Blow-up phenomena and global existence for the periodic two-component Dullin-Gottwald-Holm system

This paper is concerned with blow-up phenomena and global existence for the periodic two-component Dullin-Gottwald-Holm system. We first obtain several blow-up results and the blow-up rate of strong solutions to the system. We then present a global existence result for strong solutions to the system.


Introduction
In this paper, we consider the following periodic two-component Dullin-Gottwald-Holm (DGH) system: where m = u − u xx , A > 0 and γ are constants.
The system (1.1) has been recently derived by Zhu et al. in [1] by follow Ivanov's approach [2]. It was shown in [1] that the DGH system is completely integrable and can be written as a compatibility condition of two linear systems where ξ is a spectral parameter. Moreover, this system has the following two Hamiltonians For ρ = 0 and m = u − α 2 u xx , (1.1) becomes to the DGH equation [3] u t − α 2 u txx − Au x + 3uu x + γu xxx = α 2 (2u x u xx + uu xxx ), where A and α are two positive constants, modeling unidirectional propagation of surface waves on a shallow layer of water which is at rest at infinity, u(t, x) standing for fluid velocity. It is completely integrable with a bi-Hamiltonian and a Lax pair. Moreover, its traveling wave solutions include both the KdV solitons and the CH peakons as limiting cases [3]. The Cauchy problem of the DGH equation has been extensively studied, cf. [4,5,6,7,8,9,10,11,12,13,14,15,16].
For ρ ≡ 0, γ = 0, the system (1.1) becomes to the two-component Camassa-Holm system [2] m t − Au x + um x + 2u x m + ρρ x = 0, where ρ(t, x) in connection with the free surface elevation from scalar density (or equilibrium) and the parameter A characterizes a linear underlying shear flow. The system (1.2) describes water waves in the shallow water regime with nonzero constant vorticity, where the nonzero vorticity case indicates the presence of an underlying current. A large amount of literature was devoted to the Cauchy problem (1.2), see [17,18,19,20,21,22,23,24,25].
The Cauchy problem (1.1) has been discussed in [1]. Therein Zhu and Xu established the local well-posedness to the system (1.1), derived the precise blow-up scenario and investigated the wave breaking for the system (1.1). The aim of this paper is to study further the blow-up phenomena for strong solutions to (1.1) and to present a global existence result.
Our paper is organized as follows. In Section 2, we briefly give some needed results including the local well posedness of the system (1.1), the precise blow-up scenarios and some useful lemmas to study blow-up phenomena and global existence. In Section 3, we give several new blow-up results and the precise blow-up rate. In Section 4, we present a new global existence result of strong solutions to (1.1).
Notation Given a Banach space Z, we denote its norm by · Z . Since all space of functions are over S, for simplicity, we drop S in our notations if there is no ambiguity.

Preliminaries
In this section, we will briefly give some needed results in order to pursue our goal. With m = u − u xx , we can rewrite the system (1.1) as follows: Here we denote by * the convolution. Using this identity, we can rewrite the system (2.1) as follows: The local well-posedness of the Cauchy problem (2.1) can be obtained by applying the Kato's theorem. As a result, we have the following well-posedness result.
of (2.1). Moreover, the solution (u, ρ) depends continuously on the initial data (u 0 , ρ 0 ) and the maximal time of existence T > 0 is independent of s.

Consider now the following initial value problem
where u denotes the first component of the solution (u, ρ) to (2.1).
). Let (u, ρ) be the solution of (2.1) with initial data (u 0 , ρ 0 ) ∈ H s ×H s−1 , s ≥ 2, and T > 0 be the maximal existence. Then we have Next, we will give two useful conservation laws of strong solutions to (2.1).
Proof. By the first equation in (2.1), we have This completes the proof of the lemma.
Then, we state the following precise blow-up mechanism of (2.1).
with the best possible constant c lying within the range (1, 13 12 ]. Moreover, the best constant c is e+1 2(e−1) .

