In this paper, a family of the weakly dissipative periodic Camassa-Holm type equation cubic and quartic nonlinearities is considered. The precise blow-up scenarios of strong solutions and several conditions on the initial data to guarantee blow-up of the induced solutions are described in detail. Finally, we establish a sufficient condition for global solutions.

1. Introduction

In this paper, we are concerned with the periodic Cauchy problem of the generalized Camassa-Holm (CH) type equation the weak dissipation as follows: where , and are arbitrary constants.

When , the generalized CH type equation in (1) recovers the well-known integrable Camassa-Holm equation [1] which was derived by Camassa and Holm for shallow water waves [2] and is named after them. In fact, the CH equation was originally derived by Fokas and Fuchssiteiner [1] as a bi-Hamiltonian generalization of KdV. It is completely integrable and has infinitely many conservation laws [24]. In the recent years, the CH equation has caught a lot of attention from different perspectives, such as well-posedness, blow-up phenomena of solutions, and their stability. For example, the local well-posedness and the global strong solutions for certain class of initial data were studied [4, 5]. Existence and uniqueness results for classical solution of the periodic CH equation were established in [6]. The blow-up phenomena of the periodic CH equation were also investigated in a number of papers (see [2, 48] and references therein). Orbital stability of the peakons for the CH equation was studied by Lenells in [9, 10].

When and , we have the Dullin-Gottwald-Holm (DGH) equation [11, 12]. And if and , it reduces to the weakly dissipative CH equation, whereas if and , it reduces to the weakly dissipative DGH equation. The well-posedness and blow-up phenomena of solutions of the Cauchy problem for the weakly dissipative DGH equation were studied, see, for example, [13, 14]. The similar research of the dissipative DGH equation higher-order nonlinearities and arbitrary coefficients and the other related models was also discussed in [1519]. For some new and important developments for searching for solving numerical solutions for some PDE, the reader is referred to [20, 21] and references therein.

If the first equation of (1) becomes the equation which is related to the following physically relevant model:

with satisfying , and when . Here, is a parameter related to the rotational frequency due to Coriolis effect which is typically a manifestation of rotation when Newton’s laws are applied to model physical phenomena on Earth’s surface. The Cauchy problem of Equations (3) and (4) has been studied in ref. [16, 2226].

When the first equation of (1) can be seen as a weakly dissipative Camassa-Holm (CH) type equation. For the weakly dissipative CH type equations: the local well-posedness, global existence, and blow-up phenomena of the Cauchy problem of the weakly dissipative CH equation

on the line [27] and on the circle [27] were studied. They found that the behaviors of Equation (44) are similar to the CH equation in a finite interval of time, such as the local well-posedness and the blow-up phenomena, and that there are considerable differences between them in their long time behaviors. Thereafter, a new global existence result and a new blow-up result for strong solutions to the equation certain profiles were presented in [28]. The obtained results improve considerably the previous results. Later on, a new blow-up result for positive strong solutions of (6) was presented by Novruzov [29]. In particular, they used a condition where the initial data and its derivative are not simultaneously involved and the parameter is not bounded from above. The well-posedness and wave-breaking phenomena to the weakly dissipative CH equation quadratic and cubic nonlinearities

were considered by Freire et al. [30]. The novelty of their work is the method of group analysis was applied in order to construct conserved currents, and therefore, the conserved quantities were established as an extremely natural consequence of them. Subsequently, the periodic Cauchy problem of Equation (7) was considered by Ji and Zhou [31] and their local well-posedness was established via Kato’s theory [32]; then, a sufficient condition on the initial data to guarantee the wave breaking was given and the global existence of solutions was given finally. Recently, Freire [33] considered the Cauchy problem of the weakly dissipative CH Equation (1). In their paper, some time-dependent energy functionals of solutions were proved, then the existence of wave-breaking phenomena was investigated, and necessary conditions for its existence were also obtained.

