Stability of a Class of Coupled Systems

and Applied Analysis 3 Moreover, taking the inner product of (2) with V󸀠(t) and integrating over [0, t], we obtain


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 ;  1 () and  2 () are two nonnegative functions on [0, +∞) 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: for a constant  > 0.
We define the energy of a solution (, V) of ( 1)-(4) as (9) 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 V  (), respectively, and summing-up, we obtained the equality (21).
See more properties of the strongly positive definite kernel in [2,5].

Result and Proof
Theorem 4. Let ( 1 )-( 2 ) hold, and let  0 , V 0 ,  1 , and V 1 be as in Theorem 1.Then, the energy () satisfies where  > 0 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].
where  1 > 0 depends only on the initial data.
Proof.It follows from (1) that Abstract and Applied Analysis 3 Moreover, taking the inner product of (2) with V  () and integrating over [0, ], we obtain Combining the above two equations and using integration by parts, we get Applying Lemma 3.4 and (3.13) in [2] to the two integral terms on the left-hand side, we have Noticing ( 2 ) and Remark 2, we obtain (16).
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 (2.14) of [2], (16), and (21), we have where  3 > 0 depends only on the initial data.Moreover, in view of ( 23) and ( 1 ), we have where  4 > 0 depends only on the initial data.
Then, for any  ≥ 0, where  5 > 0 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