Exponential Stability of Two Coupled Second-Order Evolution Equations
© Q.Wan and T.-J. Xiao. 2011
Received: 30 October 2010
Accepted: 21 November 2010
Published: 2 December 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.
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:
The following is our exponential decay theorem.
Let the assumptions be satisfied. Then,
for any mild solution of (1.1)–(1.3).
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. ).
Furthermore, we need the following lemmas.
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.
This yields the estimate as desired.
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
Then, Assumption (1) is satisfied. Therefore, we claim in view of Theorem 2.1 that the energy of the system decays exponentially at infinity.
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).
- Xiao T-J, Liang J: The Cauchy Problem for Higher-Order Abstract Differential Equations, Lecture Notes in Mathematics. Volume 1701. Springer, Berlin, Germany; 1998:xii+301.View ArticleGoogle Scholar
- Yosida K: Functional Analysis, Classics in Mathematics. Springer, Berlin, Germany; 1995:xii+501.Google Scholar
- Ammar-Khodja F, Benabdallah A, Muñoz Rivera JE, Racke R: Energy decay for Timoshenko systems of memory type. Journal of Differential Equations 2003,194(1):82-115. 10.1016/S0022-0396(03)00185-2MathSciNetView ArticleMATHGoogle Scholar
- Alabau-Boussouira F, Cannarsa P: A general method for proving sharp energy decay rates for memory-dissipative evolution equations. Comptes Rendus de l'Académie des Sciences—Series I. Paris 2009,347(15-16):867-872.MathSciNetMATHGoogle Scholar
- Alabau-Boussouira F, Cannarsa P, Sforza D: Decay estimates for second order evolution equations with memory. Journal of Functional Analysis 2008,254(5):1342-1372. 10.1016/j.jfa.2007.09.012MathSciNetView ArticleMATHGoogle Scholar
- Dafermos CM: An abstract Volterra equation with applications to linear viscoelasticity. Journal of Differential Equations 1970, 7: 554-569. 10.1016/0022-0396(70)90101-4MathSciNetView ArticleMATHGoogle Scholar
- Komornik V: Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics. Masson, Paris, France; 1994:viii+156.MATHGoogle Scholar
This article is published under license to BioMed Central Ltd. 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.