Persistence properties for the Fokas-Olver-Rosenau-Qiao equation in weighted \(L^{p}\) spaces
- Shouming Zhou^{1}Email author,
- Ming Xie^{2} and
- Fuchen Zhang^{3}
Received: 20 July 2015
Accepted: 12 November 2015
Published: 2 December 2015
Abstract
In this paper, we mainly study persistence properties for a generalized Camassa-Holm equation with cubic nonlinearity, and we prove the persistence properties in weighted spaces of the solution to the equation, provided that the initial potential satisfies a certain sign condition. Our results extend the work of Brandolese (Int. Math. Res. Not. 22:5161-5181, 2012) on persistence properties to the Fokas-Olver-Rosenau-Qiao equation. In contrast to the Camassa-Holm equation with quadratic nonlinearity, the effect of cubic nonlinearity of the Fokas-Olver-Rosenau-Qiao equation on the persistence properties is rather delicate.
Keywords
nonuniform dependence persistence properties cubic nonlinearityMSC
35G25 35L05 35Q501 Introduction
The spacial decay rates for the strong solutions to the Camassa-Holm and Novikov equations were established, provided that the corresponding initial datum decays at infinity [1, 45, 46]. This kind of property is the so-called persistence property. Motivated by the recent work [1] on the nonlinear Camassa-Holm equation in weighted Sobolev spaces, the other aim of this paper is to establish the persistence properties for the modified Camassa-Holm equation (1.1) in weighted \(L^{p}\) spaces. However, there are high nonlinearity and regularity in (1.1), which makes the proof of several required nonlinear estimates very difficult.
Definition 1.1
We can now state our main result on admissible weights.
Theorem 1.1
Remark 1.1
(2) Choose \(\phi=\phi_{a,1,0,0}\) if \(x\geq0\) and \(\phi(x)=1\) if \(x\leq0\) with \(0\leq a<1\). It is easy to see that such a weight satisfies the admissibility conditions of Definition 1.1. Let further \(p=\infty\) in Theorem 1.1, then we deduce that Equation (1.1) preserve the pointwise decay \(O(e^{-ax})\) as \(x\rightarrow+\infty\) for any \(t>0\). Similarly, we have persistence of the decay \(O(e^{-ax})\) as \(x\rightarrow-\infty\). A corresponding result on persistence of strong solutions of the CH and Novikov equations and of Equation (1.1) can be found in [9, 45, 50], respectively.
Theorem 1.2
Remark 1.2
2 Analysis of the Equation (1.1) in weighted spaces
- (i)
For \(a, c,d\geq0\) and \(0\leq b\leq1\), such a weight is submultiplicative.
- (ii)
If \(a, c,d\in\mathbb{R}\) and \(0\leq b\leq1\), then ϕ is moderate. More precisely, \(\phi_{a,b,c,d}\) is \(\phi_{\alpha,\beta,\gamma ,\delta}\)-moderate for \(|a|\leq\alpha\), \(|b|\leq\beta\), \(|c|\leq\gamma \), and \(|d|\leq\delta\).
The elementary properties of submultiplicative and moderate weights can be found in [1]. Now, we prove Theorem 1.1.
Proof of Theorem 1.1
Proof of Theorem 1.2
Declarations
Acknowledgements
The authors are very grateful to the anonymous reviewers and editors for their careful reading and useful suggestions, which greatly improved the presentation of the paper. The first author is supported by National Science Fund for Young Scholars of China (Grant No. 11301573), University Young Core Teacher Foundation of Chongqing, Technology Research Foundation of Chongqing Educational Committee (Grant No. KJ1400503), Natural Science Foundation of Chongqing (Grant No. cstc2014jcyjA00008), Top-notch talent Foundation of Chongqing Normal University. The third author is supported by National Natural Science Foundation of China (Grant No. 11426047), the Basic and Advanced Research Project of CQCSTC (Grant No. cstc2014jcyjA00040) and the Research Fund of Chongqing Technology and Business University (Grant No. 2014-56-11).
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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