- Research
- Open access
- Published:
Blow-up phenomena and global existence for the periodic two-component Dullin-Gottwald-Holm system
Boundary Value Problems volume 2013, Article number: 158 (2013)
Abstract
This paper is concerned with blow-up phenomena and global existence for the periodic two-component Dullin-Gottwald-Holm system. We first obtain several blow-up results and the blow-up rate of strong solutions to the system. We then present a global existence result for strong solutions to the system.
MSC:35G25, 35L05.
1 Introduction
In this paper, we consider the following periodic two-component Dullin-Gottwald-Holm (DGH) system:
where , and γ are constants.
System (1.1) has been recently derived by Zhu et al. in [1] by following Ivanov’s approach [2]. It was shown in [1] that the DGH system is completely integrable and can be written as a compatibility condition of two linear systems
and
where ξ is a spectral parameter. Moreover, this system has the following two Hamiltonians:
and
For and , (1.1) becomes the DGH equation [3]
where A and α are two positive constants, modeling unidirectional propagation of surface waves on a shallow layer of water which is at rest at infinity, stands for fluid velocity. It is completely integrable with a bi-Hamiltonian and a Lax pair. Moreover, its traveling wave solutions include both the KdV solitons and the CH peakons as limiting cases [3]. The Cauchy problem of the DGH equation has been extensively studied, cf. [4–13].
For , , system (1.1) becomes the two-component Camassa-Holm system [2]
where is in connection with the free surface elevation from scalar density (or equilibrium), and the parameter A characterizes a linear underlying shear flow. System (1.2) describes water waves in the shallow water regime with nonzero constant vorticity, where the nonzero vorticity case indicates the presence of an underlying current. A large amount of literature was devoted to the Cauchy problem (1.2); see [14–22].
The Cauchy problem (1.1) has been discussed in [1]. Therein Zhu and Xu established the local well-posedness to system (1.1), derived the precise blow-up scenario and investigated the wave breaking for it. The aim of this paper is to further study the blow-up phenomena for strong solutions to (1.1) and to present a global existence result.
Our paper is organized as follows. In Section 2, we briefly state some needed results including the local well posedness of system (1.1), the precise blow-up scenario and some useful lemmas to study blow-up phenomena and global existence. In Section 3, we give several new blow-up results and the precise blow-up rate. In Section 4, we present a new global existence result of strong solutions to (1.1).
Notation Given a Banach space Z, we denote its norm by . Since all space of functions is over , for simplicity, we drop in our notations if there is no ambiguity.
2 Preliminaries
In this section, we will briefly give some needed results in order to pursue our goal.
With , we can rewrite system (1.1) as follows:
Note that if , is the kernel of , then for all , . Here we denote by ∗ the convolution. Using this identity, we can rewrite system (2.1) as follows:
The local well-posedness of the Cauchy problem (2.1) can be obtained by applying Kato’s theorem. As a result, we have the following well-posedness result.
Lemma 2.1 [1]
Given the initial data , , there exists a maximal and a unique solution
of (2.1). Moreover, the solution depends continuously on the initial data and the maximal time of existence is independent of s.
Consider now the following initial value problem:
where u denotes the first component of the solution to (2.1).
Lemma 2.2 [1]
Let be the solution of (2.1) with the initial data , . Then Eq. (2.3) has a unique solution . Moreover, the map is an increasing diffeomorphism of ℝ with
Lemma 2.3 [1]
Let be the solution of (2.1) with the initial data , , and be the maximal time of existence. Then we have
Moreover, if there exists an such that , then for all .
Next, we give two useful conservation laws of strong solutions to (2.1).
Lemma 2.4 [1]
Let be the solution of (2.1) with the initial data , , and let be the maximal time of existence. Then, for all , we have
Lemma 2.5 Let be the solution of (2.1) with the initial data , , and let be the maximal time of existence. Then, for all , we have
Proof By the first equation in (2.1), we have
This completes the proof of the lemma. □
Then we state the following precise blow-up mechanism of (2.1).
Lemma 2.6 [1]
Let be the solution of (2.1) with the initial data , , and let be the maximal time of existence. Then the solution blows up in finite time if and only if
Lemma 2.7 [23]
Let and . Then, for every , there exists at least one point with
and the function m is almost everywhere differentiable on with
Lemma 2.8 [24]
-
(i)
For every , we have
where the constant is sharp.
-
(ii)
For every , we have
with the best possible constant c lying within the range . Moreover, the best constant c is .
Lemma 2.9 [25]
If is such that , then, for every , we have
Moreover,
Lemma 2.10 [26]
Assume that a differentiable function satisfies
where C, K are positive constants. If the initial datum , then the solution to (2.4) goes to −∞ before t tends to .