In general, it is difficulty to avoid energy dissipation mechanics in a real world. So it is reasonable to investigate the model with energy dissipation in propagation of nonlinear waves, see [34, 35] and references therein. Inspired by the previous work, the aim of the paper is to investigate whether the periodic Cauchy problem of the equation in (1) has the similar remarkable properties as that on the entire line. The outline of the paper is as follows. In Section 2, we obtain the local well-posedness and wave-breaking criterion. In Section 3, a blow-up scenario for strong solutions is described and some sufficient conditions for wave breaking of strong solutions in finite time are established. Furthermore, the blow-up rate of blow-up solutions of (1) is also derived. In Section 4, a sufficient condition for global solutions is provided.

1.1. Notations

Throughout this paper, we identity all spaces of periodic functions with function spaces over the unit circle in , i.e., . The norm of the Sobolev space , , by . Since all space of functions are over , for simplicity, we drop in our notations of function spaces if there is no ambiguity.

2. Preliminaries and Local Well-Posedness

Let us introduce the subject of investigation of this paper. In order to establish the local well-posedness result by Kato’ theorem. Rewrite problem (1) as follows: where

Denote . Then for all . Our system (8) can be written in the following transport type:

or equivalently,

We begin by presenting the local well-posedness result for the periodic Cauchy problem (11). Concerning the generalized CH equation in (1) is suitable for applying Kato’s theory [32]; one may follow the similar argument as in [30, 33] to obtain the following theorem.

Theorem 1. Given with . Then, there exists a maximal time and unique solution to (1) satisfying the initial condition such that Moreover, the solution depends continuously on the initial data, in the sense that the mapping : is continuous and does not depend on .

Remark 2. The equation in (1), for , can be seen as the weakly dissipative periodic Camassa-Holm type equation. However, no matter the value of parameter , the problem (1) is locally well-posed as shown in Theorem 1.

Lemma 3. (see [8]). (i)For every , we havewhere the constant is sharp. (ii)For every , we havewith the best possible constant lying within the range . Moreover, the best constant is .
The next theorem will establish the time-dependent conserved quantities of solutions to problem (1), which is crucial in the investigation of wave-breaking phenomena and global existence of solutions.

Theorem 4. Assume that be the solution to problem (1) with the initial data such that and its derivatives up to second order go to as and . Let Then, for any , we have

Proof. Integrating the first equation of system (1) by parts, in view of the periodicity of , we have Then integrating (17) with respect to over implies For the proof of the other conserved quantity, multiplying both sides of the first equation in (1) by , we have Integrating (19) with respect to over yields i.e., , which completes the proof of the theorem.

It is worth mentioning that if , the results of Theorem 1 implies

which will be of great relevance in our investigation of wave breaking. Combining this observation with the Sobolev Embedding Theorem, we give the following remark.

Remark 5. If , and , then the -norm of the corresponding solution of system (1) is bounded above by .

Once we have presented conditions for having locally well-posed solutions, a natural question is whether (1) admits wave breaking, which can be assured by the following blow-up criterion. In the following proof, we only prove the case when , since the same conclusion for general case can be obtained by using denseness.

Theorem 6. Let be given and assume that is the maximal existence time of the corresponding solution to problem (1). Then, is finite if and only if

Proof. Note that , a simple computation yields Therefore, we can conclude that Multiplying to both sides of the first equation in (8), we get Integrating (24) with respect to over yields where we used the relations If differentiating the first equation in (8) with respect to , we obtain Multiplying to both sides of Equation (27), then integrating the result with respect to over , it follows that Since that we have with which yields that Then, by approximating in by function , (32) also holds for . In fact, let be the solution to problem (1) with the initial data . By local well-posedness theorem, we know that for in and as . Since we have Since in as , it follows that in as . Meanwhile, in and in as . Letting in (33), it follows that (32) holds for .
Thus, we get Note that Here, we have used the facts that Therefore, (34) is reduced to If there exists some positive constant such that then we have By using Gronwall’s inequality, we get Therefore, which implies the -norm of the solution to (1) does not blow up in finite time.
On the other hand, if then solution to (1) will blow up in finite time. This completes the proof of the theorem.

