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:(I1) is a positive self-adjoint linear operator in , satisfying for a constant .(I2) 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).