Abstract
In this paper, we consider a weakly dissipative modified two-component Camassa-Holm system. We demonstrate a simple sufficient condition on initial data to guarantee blow-up of solutions in finite time, and the solutions exist globally in time.
1. Introduction
The well-known Camassa-Holm (CH) equation which was first derived by Fokas and Fuchssteiner [1] and later derived as a model for unidirectional propagation of shallow water over a flat bottom by Camassa and Holm [2]. After the birth of the CH equation, a lot of works have been carried out to it. For example, the CH equation has travelling wave solutions of the form , called peakons, which describes an essential feature of the travelling waves of largest amplitude [3–6]. It is shown in [7] that the blow-up occurs in the form of breaking waves; namely, the solution remains bounded but its slope becomes unbounded in finite time.
In general, it is difficult to avoid energy dissipation mechanisms in our real world. Ott and Sudan [8] studied how the KdV equation was modified by the presence of dissipation and the effect of such dissipation on solitary solution of the KdV equation. Ghidaglia [9] investigated the long-time behavior of solutions to a weakly dissipative KdV equation as a finite-dimensional dynamical system. Inspired by the above works, Wu and Yin consider the following weakly dissipative CH equation [10, 11]: where is the weakly dissipative term and is a dissipative parameter. Wu and Yin show that if the initial momentum at some point satisfies some sign condition, then the corresponding solution to Equation (2) exists globally in time and blows up in finite time. Novruzov and Hagverdiyev [12] derived a condition of the changing of the sign of at some point to guarantee blow-up in finite time.
The two-component Camassa-Holm system is as follows: where , describes the horizontal velocity of the fluid, and describes the horizontal deviation of the surface from equilibrium. The system (3) appears initially in [13]; then, Constantin and Ivanov [14] give a demonstration about its derivation in view of the shallow water theory from the hydrodynamic point of view. Local well-posedness, blow-up, global existence, stability, and other mathematical properties can be seen in [15–22] and references therein.
The modified two-component Camassa-Holm system is as follows: where , , denotes the velocity field, and is taken to be a constant. The system (4) does admit peaked solutions in the velocity and average density; we refer this to Ref. [23] for details. In Ref. [23], the authors analytically identified the steepening mechanism that allows the singular solutions to emerge from smooth spatially confined initial data. They found that wave breaking in the fluid velocity does not imply singularity in the pointwise density at the point of vertical slope. Some related work can be found in [24–29]. Let ; then, the system (4) is equivalent to the following one, where and .
In this paper, we are interested in the effect of the weakly dissipative term on the system (5) as follows: where , , and is a dissipative parameter. The main difference between the systems (4) and (5) is that the system (5) does not have conservation law.
In fact, for the system (5), decays to zero as time goes to infinity (see Lemma 7). Recently, a blow-up result for the system (5) is presented in [30]. Similar to [10, 11], Ref. [30] shows that if the initial momentum at some point satisfies some sign condition, then the corresponding solution to the system (5) blows up in finite time.
The main goal of the present paper is to demonstrate a simple condition guaranteeing blow-up of solutions in finite time and guaranteeing the solutions exist globally in time by using some properties of the solution generated by initial data. Our results could be stated as follows:
Theorem 1. Suppose that with , if Then, the corresponding solution to the system (5) exists globally in time.
Theorem 2. Suppose that with . Assume that , on , and on , and assume further that for some point . Then, the corresponding solution to the system (5) blows up in finite time.
Remark 3. Note that our theorems do not need to assume that the initial momentum at some point satisfies some sign condition, so our theorems improve the results in [30]. If , our theorems improve the global existence and blow-up results in [11] and cover the results in [12].
The rest of this paper is organized as follows. In Section 2, we recall several useful results which are crucial in the proof of Theorem 1 and Theorem 2. In Section 3 and Section 4, we complete the proof of our results.
2. Preliminaries
In this section, we recall several useful results to pursue our goal. First, we recall local well-posedness for the system (5).
Theorem 4 (see [30]). Given with , there exist a maximal existence time and a unique solution to the system (5) such that Moreover, the solution depends continuously on the initial data, i.e., the mapping : is continuous.
Next, we state the following precise blow-up scenario.
Theorem 5 (see [30]). Let with and be the maximal existence time of the solution to the system (5) with initial data . Then, the corresponding solution blows up in a finite time if and only if
Consider the following initial value problem of ordinary differential equation (ODE):
The following lemma will be used to prove our theorem.
Lemma 6 (see [30]). Let with and be the maximal existence time of the solution to the system (5) with initial data . Then, we have Moreover, if there exist such that , then for all .
Lemma 7 (see [30]). Let with and be the maximal existence time of the solution to the system (5) with initial data . Then, we have or
Lemma 8. If , on , and on , then for any fixed , for all .
Proof. If , then Similarly, if , we also have Therefore, we complete the proof of Lemma 8.
3. Proof of Theorem 1
First, multiplying the first equation of the system (5) by and integrating by parts, we get
Similarly,
Multiplying (21) by yields
Note that ; we have for all and , where we denote by the convolution. Then, taking the Young inequality, one gets
Hence, we have
It is easy to derive that
Integrating from to yields
Thus, it follows that due to and
According to Theorem 5, we have that the solution exists globally in time.
4. Proof of Theorem 2
First, we introduce the following notation:
Differentiating with respect to , we get
By using the first equation of the system (5) and integrating by parts, we have
Note that
Inserting (31), (32), and (12) into (30), we have
Next, by integrating by parts, we have
Thus, from (33) and (34), it is obvious that
Multiplying (35) by , we have
Due to , adding to the left side of (36) and to the right side of (36) yields
Hence, holds as long as and . In view of Lemma 7, we have
Note that in (34) and the condition of Theorem 2, we get
Therefore, we can conclude that holds. Since is a continuous function, so we have
Lemma 7 implies that as , so there exist a such that for
Thus, we have
Note that , we have for all and , where we denote by the convolution. Then, we can rewrite the first equation of the system (5) as follows:
Differentiating (43) with respect to , we get then where we used Lemma 6 and the inequality . Then, by (42) and (45), we can infer that
Further, by Lemma 7, for sufficiently large , we have
Then, there exists a such that for ,
Thus, integrating from to , we get
This means that we can choose a such that for all , and by Lemma 7, we have for .
Then, substituting (50) and (51) into (45) yields which leads to and integrating from to gives
Thus, if we suppose that the solution exists globally, then for sufficiently large , which contradicts that the solution blows up in finite time. Therefore, we complete the proof of Theorem 2.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.
Acknowledgments
The work of is supported by National Natural Science Foundation of China (Grant No. 11461037), Yunnan Fundamental Research Projects (Grant No. KKAE202107036), and Yunnan Education Department Science Research Fund Project (Grant No. 2022Y158).