Blow-up phenomena
In this section, we discuss the blow-up phenomena of the system (2.1). Firstly, we prove that there exist strong solutions to (2.1) which do not exist globally in time.
Theorem 3.1. Let (u 0 , ρ 0 ) ∈ H s × H s−1 , s ≥ 2, and T be the maximal time of the solution (u, ρ) to (2.1) with the initial data (u 0 , ρ 0 ). If there is some x 0 ∈ S such that ρ 0 (x 0 ) = 0 and then the corresponding solution to (2.1) blows up in finite time.
Proof. Applying Lemma 2.1 and a simple density argument, we only need to show that the above theorem holds for some s ≥ 2. Here we assume s = 3 to prove the above theorem. Define now By Lemma 2.7, we let ξ(t) ∈ S be a point where this infimum is attained. It follows that m(t) = u x (t, ξ(t)) and u xx (t, ξ(t)) = 0.
Proof. By Lemma 2.5, we have S u(t, x)dx = a 0 2 . Using Lemma 2.4 and Lemma 2.9, we obtain Following the similar proof in Theorem 3.1, we have Following the same argument as in Theorem 3.1, we deduce that the solution blows up in finite time.
Letting a 0 = 0 and ǫ → 0 in Theorem 3.2, we have the following result.
Corollary 3.1. Let (u 0 , ρ 0 ) ∈ H s × H s−1 , s ≥ 2, and T be the maximal time of the solution (u, ρ) to (2.1) with the initial data (u 0 , ρ 0 ). Assume that S u 0 (x)dx = 0. If there is some x 0 ∈ S such that ρ 0 (x 0 ) = 0 and then the corresponding solution to (2.1) blows up in finite time.
Remark 3.1. Note that the system (2.1) is variational under the transformation (u, x) → (−u, −x) and (ρ, x) → (ρ, −x) even γ = 0. Thus, we can not get a blow up result according to the parity of the initial data (u 0 , ρ 0 ) as we usually do.
Next, we will give more insight into the blow-up mechanism for the wave-breaking solution to the system (2.1), that is the blow-up rate for strong solutions to (2.1). Theorem 3.3. Let (u, ρ) be the solution to the system (2.1) with the initial data (u 0 , ρ 0 ) ∈ H s × H s−1 , s ≥ 2, satisfying the assumption of Theorem 3.1, and T be the maximal time of the solution (u, ρ). Then, we have Proof. As mentioned earlier, here we only need to show that the above theorem holds for s = 3.
Define now By the proof of Theorem 3.1, we have there exists a positive constant Let ε ∈ (0, 1 2 ). Since lim inf t→T m(t) = −∞ by Theorem 3.1, there is some t 0 ∈ (0, T ) with m(t 0 ) < 0 and m 2 (t 0 ) > K ε . Since m is locally Lipschitz, it is then inferred from (3.5) that A combination of (3.5) and (3.6) enables us to infer Since m is locally Lipschitz on [0, T ) and (3.6) holds, it is easy to check that 1 m is locally Lipschitz on (t 0 , T ). Differentiating the relation m(t) · 1 m(t) = 1, t ∈ (t 0 , T ), we get with 1 m absolutely continuous on (t 0 , T ). For t ∈ (t 0 , T ). Integrating (3.7) on (t, T ) to obtain By the arbitrariness of ε ∈ (0, 1 2 ) the statement of Theorem 3.3 follows.

Global Existence
In this section, we will present a global existence result. By Lemma 2.7, we let ξ(t) ∈ S be a point where this infimum is attained. It follows that m(t) = u x (t, ξ(t)) and u xx (t, ξ(t)) = 0.
Since the map q(t, ·) given by (2.3) is an increasing diffeomorphism of R, there exists a x(t) ∈ S such that q(t, x(t)) = ξ(t).
Thus, This completes the proof by using Lemma 2.6.