# Exponential Stability of Two Coupled Second-Order Evolution Equations

- Qian Wan
^{1}and - Ti-Jun Xiao
^{1}Email author

**2011**:879649

https://doi.org/10.1155/2011/879649

© Q.Wan and T.-J. Xiao. 2011

**Received: **30 October 2010

**Accepted: **21 November 2010

**Published: **2 December 2010

## Abstract

By using the multiplier technique, we prove that the energy of a system of two coupled second order evolution equations (one is an integro-differential equation) decays exponentially if the convolution kernel decays exponentially. An example is give to illustrate that the result obtained can be applied to concrete partial differential equations.

## 1. Introduction

Here is a positive self-adjoint linear operator, , , is a nonnegative function on . Moreover, the fractional power is defined as in the well known operator theory (cf, e.g., [1, 2]).

An interesting and difficult point for it is to stabilize the whole system via the damping effect given by only one equation (1.1). We remark that there is very few work concerning the situation when the damping mechanism is given by memory terms; see [3], where a coupled Timoshenko beam system is investigated.

On the other hand, the stability of the single integro-differential equation has been studied extensively; see, for instance, [4, 5].

In this paper, through suitably choosing multipliers for the energy together with other techniques, we obtain the desired exponential decay result for the system (1.1)–(1.3). Nonlinear coupled systems with general decay rates will be discussed in a forthcoming paper.

In Section 2, we present our exponential decay theorem and its proof. An application is given in Section 3.

## 2. Exponential Decay Result

We start with stating our assumptions:

The following is our exponential decay theorem.

Theorem 2.1.

Let the assumptions be satisfied. Then,

(i)for any and , problem (1.1)–(1.3) admits a unique mild solution on . The solution is a classical one, if and ,

for any mild solution of (1.1)–(1.3).

Proof.

in . From the assumptions, one sees that— is the generator of a strongly continuous cosine function on , and is bounded from into . Therefore, we justify the assertion (i) (cf., e.g. [6]).

Furthermore, we need the following lemmas.

Lemma 2.2.

for some positive constants which only depend on , , , and .

Proof.

where is a positive constant. Those terms of the form can be similarly treated. Denote by the sum of the other terms on the right of (2.17).

where is a positive constant, small enough to satisfy . We thus verify our conclusion.

Lemma 2.3.

Proof.

The other items on the right of (2.25) can be dealt with as in the proof of Lemma 2.2. Hence, we get the conclusion.

Lemma 2.4.

Proof.

This yields the estimate as desired.

Lemma 2.5.

Proof.

we prove the conclusion.

Proof of Theorem 2.1 (continued).

By a standard approximation argument, we see that (2.45) is also true for mild solutions. From this integral inequality, we complete the proof (cf., e.g., [7, Theorem 8.1]).

## 3. An Example

Example 3.1.

Then, Assumption (1) is satisfied. Therefore, we claim in view of Theorem 2.1 that the energy of the system decays exponentially at infinity.

## Declarations

### Acknowledgments

The authors would like to thank the referees for their comments and suggestions. This work was supported partially by the NSF of China (10771202, 11071042), the Research Fund for Shanghai Key Laboratory for Contemporary Applied Mathematics (08DZ2271900) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (2007035805).

## Authors’ Affiliations

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