by and (2) by , respectively, and summing-up, we obtained the equality (21).

Remark 2. From assumption and (21), we have

For any and any , we define

Next, let us recall the concept of strongly positive definite kernel. It can be found in [2, 5].

Definition 3. Set ; is called positive definite kernel if, for any ,

Also, is said to be a strongly positive definite kernel if there exists a constant such that is positive definite, for any .

See more properties of the strongly positive definite kernel in [2, 5].

2. Result and Proof

Theorem 4. Let hold, and let , and  be as in Theorem 1. Then, the energy satisfies

where is a positive constant and depends on the initial data. Moreover,

To prove Theorem 4, we need the following lemmas.

From now on, we write

Then, is a strongly positive definite kernel; see [2, Theorem 2.1].

Lemma 5. Let hold, , and . Then, for any ,

where depends only on the initial data.

Proof. It follows from (1) that

Moreover, taking the inner product of (2) with and integrating over , we obtain

Combining the above two equations and using integration by parts, we get

Applying Lemma 3.4 and in [2] to the two integral terms on the left-hand side, we have

Noticing and Remark 2, we obtain (16).

Lemma 6. Let hold, , and . Then, for any ,

where depends only on the initial data.

Proof. Differentiating the systems (1)-(2) with respect to , we get

Thus, similar to the proof of the Lemma 5 for the above (22), we deduce (21).

In view of Lemma 2.9 and of [2], (16), and (21), we have

where depends only on the initial data.

Moreover, in view of (23) and , we have

where depends only on the initial data.

Lemma 7. Let hold, , and . Then, for any ,

where depends only on the initial data.

Proof. It follow from (1) and (2) that

Note that we have used (24)-(25) in the above calculation. Hence, we have

On the other hand, we see that

By Young’s inequality, we obtain

Putting (31)-(32) into (29), we obtain

Noticing assumption , we obtain the desired estimates (26)-(27).

Proof of Theorem 4. First, we estimate the two memory energy terms.

By a direct calculation, we have

Hence, by (26), we obtain

Similarly, we have

Thus, (24)–(27) and (35)-(36) yield

for a positive constant . As , we have

Accordingly, (37) means that

Hence, the estimate (13) follows. Furthermore, since the integral is convergent, it follows that

via the Cauchy convergence principle. Then, the proof of Theorem 4 is completed.

Conflict of Interests

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

Acknowledgment

The work was supported partly by the NSF of China (nos. 11371095 and 11271082).

References

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