The stability of strong solutions to a generalized BBM equation
© Lai and Wang; licensee Springer. 2014
Received: 25 July 2013
Accepted: 10 December 2013
Published: 2 January 2014
A nonlinear generalized Benjamin-Bona-Mahony equation is investigated. Using the estimates of strong solutions derived from the equation itself, we establish the stability of the solutions under the assumption that the initial value lies in the space .
Keywordsgeneralized BBM equation strong solutions stability
where a, b and k are constants. Equation (1) is often used as an alternative to the equation which describes unidirectional propagation of weakly long dispersive waves . As a model that characterizes long waves in nonlinear dispersive media, the equation, like the equation, was formally derived to describe an approximation for surface water waves in a uniform channel. Equation (1) covers not only the surface waves of long wavelength in liquids, but also hydromagnetic waves in cold plasma, acoustic waves in anharmonic crystals, and acoustic gravity waves in compressible fluids (see [2, 3]). Nonlinear stability of nonlinear periodic solutions of the regularized Benjamin-Ono equation and the Benjamin-Bona-Mahony equation with respect to perturbations of the same wavelength is analytically studied in . Unique continuation property and control for the Benjamin-Bona-Mahony equation on a periodic domain are discussed in . The () asymptotic property of solutions for the Benjamin-Bona-Mahony-Burgers equations is studied in  under certain assumptions on the initial data. The tanh technique is employed in  to get the compact and noncompact solutions for KP- and ZK- equations.
where , and are constants, and is an integer.
where T is the maximum existence time of solutions and and c depends on and . From our knowledge, we state that the stability of strong solutions for Eq. (2) has never been acquired in the literature.
This paper is organized as follows. Section 2 gives several lemmas and Section 3 establishes the proofs of the main result.
2 Several lemmas
At any Lebesgue point of the function , we have . Since the set of points which are not Lebesgue points of has measure zero, we get as almost everywhere.
We introduce notations connected with the concept of a characteristic cone. For any , we define . Let ℧ denote the cone . We let represent the cross section of the cone ℧ by the plane , . Let , where .
Lemma 2.1 ()
where c only depends on b.
where the operator for any .
where the constant C is independent of time t.
Using (5)-(6), the integral and (8)-(9), we obtain the proof of Lemma 2.2. □
where k is an arbitrary constant.
Let be an approximation of the function and set . Letting , we complete the proof. □
In fact, the proof of (10) can also be found in .
where c depends on and and f.
in which we have used and . Using the Fubini theorem completes the proof. □
3 Main results
where c depends on and .
From (28) and (32), we prove that inequality (21) holds.
where c is defined in Lemma 2.4.
where c is independent of ε.
from which we complete the proof of Theorem 3.1 by using the Gronwall inequality. □
The authors are very grateful to the reviewers for their helpful and valuable comments, which have led to a meaningful improvement of the paper. This work is supported by both the Fundamental Research Funds for the Central Universities (JBK120504) and the Applied and Basic Project of Sichuan Province (2012JY0020).
- Benjamin TB, Bona JL, Mahony JJ: Model equation for long waves in nonlinear dispersive system. Philos. Trans. R. Soc. Lond. Ser. A 1972, 272: 47-78. 10.1098/rsta.1972.0032MathSciNetView ArticleGoogle Scholar
- Bona JL, Smith R: The initial value problem for the Korteweg-de-Vries equation. Philos. Trans. R. Soc. Lond. Ser. A 1975, 278: 555-601. 10.1098/rsta.1975.0035MathSciNetView ArticleGoogle Scholar
- Saut JC, Tzetkov N: Global well-posedness for the KP-BBM equations. Appl. Math. Res. Express 2004, 1: 1-6.View ArticleGoogle Scholar
- Angulo J, Scialom M, Banquet C: The regularized Benjamin-Ono and BBM equations: well-posedness and nonlinear stability. J. Differ. Equ. 2011, 250: 4011-4036. 10.1016/j.jde.2010.12.016MathSciNetView ArticleGoogle Scholar
- Rosier L, Zhang BY: Unique continuation property and control for the Benjamin-Bona-Mahony equation on a periodic domain. J. Differ. Equ. 2013, 254: 141-178. 10.1016/j.jde.2012.08.014MathSciNetView ArticleGoogle Scholar
- Mei M: Decay rates of solutions for Benjamin-Bona-Mahony-Burgers equations. J. Differ. Equ. 1999, 158: 314-340. 10.1006/jdeq.1999.3638View ArticleGoogle Scholar
- Wazwaz AM: The extended tanh method for new compact and noncompact solutions for the KP-BBM and ZK-BBM equations. Chaos Solitons Fractals 2008, 38: 1505-1516. 10.1016/j.chaos.2007.01.135MathSciNetView ArticleGoogle Scholar
- Wazwaz AM: Nonlinear variants of the BBM equation with compact and noncompact physical structures. Chaos Solitons Fractals 2005, 26: 767-776. 10.1016/j.chaos.2005.01.044MathSciNetView ArticleGoogle Scholar
- Kruzkov SN: First order quasi-linear equations in several independent variables. Math. USSR Sb. 1970, 10: 217-243. 10.1070/SM1970v010n02ABEH002156View ArticleGoogle Scholar
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