A Generalized Two-Component Camassa-Holm System with Complex Nonlinear Terms and Waltzing Peakons

In this paper, we deal with the Cauchy problem for a generalized two-component Camassa-Holm system with waltzing peakons and complex higher-order nonlinear terms. By the classical Friedrichs regularization method and the transport equation theory, the local well-posedness of solutions for the generalized coupled Camassa-Holm system in nonhomogeneous Besov spaces and the critical Besov space B2,15/2×B2,15/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B^{5/2}_{2,1}\times B^{5/2}_{2,1}$$\end{document} was obtained. Besides the propagation behaviors of compactly supported solutions, the global existence and precise blow-up mechanism for the strong solutions of this system are determined. In addition to wave breaking, the another one of the most essential property of this equation is the existence of waltzing peakons and multi-peaked solitray was also obtained.


Introduction
In this paper, we propose the following Cauchy problem (1.1) where m = u − 2 u xx , n = v − 2 v xx ( ≥ 0, ≥ 0 ), the constants a, b ∈ ℝ and p, q ∈ ℤ + .Obviously, the system (1.1) has nonlinearities of degree max{p + 1, q + 1} .If choosing m = u, n = v , then Equ.(1.1) is a generalized two- component Burgers type system; if ordering m = u xx , n = v xx , then Equ.(1.1) becomes a generalized two-component Hunter-Saxton type system; and if selecting = = 1 , namely, m = u − u xx , n = v − v xx , then Equ.(1.1) reads as a gen- eralized two-component Camassa-Holm type system.In this paper, to keep our paper concise, we only focus on the coupled Camassa-Holm type system, and During last few decades, due to various mathematical problems and nonlinear physics phenomena interfered, the water wave and fluid dynamics have been attracting much attention [2,4,36,44].Since the raw water wave governing equations have proven to be nearly intractable, the request for suitably simplified model equations was initiated at the early stage of hydrodynamics development.Until the early twentieth century, the study of water waves was restricted almost exclusively to the linear theory.Due to the linearization approach losing some important properties, such us the rare wave breaking, then people usually propose some nonlinear models to explain practical behaviors liking breaking waves and solitary waves [7].The most marked example is the following dispersive nonlinear PDEs where the constants , , c 1 , c 2 c 3 ∈ ℝ .The Painlevé analysis method (cf.[14,16,28]) shown that there are only three asymptotically integrable members in this family, i.e., the famous KdV equation, the Camassa-Holm equation and the Degasperis-Procesi equation.Recently, such integrable peakon equations with cubic nonlinearity and wave breaking have been initialed: one is the Novikov equation, and the other one is the FORQ equation.
Integrable equations with soliton has been studied extensively since they usually have very delicate properties including infinite higher-order symmetries, infinitely many conservation laws, Lax pair, bi-Hamiltonian structure, which can be solved by the inverse scattering method, and so on.Discovering a new integrable equation may be accomplished via different methods.One of ways is the approach proposed by Fokas and Fuchssteiner [20] where the Korteweg-de Vries equation, the Camassa-Holm equation, and the Hunter-Saxton equation are derived in a unified way.The approach is based on the following fact: If 1 , 2 are two Hamiltonian operators and for arbitrary number k their combination 1 + k 2 is also Hamiltonian, then is an integrable equation.Now, letting 1 = x and 2 = x + 3 x + 3 (q x + x q) , where and are constants, then Equ.(1.3) reads as the celebrated KdV equation (see [30]).If choosing 1 = x + 3 x and 2 as above with q = u + u xx , then Equ.(1.3) yields the following equation
The other attractive feature of the CH types equation (1.8) with = = 0 and = −1 is: it admits the following peakon solutions [22]: with (1.4) (1.6) q t + x q + 3 x q + q k q x = 0. (1.7) (1.8) (1.9) q t + 2q 2 u x + q x (u 2 − u 2 x ) = 0, Equations (1.8) also has the multi-peakon solutions by the following unified form (cf. [22]): here the peak positions q i (t) and amplitudes p i (t) satisfy In recent years, the famous CH equation has been generalized to integrable two component Camassa-Holm models.One of them is the following form Obviously, the 2CH and the 2DP are contained in Equ.(1.11) as two special cases with k 1 = 2, k 2 = = ±1 and with k 1 = 3 , respectively.Constantin and Ivanov [8]  derived the 2CH in the condition of shallow water theory.where the variable (x, t) is in connection with the horizontal deviation of the surface from equilibrium and the variable u(x, t) describes the horizontal velocity of the fluid, and all are measured in dimensionless units [8].The 2DP was shown to have solitons, kink, and antikink solutions [41].Escher, Kohlmann and Lenells studied the geometric properties of the 2DP and local well-posedness in various function spaces [18].However, peakon and superposition of multi-peakons were not investigated yet.Motivated by the work of Cotter, Holm, Ivanov and Percival for the Cross-Coupled Camassa-Holm in [11] (called the CCCH equation, i.e., Equ.(1.1) with the choice of a = b = 2 and p = q = 1 ), we first deduce Equ.(1.1), and then study its wave-breaking criteria and peakon dynamical system.The CCCH can be derived from a variational principle by an Euler-Lagrange system with the following Lagrangian [11] (1.10) on the line: (1.11)

