Blow-up for the weakly dissipative generalized Camassa-Holm equation
© Guo and Tang; licensee Springer. 2014
Received: 17 September 2014
Accepted: 8 December 2014
Published: 18 December 2014
The main goal of this paper is to investigate the blow-up phenomena of solutions to a weakly dissipative generalized Camassa-Holm equation, which contains higher power nonlinear dispersion terms and a convection term. We give a sufficient condition on the initial data such that the strong solution blows up at a finite time, and then we establish an estimate of the blow-up time. Finally, we give a global existence result of the strong solution.
MSC:35A35, 35B30, 35G25, 35Q53.
has attracted much attention in the study of mathematical physics, where , .
which describes the unidirectional propagation of waves at the free surface of shallow water. stands for the fluid velocity at time t in the spatial direction x. The Camassa-Holm equation (1.2) is bi-Hamiltonian and admits an infinite number of conservation laws . The Camassa-Holm equation (1.2) has been extensively studied by Constantin and Escher [3–6], Lai and Wu , and so on. The well-posedness of the Camassa-Holm shallow water equation has been established, and some blow-up scenarios were derived by Constantin and Escher , Wu and Yin [9, 10], Lai and Wu , Zhou , Xin and Zhang [13, 14].
which was recently discovered by Novikov . Since the Novikov equation possesses a matrix Lax pair and has a bi-Hamiltonian structure as the Camassa-Holm equation, this equation has been studied by many researchers in the past few years, the well-posedness and persistence properties were studied by Lai et al. , Ni and Zhou , Zhao et al. . Jiang and Ni  considered the blow-up phenomena for the integrable Novikov equation, Yan et al.  gave the global existence and blow-up phenomena for the weakly dissipative Novikov equation.
where , is a positive integer. Equations (1.4) and (1.1) have similar properties as regards the local well-posedness and blow-up phenomena, but they are different as regards the long time behavior. For example, when , (1.1) is completely integrable and has an infinite number of conservation laws, but for the corresponding equation (1.4), is not conservative.
it is very similar with (1.4), but (1.4) contains the higher power nonlinear dispersion terms , , and the nonlinear convection term .
Compared to , the main difficulty in this paper comes from the nonlinear effect of higher power nonlinear dispersion terms , , and the nonlinear convection term . On the other hand, in the proof of the blow-up property of the solution to (1.4), we need the sign of the term , but changes the sign for . Compared to the classical Camassa-Holm equation () and the classical Novikov equation (), the term disappears, accordingly. Therefore, we generalized the blow-up property of solutions to the Cauchy problem (1.4).
We first give a sufficient condition on the initial data such that the strong solution of (1.4) blows up at a finite time, and then we establish an estimate of the blow-up time. Finally, we give a global existence result of the strong solution of (1.4).
The paper is organized as follows. In Section 2, we give some preliminaries used in our investigation. In Section 3, we give in our main conclusion the blow-up scenario and global existence result.
Lemma 2.1 
Moreover, the mapping is Hölder continuous.
where is a solution to the Cauchy problem (2.2). The following important properties are immediate consequence of the classical results in the theory of ordinary differential equations.
This concludes the proof. □
Proof When , Lemma 2.4 is a case of Lemma 2.8 in Zhao et al. . The proof carries over with a slight modification and we present it here for the reader’s convenience.
Integrating with respect to t from 0 to t, we get the desired conclusion. □
Following the proof of Lemma 2.9 given by Zhao et al. in , we can obtain a similar blow-up result of the solution to the Cauchy problem (2.2).
3 Blow-up and global existence
Following the local existence Theorem 2.1, we will give our main result on the blow-up property of solution to (2.2). We first give a sufficient condition to guarantee that the solution blows up at a finite time.
Proof For , the result can be found in Wu and Yin . We just show that the results hold for , , and the initial data , for the general case we can use the smooth approximate technique and denseness.
then , , and .
we have .
we have .
Recalling (3.24), we get the desired contradiction, which concludes the proof of the claim.
This completes the proof. □
Remark 3.1 The result in Theorem 3.1 contains the cases for : the weakly dissipative Camassa-Holm equation and : the weakly dissipative Novikov equation. We used the method developed by Liu and Yin  to deal with the Degasperis-Procesi equation (1.5): , but (1.4) contains higher power nonlinear dispersion terms , , and the nonlinear convection term . When the local solution of (2.2) exists, in the proof of its blow-up property we need the sign of ; see the last term in (3.13). In general, changes the sign for so we give the condition on the power of nonlinear term , in (1.4). For , the last term in (3.13) disappears; for , the last term in (3.13) does not contain . Therefore, we generalized the blow-up property of the solutions to the Cauchy problem (1.4).
Finally we give a global existence result, thanks to Theorem 2.1, this will be done if we can estimate is finite.
Proof We just consider , otherwise we can use the smooth approximate technique and denseness. When , then from Lemma 2.2 and Lemma 2.3, we can derive that , for all .
we obtain by Theorem 2.1.
we obtain by Theorem 2.1.
