On the Cauchy problem for a four-component Camassa-Holm type system

This paper is concerned with a four-component Camassa-Holm type system proposed in [37], where its bi-Hamiltonian structure and infinitely many conserved quantities were constructed. In the paper, we first establish the local well-posedness for the system. Then we present several global existence and blow-up results for two integrable two-component subsystems.

(1) is reduced to the generalized Camassa-Holm equation with both quadratic and cubic nonlinearly [29] One can refer to [39,45] for the study of its Lax pair, peakons, weak kinks, kinkpeakon interaction, and local well-posedness. As m 1 = n 2 , m 2 = n 1 , making the change of variable Recently, Yan, Qiao and Yin [56] studied the local well-posedness and derived a precise blow-up scenario and a blow-up result for the strong solutions to Eq.(8).
Later, Zhang and Yin [59] improved the results stated in [56] for Eq.(8) and obtained several global existence or blow-up results for Eq. (9). For more research on other multi-component Camassa-Holm type systems, one can refer to [34,36,46,53].
The aim of this paper is to establish the local well-posedness for the Cauchy problem of Eq.(1) in Besov spaces, and present several global existence and blowup results for Eq.(10)-Eq. (11).
The rest of our paper is then organized as follows. In Section 2, we recall the Littlewood-Paley decomposition and some basic properties of the Besov spaces. In Section 3, we establish the local well-posedness for Eq.(1). The last section is devoted to proving several global existence and blow-up results for Eq.(10)-Eq.(11).
Throughout the paper, C > 0 stands for a generic constant, A B denotes the relation A ≤ CB. Since all function spaces in this paper are over R, for simplicity, we drop R in the notations of function spaces if there is no ambiguity.

2.
Preliminaries. In this section, we recall several useful lemmas which will be used in the sequel.
there j is the Littlewood-Paley decomposition operator [2].
Let us give some classical properties of the Besov spaces.

Lemma 2.2. [2]
The set B s p,r is a Banach space, and satisfies the Fatou property, namely, if (u n ) n∈N is a bounded sequence of B s p,r , then an element u of B s p,r and a subsequence u ψ(n) exist such that Let m ∈ R and f be an S m -multiplier (i.e. f : R → R is smooth and satisfies that for each multi-index α, there exists a constant C α such that |∂ α f (ξ)| ≤ C α (1 + |ξ|) m−|α| , ∀ξ ∈ R). Then the operator F (D) is continuous from B s p,r to B s−m p,r . Lemma 2.4. [23] (i) For s > 0 and 1 ≤ p, r ≤ ∞, there exists C = C(d, s) such that (ii) If 1 ≤ p, r ≤ ∞, s 1 ≤ 1 p , s 2 > 1 p (s 2 ≥ 1 p , if r = 1) and s 1 + s 2 > max{0, 2 p − 1}, there exists C = C(s 1 , s 2 , p, r) such that for some ρ > 1 and M > 0, and Then the following transport equation has a unique solution Moreover, the following inequality holds true: As a consequence of Lemma 2.3 and the Young inequality, we have the following inequalities which will be frequently used in the sequel.
3. Local well-posedness. In this section, we study the local well-posedness for Eq.(1). We rewrite Eq.(1) as follows: where Further, suppose that T * > 0 is the maximal existence time of the corresponding solution M to Eq. (17). If T * is finite, then we have where q = max{l, 2}.

