Persistence properties of solutions to the dissipative 2-component Degasperis-Procesi system
© Ming et al.; licensee Springer. 2014
Received: 29 March 2014
Accepted: 28 April 2014
Published: 9 May 2014
The persistence properties of solutions to the dissipative 2-component Degasperis-Procesi system are investigated. We find that if the initial data with their derivatives of the system exponentially decay at infinity, then the corresponding solution also exponentially decays at infinity.
MSC:35G25, 35L15, 35Q58.
where λ, are nonnegative constants, , ().
where () is the dissipative term. They obtained the global weak solutions to (1.3). Guo  established the local well-posedness for (1.3), and also obtained the global existence, persistence properties and propagation speed of solutions. Wu and Yin  obtained the local well-posedness for (1.3), and also studied the blow-up scenarios of solutions in periodic case.
where . They established the local well-posedness for system (1.4) in Besov space with , and also derived the precise blow-up scenarios of strong solutions in Sobolev space with . Zhou et al.  investigated the traveling wave solutions to the 2-component Degasperis-Procesi system. Manwai  studied the self-similar solutions to the 2-component Degasperis-Procesi system. Fu and Qu  obtained the persistence properties of solutions to the 2-component Degasperis-Procesi system in Sobolev space with . For system (1.4), Jin and Guo  studied the blow-up mechanisms and persistence properties of strong solutions.
where . The author not only established the local well-posedness for system (1.5) in Besov space with , but she also presented global existence results and the exact blow-up scenarios of strong solutions in Sobolev space with . For in system (1.5), Jin and Guo  considered the persistence properties of solutions to the modified 2-component Camassa-Holm system. Zhu  considered the persistence property of solutions to the coupled Camassa-Holm system, and also established the global existence and blow-up mechanisms of solutions. Guo [21, 22] studied the persistence properties and unique continuation of solutions to the 2-component Camassa-Holm system in the case . It was shown in  that the dissipative Camassa-Holm, Degasperis-Procesi, Hunter-Saxton and Novikov equations could be reduced to the non-dissipative versions by means of an exponentially time-dependent scaling. One may refer to [30–34] and the references therein for more details in this direction.
Motivated by the work in [13, 20, 35], we study the dissipative 2-component Degasperis-Procesi system (1.1). We note that the persistence properties of solutions to system (1.1) have not been discussed yet. The aim of this paper is to investigate the persistence properties of solutions in Sobolev space . The main idea of this work comes from .
where the operator .
The main results are presented as follows.
Theorem 1.1 Assume and with . Then the Cauchy problem (1.1) has a unique solution .
uniformly on the interval .
- (1)For ,
- (2)For ,
uniformly on the interval .
The remainder of this paper is organized as follows. In Section 2, the proofs of Theorems 1.1 and 1.2 are presented. Section 3 is devoted to the proofs of Theorems 1.3 and 1.4. The proofs of Theorems 1.5 and 1.6 are given in Section 4.
2 Proofs of Theorems 1.1 and 1.2
Proposition 2.1 
2.1 Proof of Theorem 1.1
Using the Littlewood-Paley theory and estimates for solutions to the transport equation, one may follow similar arguments as in  to establish the local well-posedness for system (1.1) with some modification. Here we omit the detailed proof. For system (1.1) with initial data (), we see that the corresponding solution . Thus we complete the proof of Theorem 1.1.
2.2 Proof of Theorem 1.2
uniformly on the interval . This completes the proof of Theorem 1.2.
3 Proofs of Theorems 1.3 and 1.4
3.1 Proof of Theorem 1.3
which combined with the above estimates yields a contradiction. We obtain . Consequently, , .
From (3.5), we have . This completes the proof of case (2) in Theorem 1.3.
3.2 Proof of Theorem 1.4
Then as . From Theorem 1.2, if as , we have as . This completes the proof of Theorem 1.4.
4 Proofs of Theorems 1.5 and 1.6
4.1 Proof of Theorem 1.5
Using the assumption in Theorem 1.5, we complete the proof.
4.2 Proof of Theorem 1.6
The proof of Theorem 1.6 is similar to the proof of Theorem 1.3, here we omit it.
The authors would like to express sincere gratitude to the anonymous referees for a number of valuable comments and suggestions. This work was partially supported by National Natural Science Foundation of P.R. China (71003082) and Fundamental Research Funds for the Central Universities (SWJTU12CX061 and SWJTU09ZT36).