3 Blow-up phenomena
In this section, we discuss the blow-up phenomena of system (2.1). Firstly, we prove that there exist strong solutions to (2.1) which do not exist globally in time.
Theorem 3.1 Let , , and T be the maximal time of the solution to (2.1) with the initial data . If there is some such that and
then the corresponding solution to (2.1) blows up in finite time.
Proof Applying Lemma 2.1 and a simple density argument, we only need to show that the above theorem holds for some . Here we assume to prove the above theorem.
Define now
By Lemma 2.7, we let be a point where this infimum is attained. It follows that
Differentiating the first equation in (2.2) with respect to x and using the identity , we have
Since the map given by (2.3) is an increasing diffeomorphism of ℝ, there exists a such that . In particular, . Note that , we can choose . It follows that . By Lemma 2.3 and the condition , we have
Thus .
Valuating (3.1) at and using Lemma 2.7, we obtain
here we used the relations and . Note that . By Lemma 2.4 and Lemma 2.8, we get
and
It follows that
where . Since , Lemma 2.10 implies
Applying Lemma 2.6, the solution blows up in finite time. □
Theorem 3.2 Let , , and T be the maximal time of the solution to (2.1) with the initial data . Assume that . If there is some such that and for any ,
then the corresponding solution to (2.1) blows up in finite time.
Proof By Lemma 2.5, we have . Using Lemma 2.4 and Lemma 2.9, we obtain
and
Following a similar proof in Theorem 3.1, we have
where . Following the same argument as in Theorem 3.1, we deduce that the solution blows up in finite time. □
Letting and in Theorem 3.2, we have the following result.
Corollary 3.1 Let , , and T be the maximal time of the solution to (2.1) with the initial data . Assume that . If there is some such that and
then the corresponding solution to (2.1) blows up in finite time.
Remark 3.1 Note that system (2.1) is variational under the transformation and even . Thus, we cannot get a blow-up result according to the parity of the initial data as we usually do.
Next, we will give more insight into the blow-up mechanism for the wave-breaking solution to system (2.1), that is, the blow-up rate for strong solutions to (2.1).
Theorem 3.3 Let be the solution to system (2.1) with the initial data , , satisfying the assumption of Theorem 3.1, and T be the maximal time of the solution . Then we have
Proof As mentioned earlier, here we only need to show that the above theorem holds for .
Define now
By the proof of Theorem 3.1, there exists a positive constant such that
Let . Since by Theorem 3.1, there is some with and . Since m is locally Lipschitz, it is then inferred from (3.5) that
A combination of (3.5) and (3.6) enables us to infer
Since m is locally Lipschitz on and (3.6) holds, it is easy to check that is locally Lipschitz on . Differentiating the relation , , we get
with absolutely continuous on . For . Integrating (3.7) on to obtain
that is,
By the arbitrariness of the statement of Theorem 3.3 follows. □
4 Global existence
In this section, we will present a global existence result.
Theorem 4.1 Let , , and T be the maximal time of the solution to (2.1) with the initial data . If for all , then the corresponding solution exists globally in time.
Proof Define
By Lemma 2.7, we let be a point where this infimum is attained. It follows that
Since the map given by (2.3) is an increasing diffeomorphism of ℝ, there exists an such that .
Set and . Valuating (3.1) at and using Lemma 2.7, we obtain
where . By Lemma 2.4, Lemma 2.8 and , we have
By Lemmas 2.2-2.3, we know that has the same sign with for every . Moreover, there is a constant such that because of for all . Next, we consider the following Lyapunov positive function:
Letting in (4.2), we have
Differentiating (4.2) with respect to t and using (4.1), we obtain
By Gronwall’s inequality, we have
for all . On the other hand,
Thus,
for all . It follows that
This completes the proof by using Lemma 2.6. □
References
Zhu M, Xu J: On the wave-breaking phenomena for the periodic two-component Dullin-Gottwald-Holm system. J. Math. Anal. Appl. 2012, 391: 415–428. 10.1016/j.jmaa.2012.02.058
Ivanov R: Two-component integrable systems modelling shallow water waves: the constant vorticity case. Wave Motion 2009, 46: 389–396. 10.1016/j.wavemoti.2009.06.012
Dullin HR, Gottwald GA, Holm DD: An integral shallow water equation with linear and nonlinear dispersion. Phys. Rev. Lett. 2001, 87: 4501–4504.
