- Research Article
- Open Access
Exponential Stability of Two Coupled Second-Order Evolution Equations
Advances in Difference Equations volume 2011, Article number: 879649 (2010)
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.
Of concern is the exponential stability of two coupled second-order evolution equations (one is an integro-differential equation) in Hilbert space
with initial data
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 , where a coupled Timoshenko beam system is investigated.
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:
(1) is a self-adjoint linear operator in , satisfying
(2), are constants. is locally absolutely continuous, satisfying
and there exists a positive constant , such that
We define the energy of a mild solution of (1.1)–(1.3) as
The following is our exponential decay theorem.
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 ,
(ii)there exists a constant such that the energy
for any mild solution of (1.1)–(1.3).
Then, (1.1)–(1.3) becomes
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. ).
Suppose now that is a classical solution of (1.1)–(1.3). We observe
by Assumption (2) and so
and take . We have
Furthermore, we need the following lemmas.
For any and for any , there exist positive numbers , such that
for some positive constants which only depend on , , , and .
At first, let us take the inner product of both sides of (1.1) with and integrate over . Then, noticing (1.2), we obtain
For the first item, integrating by parts, we have
The second and the fifth items can be treated similarly. Therefore,
Then, taking the inner product of both sides of (1.1) with and integrating over , we obtain
Equation (2.15) × + (2.16) yields that
Next, we will estimate all the terms on the right side of (2.17). From (2.11), we have the following estimate:
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).
Using Young's inequality and noting (2.8), we get, for ,
The treatment of the other terms of is similar, giving
Thus, we obtain
where . Make use of the estimate
where is a positive constant, small enough to satisfy . We thus verify our conclusion.
For any and for any , there exist positive numbers , , such that
We denote and take the inner product of both sides of (1.2) with , and integrate over . It follows that
Plugging this equation into (2.16), we find
where , and for
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.
For any , there exist positive numbers , such that
Taking the inner product of both sides of (1.2) with and integrating over , we see
Combining this equation and (2.16) gives
This yields the estimate as desired.
Let be fixed. For any and for any , there exist positive numbers , such that
Take the inner product of both sides of (1.1) with and integrate over . This leads to
Just as in the proofs of the above lemmas, using Young's inequality and noting that
we prove the conclusion.
Proof of Theorem 2.1 (continued).
From Assumption (2) and (2.8), we have
Now, fix . Thanks to Lemmas 2.2 and 2.3, we know that for any and for ,
Moreover, by the use of Lemmas 2.4 and 2.5, we have
Taking small enough gives
Therefore, there is a constant such that
by (2.36). Using Lemma 2.4 and (2.34), we deduce that for some ,
It is easy to see that there exist such that . Therefore,
On the other hand, when ,
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
Consider a coupled system of Petrovsky equations with a memory term
where is a bounded open domain in , with sufficiently smooth boundary and as in Assumption (2). Let with the usual inner product and norm. Here, we denote by the time derivative of and by the Laplacian of with respect to space variable . Define by
Then, Assumption (1) is satisfied. Therefore, we claim in view of Theorem 2.1 that the energy of the system decays exponentially at infinity.
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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).