On Blow-Up of Solutions to a Weakly Dissipative Two-Component Camassa–Holm System

In this paper, we consider a weakly dissipative two-component Camassa–Holm system. We demonstrate a simple sufficient condition on initial date to guarantee blow-up of solutions in finite time and guarantee the solutions exist globally in time. The results improve considerable the previous results.

Obviously, for = 0 , the system (1) is reduced to the Camassa-Holm equation (2) m t + um x + 2u x m = 0, m = u − u xx , which was first derived formally by Fokas and Fuchssteiner [18], and later derived as a model for unidirectional propagation of shallow water over a flat bottom by Camassa and Holm in [19]. Moreover, the Camassa-Holm equation could be also derived as a model for the propagation of axially symmetric waves in hyperelastic rods [20]. The Camassa-Holm equation has a bi-Hamilton structure [18] and is completely integrable [19,21]. In particular, it possesses an infinity of conservation laws and is solvable by its corresponding inverse scattering transform [21]. Local in space blow-up for the Camassa-Holm equation has been introduced in [23].
In general, it is difficult to avoid energy dissipation mechanisms in a real world. Recently, Wu and Yin discussed the blow-up, blow-up rate and decay of the solution of the weakly dissipative Camassa-Holm equation [13,14]. Hu and Yin studied the blow-up and blow-up rate of solutions to a weakly dissipative periodic rod equation [15]. Motivated by the above results, we are interested in the effect of the weakly dissipative term on the system (1). In this paper, we consider the following weakly dissipative two-component Camassa -Holm system where m = u − u xx and > 0 is a dissipative parameter. Hu [16] considered global existence and blow-up phenomena for the periodic system (3). Chen et al [17] showed that the system (3) can exhibit the wave-breaking phenomenon. The main difference between the systems (1) and (3) is that the system (3) does not have conservation law In fact, for the system (3), E(t) decays to zero as time goes to infinity (see Lemma 2.2). When = 0 , the system (3) is reduced to the weakly dissipative Camassa-Holm equation In Ref. [13], Wu and Yin present a global existence result and a blow-up result for Eq. (4). The results shows that if the initial potential m 0 = u 0 − u 0xx at some point x 0 ∈ ℝ satisfies some sign condition, then the corresponding solution to Eq. (4) exists globally in time or blows up in finite time. Some properties of solutions to the weakly dissipative equation have been investigated in [14,25].
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. Unlike previous results, we do not use a condition of the changing of the sign of m 0 at some point x 0 ∈ ℝ . Our results could be stated as follows: Then the corresponding solution u(t, x) to the system (3) exists globally in time.
Then the corresponding solution u(t, x) to the system (3) blows up in finite time. Remark 1.1 Theorems 1.1 and 1.2 improve or cover the previous global existence and blow-up results in [13,14,17,22].
The rest of this paper is organized as follows. In Sect. 2, we recall several useful results which are crucial in the proof of Theorems 1.1 and 1.2. In Sects. 3 and 4, we complete the proof of our results.

Preliminaries
In this section, we recall several useful results to purse our goal.
To begin with, applying transport equation theory combined with the method of Besov spaces, and going along the similar line of the proof in [3], one can readily prove the following local well-posedness result. The proof of the theorem is similar to that of the case = 0 in [3], so we omit it here.
Moreover, the solution depends continuously on the initial data, i.e., the mapping Next, we state the following precise blow-up scenario. The proof of the theorem is similar to the proof of Theorem 4.2 in [4], we omit it here. 3 2 , and T > 0 be the maximal existence time of the solution (u, ) to the system (3) with initial data (u 0 , 0 ) . 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. 3 2 , and T > 0 be the maximal existence time of the solution (u, ) to the system (3) with initial data (u 0 , 0 ) . Then we have Proof Form the second equation of the system (3), it's obvious that Solving above ODE (7) yields (6). ◻ 3 2 , and T > 0 be the maximal existence time of the solution (u, ) to the system (3) with initial data (u 0 , 0 ) . Then we have or Proof Multiplying the first equation of the system (3) by 2u and integrating by ℝ , by a simple calculation, we get Similarly, Adding (9) and (10) where we denote by * the convolution. Then taking the Young inequality, one get According to Theorem 2.2, we have that the solution exists globally in time.

Proof of Theorem 1.2
First, we introduce the following notation: Differentiating I(t) with respect to t, we get By using the first equation of the system (3) and integrating by parts, we have Note that Journal of Nonlinear Mathematical Physics (2022) 29:588-600 Inserting (15), (16) and (5) into (14), we have Next, by integrating by parts, we have Thus, from (17) and (18), it obvious that Multiplying (19) by e −q(t,x 0 ) , we have Due to q t t, x 0 = u t, q(t, x 0 ) , adding −q t t, x 0 e −q(t,x 0 ) I(t) to the left-side of (20) and −u t, q(t, x 0 ) e −q(t,x 0 ) I(t) to the right-side of (20) yields Hence, d dt e −q(t,x 0 ) I(t) > 0 holds as long as 1 2 e −q(t,x 0 ) I(t) > + u and e −q(t,x 0 ) I(t) > 0 . In view of Lemma 2.2, we have (17) Note that (18) and the condition of Theorem 1.2, we get Therefore, we can conclude that d holds. Since u t, q(t, x 0 ) − u x t, q(t, x 0 ) is a continuous function, so we have , p * m = u , where we denote by * the convolution. Then we can rewrite the first equation of the system (3) as follows: Differentiating (25) with respect to x, we get then where we used Lemma 2.1 and the inequality p * (u 2 + 1 2 u 2 x ) ≥ 1 2 u 2 . Then by (24) and (26)  .