Ai X, Gui G: On the inverse scattering problem and the low regularity solutions for the Dullin-Gottwald-Holm equation. Nonlinear Anal., Real World Appl. 2010, 11: 888–894. 10.1016/j.nonrwa.2009.01.031
Liu Y: Global existence and blow-up solutions for a nonlinear shallow water equation. Math. Ann. 2006, 335: 717–735. 10.1007/s00208-006-0768-1
Meng Q, He B, Long Y, Li Z: New exact periodic wave solutions for the Dullin-Gottwald-Holm equation. Appl. Math. Comput. 2011, 218: 4533–4537.
Mustafa OG: Global conservative solutions of the Dullin-Gottwald-Holm equation. Discrete Contin. Dyn. Syst. 2007, 19: 575–594.
Tian L, Gui G, Liu Y: On the Cauchy problem and the scattering problem for the Dullin-Gottwald-Holm equation. Commun. Math. Phys. 2005, 257: 667–701. 10.1007/s00220-005-1356-z
Yan K, Yin Z: On the solutions of the Dullin-Gottwald-Holm equation in Besov spaces. Nonlinear Anal., Real World Appl. 2012, 13: 2580–2592. 10.1016/j.nonrwa.2012.03.004
Yin Z: Well-posedness, blowup, and global existence for an integrable shallow water equation. Discrete Contin. Dyn. Syst. 2004, 11: 393–411.
Yin Z: Global existence and blow-up for a periodic integrable shallow water equation with linear and nonlinear dispersion. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal. 2005, 12: 87–101.
Zhang S, Yin Z: On the blow-up phenomena of the periodic Dullin-Gottwald-Holm equation. J. Math. Phys. 2008, 49: 1–16.
Zhang S, Yin Z: Global weak solutions for the Dullin-Gottwald-Holm equation. Nonlinear Anal. 2010, 72: 1690–1700. 10.1016/j.na.2009.09.008
Chen RM, Liu Y: Wave-breaking and global existence for a generalized two-component Camassa-Holm system. Int. Math. Res. Not. 2011, 2011: 1381–1416.
Chen RM, Liu Y, Qiao Z: Stability of solitary waves of a generalized two-component Camassa-Holm system. Commun. Partial Differ. Equ. 2011, 36: 2162–2188. 10.1080/03605302.2011.556695
Constantin A, Ivanov R: On the integrable two-component Camassa-Holm shallow water system. Phys. Lett. A 2008, 372: 7129–7132. 10.1016/j.physleta.2008.10.050
Escher J, Lechtenfeld O, Yin Z: Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation. Discrete Contin. Dyn. Syst. 2007, 19: 493–513.
Fu Y, Liu Y, Qu C: Well-posedness and blow-up solution for a modified two-component periodic Camassa-Holm system with peakons. Math. Ann. 2010, 348: 415–448. 10.1007/s00208-010-0483-9
Gui G, Liu Y: On the Cauchy problem for the two-component Camassa-Holm system. Math. Z. 2010, 268: 45–66.
Gui G, Liu Y: On the global existence and wave-breaking criteria for the two-component Camassa-Holm system. J. Funct. Anal. 2010, 258: 4251–4278. 10.1016/j.jfa.2010.02.008
Hu Q: On a periodic 2-component Camassa-Holm equation with vorticity. J. Nonlinear Math. Phys. 2011, 18: 541–556. 10.1142/S1402925111001787
Zhang P, Liu Y: Stability of solitary waves and wave-breaking phenomena for the two-component Camassa-Holm system. Int. Math. Res. Not. 2010, 11: 1981–2021.
Constantin A, Escher J: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 1998, 181: 229–243. 10.1007/BF02392586
Yin Z: On the blow-up of solutions of the periodic Camassa-Holm equation. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal. 2005, 12: 375–381.
Hu Q, Yin Z: Blowup phenomena for a new periodic nonlinearly dispersive wave equation. Math. Nachr. 2010, 283(11):1613–1628. 10.1002/mana.200810075
Zhou Y: Blow-up of solutions to a nonlinear dispersive rod equation. Calc. Var. Partial Differ. Equ. 2005, 25: 63–77.
Acknowledgements
The authors would like to thank the editors and the referees for their valuable suggestions to improve the quality of this paper. This research is partially supported by the Doctoral Research Foundation of Zhengzhou University of Light Industry.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
This paper is the result of joint work of all authors who contributed equally to the final version of this paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Liu, J., Zhang, D. Blow-up phenomena and global existence for the periodic two-component Dullin-Gottwald-Holm system. Bound Value Probl 2013, 158 (2013). https://doi.org/10.1186/1687-2770-2013-158
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-2770-2013-158