Global Existence and Blow-Up for a Weakly Dissipative Modified Two-Component Camassa-Holm System

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 ce−jx−ctj, 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]:


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 ce −jx−ctj , called peakons, which describes an essential feature of the travelling waves of largest amplitude [3][4][5][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 λm is the weakly dissipative term and λ > 0 is a dissipative parameter. Wu and Yin show that if the initial momentum m 0 = u 0 − u 0xx at some point x 0 ∈ ℝ 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 m 0 at some point x 0 ∈ ℝ to guarantee blow-up in finite time.
The two-component Camassa-Holm system is as follows: where m = u − u xx , uðt, xÞ describes the horizontal velocity of the fluid, and ρðt, xÞ 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, blowup, global existence, stability, and other mathematical properties can be seen in [15][16][17][18][19][20][21][22] and references therein. The modified two-component Camassa-Holm system is as follows: x Þð ρ − ρ 0 Þ, u denotes the velocity field, and ρ 0 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 ρ at the point of vertical slope. Some related work can be found in [24][25][26][27][28][29]. Let γ = ρ − ρ 0 ; then, the system (4) is equivalent to the following one, where m = u − u xx and ρ = γ − γ xx .
In this paper, we are interested in the effect of the weakly dissipative term on the system (5) as follows: where m = u − u xx , ρ = γ − γ xx , and λ > 0 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), EðtÞ 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 m 0 = u 0 − u 0xx at some point x 0 ∈ ℝ 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: Then, the corresponding solution uðt, xÞ to the system (5) exists globally in time.
and ρ 0 ðxÞ ≤ 0 on ðx 0 , ∞Þ, and assume further that for some point x 0 . 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 m 0 = u 0 − u 0xx at some point x 0 ∈ ℝ satisfies some sign condition, so our theorems improve the results in [30]. If ρ = 0, 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.

Preliminaries
In this section, we recall several useful results to pursue our goal. First, we recall local well-posedness for the system (5).
Next, we state the following precise blow-up scenario.
Theorem 5 (see [30]). Let ðu 0 , γ 0 Þ ∈ H s ðℝÞ × H s−1 ðℝÞ with s ≥ 5/2 and T > 0 be the maximal existence time of the solution ðu, γÞ to the system (5) with initial data ðu 0 , γ 0 Þ. Then, the corresponding solution blows up in a finite time T < ∞ 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.
Proof. If x ≤ qðt, x 0 Þ, then Similarly, if x ≥ qðt, x 0 Þ, we also have Therefore, we complete the proof of Lemma 8.

Proof of Theorem 1
First, multiplying the first equation of the system (5) by 2m and integrating by parts, we get Similarly, Adding (19) and (20), we have d dt Multiplying (21) by e 2λt yields Note that pðxÞ for all f ∈ L 2 ðℝÞ and p * m = u, where we denote by $ * $ the convolution. Then, taking the Young inequality, one gets Hence, we have

Advances in Mathematical Physics
It is easy to derive that Integrating from 0 to t yields Thus, it follows that due to ðkm 0 k 2 L 2 + kρ 0 k 2 L 2 Þ 1/2 < ð4λ/3Þ and According to Theorem 5, we have that the solution exists globally in time.