- Degasperis A, Procesi M: Asymptotic integrability. In Symmetry and Perturbation Theory. World Scientific, Singapore; 1999:23-37.Google Scholar
- Escher J, Liu Y, Yin Z: Global weak solutions and blow-up structure for the Degasperis-Procesi equation. J. Funct. Anal. 2006, 241: 457-485. 10.1016/j.jfa.2006.03.022MathSciNetView ArticleGoogle Scholar
- Liu Y, Yin Z: Global existence and blow-up phenomena for the Degasperis-Procesi equation. Commun. Math. Phys. 2006, 267: 801-820. 10.1007/s00220-006-0082-5MathSciNetView ArticleGoogle Scholar
- Yin Z: On the Cauchy problem for an integrable equation with peakon solutions. Ill. J. Math. 2003, 47: 649-666.Google Scholar
- Yin Z: Global existence for a new periodic integrable equation. J. Math. Anal. Appl. 2003, 283: 129-139. 10.1016/S0022-247X(03)00250-6MathSciNetView ArticleGoogle Scholar
- Yin Z: Global solutions to a new integrable equation with peakons. Indiana Univ. Math. J. 2004, 53: 1189-1210. 10.1512/iumj.2004.53.2479MathSciNetView ArticleGoogle Scholar
- Yin Z: Global weak solutions to a new periodic integrable equation with peakon solutions. J. Funct. Anal. 2004, 212: 182-194. 10.1016/j.jfa.2003.07.010MathSciNetView ArticleGoogle Scholar
- Wu S, Yin Z: Blow-up phenomena and decay for the periodic Degasperis-Procesi equation with weak dissipation. J. Nonlinear Math. Phys. 2008, 15: 28-49. 10.2991/jnmp.2008.15.s2.3MathSciNetView ArticleGoogle Scholar
- Guo Y, Lai S, Wang Y: Global weak solutions to the weakly dissipative Degasperis-Procesi equation. Nonlinear Anal. 2011, 74: 4961-4973. 10.1016/j.na.2011.04.051MathSciNetView ArticleGoogle Scholar
- Guo Z: Some properties of solutions to the weakly dissipative Degasperis-Procesi equation. J. Differ. Equ. 2009, 246: 4332-4344. 10.1016/j.jde.2009.01.032View ArticleGoogle Scholar
- Yan K, Yin Z: On the Cauchy problem for a 2-component Degasperis-Procesi system. J. Differ. Equ. 2012, 252: 2131-2159. 10.1016/j.jde.2011.08.003MathSciNetView ArticleGoogle Scholar
- Zhou J, Tian L, Fan X: Soliton, kink and antikink solutions of a 2-component of the Degasperis-Procesi equation. Nonlinear Anal., Real World Appl. 2012, 11: 2529-2536.MathSciNetView ArticleGoogle Scholar
- Fu Y, Qu C: Unique continuation and persistence properties of solutions of the 2-component Degasperis-Procesi equations. Acta Math. Sci. 2012, 32: 652-662. 10.1016/S0252-9602(12)60046-0MathSciNetView ArticleGoogle Scholar
- Jin L, Guo Z: On a 2-component Degasperis-Procesi shallow water system. Nonlinear Anal., Real World Appl. 2010, 11: 4164-4173. 10.1016/j.nonrwa.2010.05.003MathSciNetView ArticleGoogle Scholar
- Yu L, Tian L: Loop solutions, breaking kink wave solutions, solitary wave solutions and periodic wave solutions for the 2-component Degasperis-Procesi equation. Nonlinear Anal., Real World Appl. 2014, 15: 140-148.MathSciNetView ArticleGoogle Scholar
- Manwai Y: Self-similar blow-up solutions to the 2-component Degasperis-Procesi shallow water system. Commun. Nonlinear Sci. Numer. Simul. 2011, 16: 3463-3469. 10.1016/j.cnsns.2010.12.039MathSciNetView ArticleGoogle Scholar
- Hu Q: Global existence and blow-up phenomena for a weakly dissipative 2-component Camassa-Holm system. Appl. Anal. 2013, 92: 398-410. 10.1080/00036811.2011.621893MathSciNetView ArticleGoogle Scholar
- Hu Q: Global existence and blow-up phenomena for a weakly dissipative periodic 2-component Camassa-Holm system. J. Math. Phys. 2011., 52: Article ID 103701Google Scholar
- Jin L, Guo Z: A note on a modified 2-component Camassa-Holm system. Nonlinear Anal., Real World Appl. 2012, 13: 887-892. 10.1016/j.nonrwa.2011.08.024MathSciNetView ArticleGoogle Scholar