Remark 7. Similar to the case of nonperiodic, we prove the following result: Let be given and . And be the corresponding solution to problem (1). Assume that for some positive constant . Then, we obtain that , for a certain positive constant .

3. Blow-up Solutions and Blow-up Rate

In this section, we establish some sufficient conditions for the breaking of waves for the initial-value problem (1). To this end, we need the following lemma.

Lemma 8. Let and be a given function. Then, for any there exists at least one point such that and the function is almost everywhere differentiable in , with

Now we are in position to state the following that provide a case that wave breaks in finite time.

Theorem 9. Let with and . Assume and . If there exists some such that Then, the corresponding solution to (1) blows up in finite time in the following sense: there exists a with such that

Proof. . Differentiating the first equation in (11) with respect to , we get where we have used the relation
According to Lemma 8 and the local well-posedness theorem, there is at least one point satisfying . Hence, Thus, we get where is given by Note that then Let , it follows from (9) that with is a positive constant. Therefore, we obtain Combining (49) and (54), we have with According to assumption (44), we have , hence . Considering the continuity of with respect to , we can obtain that for any , and .
Then by solving the inequality (54), it follows that then Notice that there exists such that which demonstrates that the solution blows up at a time

Theorem 10. If is the blow-up time of the solution to (1) with initial data satisfying the assumption of Theorem 9. Then,

Proof. From (49) and (54), we know that therefore, Choose . Since , we get ; there is some point such that Since is absolutely continuous on . By the above differential inequality, it follows that is strictly decreasing on and hence Combining (63) and (63), we get Since is arbitrary, the above relation implies namely, This implies by in view of the definition of .

4. Global Existence

In this section, we turn our attention to existence of the global solution of system (1). We begin with the related results as follows.

Lemma 11. Let be the solution of problem (1) with , and be the maximal time of existence. Then, the problem has a unique solution and is an increasing diffeomorphism of the line with Furthermore, where is the solution of the problem (72) and is the function given in (44).

Proof. The proof of Lemma 11 is similar as for the classic CH Equation (2), see ref. [36] for details.

Theorem 12. Let , and . Assume that
on or on does not change sign Then, the solution of (1) possesses bounded from blow -derivative, which implies the global existence of the solution in time .

Proof. Since with , we get Differentiating this representation of with respect to gives By combining (72) and (73), we have Since that it follows from (72), (74), and (75) that if , then and or if , then and
The above facts are enough to ensure that Consider that we conclude that The above inequality and Theorem 10 imply which shows the solution or exists globally in time.

Remark 13. It is observed that if the conditions of Theorem 12 are satisfied and if , then does not blow up in finite time by considering Theorem 9. This is similar to the previous results on the line [33].

Remark 14. From Lemma 11, we can observe that the presence of function is the main obstacle of our investigation of existence of the global solutions. In fact, if , it follows from Lemma 11 that the second condition of Theorem 12 would hold automatically as a consequence of its first condition. Therefore, the sign invariance of is essential to prove the global existence of solutions.

Remark 15. Similarly for the Camassa-Holm equation and other analogous equations (see [4, 5, 7, 25, 33]), (1) admits the wave-breaking phenomenon, but differently from the Camassa-Holm equation, we cannot assure the global existence of solutions through the way we followed here [30, 33].

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Furthermore, all data in the paper are available.


We would like to thank the referees for valuable comments and suggestions. The first author Zhang is thankful to Longyuan Youth Innovation and Entrepreneurship Talent Project and Tianshui Normal University ‘Qinglan Talents’ Project for all support provided. The work of Zhang is partially supported by the NSF of China under the grant 11561059, the NSF of Gansu Province (China) grant 21JR7RE176, and the Innovation Improvement Project for Colleges and Universities in Gansu Province in 2021 (2021B-192). The work of Peng is partially supported by the NSF of Gansu Province (China) grant 20JR5RA498.