LOCAL-IN-SPACE BLOW-UP CRITERIA FOR TWO-COMPONENT NONLINEAR DISPERSIVE WAVE SYSTEM

. We investigate the blow-up phenomena for the two-component generalizations of Camassa-Holm equation on the real line. We establish some a local-in-space blow-up criterion for system of coupled equations under certain natural initial proﬁles. Presented result extends and speciﬁes the earlier blow-up criteria for such type systems.

1. Introduction. In this paper, we consider the following two-component Cauchy problem for the generalized Camassa-Holm equation ρ t + (ρu) x = 0 (2) u (x, 0) = u 0 (x) (3) ρ (x, 0) = ρ 0 (x) (4) where u(t, x) denotes the horizontal velocity of the fluid and ρ(t, x) is a parameter related to the free surface elevation from equilibrium (or scalar density). When g(u) = ku and ρ ≡ 0 (1) becomes the Camassa-Holm equation where k is a dispersive coefficient related to the crtical shallow water speed. Twocomponent model for C-H was first derived in [49] and can be viewed as an important model in the context of shallow water theory [13,26,36]. It is worth to note that the papers [13] and [26] incorporate vorticity into the fluid model (physically, vorticity is vital for incorporating the ubiquitous effects of currents and wave-current interactions in fluid motion, also the mathematical analysis of the full-governing equations for water waves with vorticity is of particular interest). In the paper [35] a classification of integrable two-component systems of non-evolutionary partial differential equations that are analogous to the Camassa-Holm equation is carried The local well-posedness of Cauchy problem for equation (1) has been considered in [52] (also, see [53,55,50,6]). It should be noted that blowup criteria for such type equation systematically involved the computation of some global quantities or other global conditions like antisymmetry assumptions or sign conditions on the associate potential. In this context, it is interesting that in [4] (see also [5,6]), contrary to previously known blow-up criteria, Brandolese and Cortez suggested a local-in space blow-up criteria which only involves the values of u 0 (x 0 ) and u 0 (x 0 ) in a single point x 0 of the real line. In the paper [48] author improved the blow-up criteria which established in the paper [4].
Motivated by the above-mentioned papers [4,5,6,48] we would like to establish a "local-in-space" blowup criterion for two-component systems (1) − (4). The presented result extend and specify the earlier blow-up criteria for such type the system.
Our paper is organized as follows. In section 2, we recall several useful results which are crucial in the proof of the new blowup result. We give this proof in Section 3.

2.
Preliminaries. In this section, we present the local well-posedness result, the precise blow-up scenario of the nonlinear dispersive wave equation and one result from [4] in order to pursue our goal.
Note that, the local well-posedness of the Cauchy problem for (1) was proved in [50,52] by classical Kato's approach [39]. For the system (1) − (4) the following theorem holds Then there exists T > 0, with T = T (u 0 , ρ 0 , g) and a unique solution to the Cauchy problem (1)- (4) . Moreover, the solution depends continuously on the initial data.
Since the proof of this theorem is similar to results from [25,11,56], with corresponding changing according to proof of Theorem 3.1 from [50], we omit this proof.
Also, by standard way we obtain the following 5 2 , and T the maximal time of existence. Then for all t ∈ [0, T ) we have Analogical reasoning to those made in [25] for a similar system show that the following result holds Theorem 2.3. Assume that g ∈ C ∞ (R) . Let (u, ρ) be the solution of the system (1)-(4) with initial date (u 0 , ρ 0 − 1) ∈ H s (R) × H s−1 (R), s > 5 2 and T * > 0. Then (u, ρ) blows up in finite time if and only if We finally recall the following useful lemma which will be used in the sequel.
Then the following estimate holds: where p (x) = 1 2 e −|x| and α = 1 Then we have: In the case g 1 = m = M be a constant function (this corresponds to K = 0), the right-hand side of the above convolution estimates reads p1 R± * g 1 (u) 3. Main result. We are now ready to formulate and prove our main result.
Proof. We introduce the following notation: y = u − u xx and where q (x 0 , t) is the solution of the following problem : q (x, 0) = x. Notice that the assumptions made on u imply that q ∈ C 1 ([0, T * ) × R, R) is well defined on the whole time interval [0, T * ) (see [45,21,50,4]).
Then, differentiating E 1 (t) we get Integration by parts yields the following identities Then, inserting (13) into (12) we have Bearing in the mind equality (11) we obtain that Multiplying the inequality (14) by e −q(x0,t) we get Adding the expression −q t (x 0 , t) e −q(x0,t) E 1 (t) to the right-hand side and lefthand side of the latter inequality, by virtue of equality (11) we have Besides, it is easy to see that Then, using (7) (precisely, inequality (3.5) in [4]) for f (s) = 1 2 s 2 and g 1 (s) = s 2 + g (s) we obtain that Obviously, Next, we would like to obtain the similar inequality for the following integral: Indeed, by differentiating we get: Integration by parts yields the following identities Further, according to (16) and (17) we Meanwhile, integration by part gives Next, adding the expression q t (x 0 , t) e q(x0,t) E 2 (t) to the right-hand side and lefthand side of the inequality (18), by virtue of equality (11) and (7) we have In virtue of (2) and (11) we have −ux(q(x0,τ ),τ )dτ and since ρ 0 (x 0 ) = 0 we obtain that ρ (q (x 0 , t) , t) = 0 for every t.

VURAL BAYRAK, EMIL NOVRUZOV AND IBRAHIM OZKOL
The necessary changes to deal with the conditions of Part (ii) of the theorem are slight. In this case, instead of (15) and (19) we have ( by using (8) ) The last part of the proof proceeds in the same manner, by applying (10). Theorem is proved.
From the proof of Theorem 3.1 it is easy to see that the following Corollaries holds Corollary 1. Let g ∈ C ∞ (R), ρ ≡ 0 and u ∈ C ([0, ∞) , H s (R)) be global smooth solution of the problem (1) , (3) (with s > 5/2). Also suppose that at least one of the two following conditions (i) or (ii) is fulfilled: Then, for all t ≥ 0, the two following (i') and (i") conclusions holds: (i") Or, under condition (ii) Corollary 2. Under the conditions of Corollary 1 (i') let c < 0, then u ≥ c for all x ∈ R (i") let c > 0, then u ≤ c for all x ∈ R.
Proof. If However, c < 0 and u (s, t) → 0 as s → −∞. Hence, left-hand side of the latter inequality sholud nonnegative for some s < p, while (u (s, t) − c) < 0. Thus, we obtain a contradiction which show that u (x, t) − c > 0 for all x ∈ R.
The proof proceeds in the same manner with the necessary slight changes.