Asymptotic behavior of solutions for the time-delayed equations of Benjamin-Bona-Mahony’s type
- Feng Wei^{1},
- Pu Zhilin^{2} and
- Zhu Chaosheng^{3}Email author
Received: 6 January 2015
Accepted: 3 July 2015
Published: 1 August 2015
Abstract
In this paper, we investigate the asymptotic behavior of the solutions for the equations of Benjamin-Bona-Mahony’s type with a time delay. We prove the global existence of solutions and energy decay. By using the Liapunov function method, we shall show that the solution is exponentially decay if the delay parameter τ is sufficiently small.
Keywords
Benjamin-Bona-Mahony equation time delay exponential decay Liapunov functionMSC
35R10 35B35 35Q531 Introduction
Let X be a Banach space and \(a< b\). We denote by \(C^{n}([a,b];X)\) the space of n times continuously differentiable functions defined on \([a,b]\) with values in X with the supremum norm and we write \(C([a,b];X)\) for \(C^{0}([a,b];X)\).
The main result of this paper is stated as follows.
Theorem 1.1
Theorem 1.2
This paper is organized as follows. In next section, we prove the existence of the solution. Furthermore, we show that the solution is exponentially decay by using the Liapunov function method.
2 Exponential decay estimates
Firstly, we briefly show that problem (1.1)-(1.3) is well posed. To conveniences, we denote \(\|u\|\leq\lambda_{1}\|L^{\frac{1}{2}}u\|\), \(\|u_{x}\|\leq\lambda_{2}\|L^{\frac{1}{2}}u\|\), \(\|u\|\leq\lambda_{3}\|M^{\frac{1}{2}}u\|\), \(\|u_{x}\|\leq\lambda_{4}\| M^{\frac{1}{2}}u\|\), \(\|M^{\frac{1}{2}}u\|\leq\lambda_{5}\|L^{\frac{1}{2}}u\|\).
Proof of Theorem 1.1
Proof of Theorem 1.2
Declarations
Acknowledgements
This work was supported by the Postdoctor Research Fund of Chongqing (No. YuXM201102006), the National Natural Science Foundation of China (No. 61273020), and the Fundamental Research Funds for the Central Universities (No. XDJK2009C070).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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