ZENG ZHANG AND ZHAOYANG YIN
Proof of Theorem 3.1. We shall proceed as follows.
Step 1. Uniqueness and continuity with respect to the initial data. We can get where θ ∈ (0, 1).
Step 2. Existence. We can obtain (i). constructing approximate solutions where p,r ). (iv). passing to the limit and to concluding that M is a solution of Eq.(17).
Step 3. If T * < ∞, and Thus we can extent the solution M beyond T * , which is a contradiction with the assumption of T * .
For more details about the proof, one can refer to the proofs of Theorem 3.1 and Theorem 3.2 in [59], where the system we deal with is where M = (m 1 , · · · , m N , n 1 , · · · , n N ) T , M 0 = (m 10 , · · · , m N 0 , n 10 · · · , n N 0 ) T , The terms A(Γ, Γ x ) and B(U, U x ) in this paper can be estimated exactly in the same way as we did for the terms A(H, H x ) and B(U, U x ) in [59]. Thus following along exactly the same lines of the proofs of Theorem 3.1 and Theorem 3.2 in [59], one can prove the theorem. For the sake of simplicity, we omit the details here.
Note that, if Γ = 0, then Eq.(1) is reduced to the following ODE system: where M = (m 1 , m 2 , n 1 , n 2 ) T , M 0 = (m 10 , m 20 , n 10 , n 20 ) We can use the classical ODE theory to set up the local well-posedness of Eq. (22). More precisely, we have Proof of Theorem 3.2. If X = L ∞ , using Proposition 1, one can readily have that On the other hand, if X = B s p.r ∩ L ∞ with s > 0, using Lemma 2.4 and Proposition 1, we get M is locally Lipschitz with respect to the norm X. Therefore, using the classical Picard scheme, or using the Banach fixed point theorem, or applying Theorem 3.2 in [2], one can readily prove the existence and uniqueness of this theorem.
As for the blow-up criterion, since the case X = L ∞ is much more simple, we only consider the case X = B s p.r ∩ L ∞ with s > 0. Using Lemma 2.4, we get The Gronwall lemma then implies that if T * < ∞, and L ∞ dτ < ∞, then we have lim sup t→T * M (t) B s p,r ∩L ∞ < ∞. We can extent the solution M beyond T * , which is a contradiction with the assumption of T * . This completes the proof of the theorem.
Utilizing Theorem 3.1-3.2, we readily obtain the following corollaries.   Proof of Theorem 4.1. Note that, we have On the other hand, since we have Thus, (u − u x )(v + v x ) = 0. Therefore (m, n)(t, x) = (m 0 (x), n 0 (x)) exists globally in time. Proof of Theorem 4.2. Let T > 0 be the maximal existence time of the corresponding solution (m, n) to Eq.(10). One can assume without loss of generality that m 0 ≥ 0, n 0 ≥ 0 for all x ∈ R. Thus, (23) and (24) imply that Noticing we obtain which leads to Similar arguments yield Using Eq.(10), we get Using Eq.(10) again, and combining the above equality with (25)- (28) give rise to The Gronwall inequality then yields that ∀ t ∈ [0, T ), which leads to Therefore, in view of Theorem 3.2 and Corollary 2, we conclude that the solution (m, n) exists globally in time.
For Eq.(11), we first consider the following initial value problem Lemma 4.3. Let m 0 , n 0 ∈ H s (s > 1 2 ), and let T > 0 be the maximal existence time of the corresponding solution (m, n) to Eq. (11). Then Eq.(32) has a unique solution q ∈ C 1 ([0, T ] × R; R). Moreover, the mapping q(t, ·) (t ∈ [0, T )) is an increasing diffeomorphism of R, with Proof. According to Theorem 3.1, we get that m, n ∈ C([0, T ]; H s )∩C 1 ([0, T ]; H s−1 ) with s > 1 2 , from which we deduce that 1 2 (uv x − vu x ) is bounded and Lipschitz continuous in the space variable x and of class C 1 in time variable t, then the classical ODE theory ensures that Eq.(32) has a unique solution q ∈ C 1 ([0, T ] × R; R). Differentiating Eq.(32) with respect to x gives which leads to (33). So, the mapping q(t, ·) (t ∈ [0, T )) is an increasing diffeomorphism of R.
Proof. Without loss of generality, we assume that m 0 ≥ 0, n 0 ≥ 0. Next, from (35) and (36) Finally, using Eq.(11) again and using the above conservation laws, we have Then the Gronwall lemma yields the desired inequality (38). This completes the proof of the lemma.
Prof of Theorem 4.6. It follows from Eq.(11) that Applying Lemma 4.5 to the first term and the second term on the right hand side of the above inequality yields where we have used the fact that m, n do not change sign. The left terms can be treated in the same way. We have Plunging the above two inequalities into (39) yields Again using Lemma 4.5, we have Next, we consider the following function where g(x) = g(x) = 1 C log log(x + 1) + 1 , x ≥ 0, is the inverse function of f . Differentiating G(a) with respect to a, we obtain N (0, x 0 ) ) > 0, a < 0.
Thus, we can get lim sup t→T0 m(t) L ∞ = ∞ or lim sup t→T0 n(t) L ∞ = ∞. According to Corollary 1, we conclude that the solution (m, n) blows up at the time T 0 .