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.
Keywordsgeneralized Camassa-Holm equation weak dissipativity blow-up global existence
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).
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