- Zhu M: Blow-up, global existence and persistence properties for the coupled Camassa-Holm equations. Math. Phys. Anal. Geom. 2011, 14: 197-209. 10.1007/s11040-011-9094-2MathSciNetView ArticleGoogle Scholar
- Guo Z: Asymptotic profiles of solutions to the 2-component Camassa-Holm system. Nonlinear Anal. 2012, 75: 1-6. 10.1016/j.na.2011.01.030MathSciNetView ArticleGoogle Scholar
- Guo Z, Ni L: Persistence properties and unique continuation of solutions to a 2-component Camassa-Holm equation. Math. Phys. Anal. Geom. 2011, 14: 101-114. 10.1007/s11040-011-9089-zMathSciNetView ArticleGoogle Scholar
- Constantin A, Ivanov R: On an integrable 2-component Camassa-Holm shallow water system. Phys. Lett. A 2008, 372: 7129-7132. 10.1016/j.physleta.2008.10.050MathSciNetView ArticleGoogle Scholar
- Gui G, Liu Y: On the global existence and wave-breaking criteria for the 2-component Camassa-Holm system. J. Funct. Anal. 2010, 258: 4251-4278. 10.1016/j.jfa.2010.02.008MathSciNetView ArticleGoogle Scholar
- Gui G, Liu Y: On the Cauchy problem for the 2-component Camassa-Holm system. Math. Z. 2011, 268: 45-66. 10.1007/s00209-009-0660-2MathSciNetView ArticleGoogle Scholar
- Ai X, Gui G: Global well-posedness for the Cauchy problem of the viscous Degasperis-Procesi equation. J. Math. Anal. Appl. 2010, 361: 457-465. 10.1016/j.jmaa.2009.07.031MathSciNetView ArticleGoogle Scholar
- Chen W, Tian L, Deng X, Zhang J: Wave breaking for a generalized weakly dissipative 2-component Camassa-Holm system. J. Math. Anal. Appl. 2013, 400: 406-417. 10.1016/j.jmaa.2012.10.063MathSciNetView ArticleGoogle Scholar
- Jin L, Jiang Z: Wave breaking of an integrable Camassa-Holm system with 2-component. Nonlinear Anal. 2014, 95: 107-116.MathSciNetView ArticleGoogle Scholar
- Lenells J, Wunsch M: On the weakly dissipative Camassa-Holm, Degasperis-Procesi, and Novikov equations. J. Differ. Equ. 2013, 255: 441-448. 10.1016/j.jde.2013.04.015MathSciNetView ArticleGoogle Scholar
- Zhu M: On a shallow water equation perturbed in Schwartz class. Math. Phys. Anal. Geom. 2012. 10.1007/s11040-012-9112-zGoogle Scholar
- Zhu M, Jiang Z: Some properties of solutions to the weakly dissipative b -family equation. Nonlinear Anal., Real World Appl. 2012, 13: 158-167. 10.1016/j.nonrwa.2011.07.020MathSciNetView ArticleGoogle Scholar
- Ni L, Zhou Y: Well-posedness and persistence properties for the Novikov equation. J. Differ. Equ. 2011, 250: 3002-3021. 10.1016/j.jde.2011.01.030MathSciNetView ArticleGoogle Scholar
- Zong X: Properties of the solutions to the 2-component b -family systems. Nonlinear Anal. 2012, 75: 6250-6259. 10.1016/j.na.2012.07.001MathSciNetView ArticleGoogle Scholar
- Zhou S, Mu C: The properties of solutions for a generalized b -family equation with peakons. J. Nonlinear Sci. 2013, 23: 863-889. 10.1007/s00332-013-9171-8MathSciNetView ArticleGoogle Scholar
- Himonas A, Misiolek G, Ponce G, Zhou Y: Persistence properties and unique continuation of solutions to the Camassa-Holm equation. Math. Phys. Anal. Geom. 2007, 271: 511-522.MathSciNetGoogle Scholar
- Danchin R: A few remarks on the Camassa-Holm equation. Differ. Integral Equ. 2001, 14: 953-988.MathSciNetGoogle Scholar
- Danchin R: A note on well-posedness for Camassa-Holm equation. J. Differ. Equ. 2003, 192: 429-444. 10.1016/S0022-0396(03)00096-2MathSciNetView ArticleGoogle Scholar
- Danchin, R: Fourier analysis methods for PDEs. Lecture Notes, 14 Nov. Preprint (2005)Google Scholar
- Bahouri H, Chemin J, Danchin R Grun. der Math. Wiss. In Fourier Analysis and Nonlinear Partial Differential Equations. Springer, Berlin; 2010.Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.