Therefore, we find that the solution exists globally in time. □
This work was supported by National Natural Science Foundation of China (grant Nos. 11471129, 11272277).
- Guo YT, Wang M, Tang YB: Higher regularity of global attractor for a damped Benjamin-Bona-Mahony equation on R . Appl. Anal. 2014. 10.1080/00036811.2014.946561Google Scholar
- Camassa R, Holm D: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 1993, 71: 1661-1664. 10.1103/PhysRevLett.71.1661MathSciNetView ArticleMATHGoogle Scholar
- Constantin A, Escher J: Global weak solutions for a shallow water equation. Indiana Univ. Math. J. 1998, 47: 1527-1545.MathSciNetView ArticleMATHGoogle Scholar
- Constantin A, Escher J: Wave breaking for nonlocal shallow water equations. Acta Math. 1998, 181: 229-243. 10.1007/BF02392586MathSciNetView ArticleMATHGoogle Scholar
- Constantin A, Escher J: Global existence and blow-up for a shallow water equation. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 1998,XXVI(4):303-328.MathSciNetGoogle Scholar
- Constantin A: Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier (Grenoble) 2000, 50: 321-362. 10.5802/aif.1757MathSciNetView ArticleMATHGoogle Scholar
- Lai S, Wu Y: The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation. J. Differ. Equ. 2010, 248: 2038-2063. 10.1016/j.jde.2010.01.008MathSciNetView ArticleMATHGoogle Scholar
- Constantin A, Escher J: On the blow-up rate and the blow-up set of breaking waves for a shallow water equation. Math. Z. 2000, 233: 75-91. 10.1007/PL00004793MathSciNetView ArticleMATHGoogle Scholar
- Wu S, Yin Z: Blowup, blowup rate and decay of the solution of the weakly dissipative Camassa-Holm equation. J. Math. Phys. 2006, 47: 1-12.MathSciNetGoogle Scholar
- Wu S, Yin Z: Global existence and blow-up phenomena for the weakly dissipative Camassa-Holm equation. J. Differ. Equ. 2009, 246: 4309-4321. 10.1016/j.jde.2008.12.008MathSciNetView ArticleMATHGoogle Scholar
- Lai S, Wu Y: Global solutions and blow-up phenomena to a shallow water equations. J. Differ. Equ. 2010, 249: 693-706. 10.1016/j.jde.2010.03.008MathSciNetView ArticleMATHGoogle Scholar
- Zhou Y: Blowup of solutions to the DGH equation. J. Funct. Anal. 2007, 250: 227-248. 10.1016/j.jfa.2007.04.019MathSciNetView ArticleMATHGoogle Scholar
- Xin Z, Zhang P: On the uniqueness and large time behavior of the weak solution to a shallow water equation. Commun. Partial Differ. Equ. 2002,27(9-10):1815-1844. 10.1081/PDE-120016129MathSciNetView ArticleMATHGoogle Scholar
- Xin Z, Zhang P: On the weak solution to a shallow water equation. Commun. Pure Appl. Math. 2000, 53: 1411-1433. 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5MathSciNetView ArticleMATHGoogle Scholar
- Novikov V: Generalizations of the Camassa-Holm equation. J. Phys. A 2009. 42: Article ID 342002Google Scholar
- Lai S, Li N, Wu Y: The existence of global strong and weak solutions for Novikov equation. J. Math. Anal. Appl. 2013, 399: 682-691. 10.1016/j.jmaa.2012.10.048MathSciNetView ArticleMATHGoogle Scholar
- Ni L, Zhou Y: Well-posedness and persistence properties for the Novikov equation. J. Differ. Equ. 2011, 250: 3002-3201. 10.1016/j.jde.2011.01.030MathSciNetView ArticleMATHGoogle Scholar
- Zhao Y, Li Y, Yan W: Local well-posedness and persistence property for the generalized Novikov equation. Discrete Contin. Dyn. Syst., Ser. A 2014, 34: 803-820.MathSciNetView ArticleMATHGoogle Scholar
- Jiang Z, Ni L: Blow-up phenomena for the integrable Novikov equation. J. Math. Anal. Appl. 2012, 385: 551-558. 10.1016/j.jmaa.2011.06.067MathSciNetView ArticleMATHGoogle Scholar
- Yan W, Li Y, Zhang Y: Global existence and blow-up phenomena for the weakly dissipative Novikov equation. Nonlinear Anal. 2012, 75: 2464-2473. 10.1016/j.na.2011.10.044MathSciNetView ArticleMATHGoogle Scholar
- Zhao Y, Li Y, Yan W: The global weak solutions to the Cauchy problem of the generalized Novikov equation. Appl. Anal. 2014. 10.1080/00036811.2014.930826Google Scholar
- Liu Y, Yin Z: On the blow-up phenomena for the Degasperis-Procesi equation. Int. Math. Res. Not. 2007. 10.1093/imrn/rnm117Google 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.