3
Journal of Nonlinear Mathematical Physics (2023) 30:1153-1189 And the Euler-Poincaré system in one dimension defined as follows, with K(x, y) = 1 2 e −|x−y| and the Hamiltonian h(n, m) = ∫ ℝ nK * mdx = ∫ ℝ mK * ndx, this Hamiltonian system own a two-component singular momentum map [11] Such a formal waltzing peakons, multi-peakon and compactons of the CCCH are given in [11].In [17], the authors given a geometrical interpretation for the CCCH system along with a large class of peakon equations.Recently, the Cauchy problem of Equ.(1.1) has been studied extensively.The local well-posedness, the condition lead to global existence or wave-breaking, continuity and analyticity of the data-tosolution map, and persistence properties for the CCCH system were discussed in [24,31,37,38].
Inspired by the argument on the approximate solutions for the CH-type equations in [12] and [39,40], we want to obtain the local well-posedness for Equ.(1.1) by the transport equations theory and classical Friedrichs regularization method.However, comparing with the one appearing in [12,39,40], the nonlinear terms of Equ.(1.1) is very complicated.Unlike the regular procedure, we will use the original Equ.
(1.1) rather than the nonlocal form (see (3.18) below) since the fact: . The key to show the local well-posedness through the Littlewood-Paley decomposition and nonhomogeneous Besov spaces is to prove the following inequality and we obtain this inequality by mathematical induction, which involved the degree of the nonlinearities.This result specifically reads as follows.
Then there exists a lifespan T > 0 such that the Equ.(1.1) for every s ′ < s when r = +∞ , and s � = s whereas r < +∞. (1.12) ) 1 We known that B s 2,2 (ℝ) = H s .Thus, under the condition m 0 , n 0 ∈ H s with s > 1 2 , i.e., (u 0 , v 0 ) ∈ H s × H s with s > 5 2 , the above theorem implies that there exists a lifespan T > 0 such that the Cauchy problem (1.1) has a unique solution m, n ∈ C([0, T];H s ) ∩ C 1 ([0, T];H s−1 ) , and the map For any s � < 5∕2 < s , we have the following imbed relationship: which implies that H s and B s 2,1 are very close, so, we next establish the local wellposedness solution for Equ.(1.1) . Then there exists a lifespan T = T(z 0 ) > 0 and a unique solution z = (u, v) verify that the Cauchy problem (1.1) Furthermore, the solutions continuously depend on the initial data, i.e., the mapping is continuous.
In order to get the precise blow-up scenario, we need the following equivalent theorem: Theorem 1.3 Suppose that the initial data (m 0 , n 0 ) ∈ H s (ℝ) × H s (ℝ) with s > 1  2 , and (m, n) be the corresponding solution to the Cauchy problem (1.1), and T * m 0 ,n 0 > 0 is the maximum time of existence for the solution of the Equ.(1.1).Then It is will known that the solution of Camassa-Holm type equations occurs blowup only in the form of breaking waves, namely, the solution remains bounded but its slope about the space becomes unbounded in finite time [40].Next, we establish the accurate blowup scenarios for sufficiently regular solutions to the Equ.(1.1).
Journal of Nonlinear Mathematical Physics (2023) 30:1153-1189 Theorem 1.4 Let z 0 = (u 0 , v 0 ) ∈ L 1 ∩ H s with s > 5∕2 , and T be the lifespan of the solution z(x, t) = (u(x, t), v(x, t)) to Equ. (1.1) with the initial data z 0 .If p = 2a, q = 2b , then every solution z(x, t) to Equ. (1.1) remains globally regular in all time.If p > 2a (or q > 2b ), then the corresponding solutions z(x, t) blow up in a finite time iff (v p ) x (or (u q ) x ) are unbounded at −∞ in a finite time.If p < 2a (or q < 2b ), then the corresponding solutions z(x, t) blow up in a finite time iff (v p ) x (or (u q ) x ) tends to +∞ in a finite time.
Let us now give a sufficient condition for the global existence of the solutions to Equ. (1.1).
Then the solution to Equ. (1.1) remains smooth for all time.
As per [11,43], the CCCH system might not be completely integrable.However, it does have peakon and multi-peakon solutions which display interesting dynamics property with both oscillation and propagation.Furthermore, if the two initial values u 0 or v 0 of in Equ.(1.1) have a compact support, then the compact property will be succeed to u and v at all times t ∈ [0, T).Theorem 1.6 Supposed that the initial data (u 0 , v 0 ) ∈ H s × H s with s > 5∕2 , and ) have a compact support, and T = T(u 0 , v 0 ) > 0 be the maximal existence time to the corresponding initial data.Then the C 1 func- tions x ↦ m(x, t) (or x ↦ n(x, t) ) also have a compact support, for any t ∈ [0, T).
Finally, we will exhibit that Equ.(1.1) not only admits peaked solitary wave but also possesses multi-peaked solitray wave solutions.
Theorem 1.7 Let the constant c > 0 , Equ. (1.1) has the single peaked solitary wave in the form which are global weak solutions iff = c 1∕q , = c 1∕p .Moreover, the multi-peaked solitary wave solutions for Equ.(1.1) takes on the form of (1.14) on the line: in the non-periodic case: whose peaked positions g i (t), k j (t) and amplitudes f i (t), h j (t) satisfy the following dynamical system where ⟨f (x)⟩ = 1 2 (f (x − ) + f (x + )) , and the notation [x − ct] defined by Equ.(1.10).
The entire paper is organized as follows.In next section, we obtain the local wellposedness solution in Besov spaces of Equ.(1.1) through proving Theorems 1.1 − 1.2.In section 3, our goal is twofold, one is to get the condition leads to global existence and blow up phenomena, and the another is to analyze the propagation behaviors start from compactly supported solutions to the problem (1.1), see Theorems 1.3 − 1.6 for the details.In section 4, the peakon and multi-peakons are derived through proving Theorem 1.7.

Local Well-Posedness to Equ. (1.1) in the Besov Spaces
In present section, we will establish the local well-posedness for the initial-value problem Equ.(1.1) in the Besov spaces, i.e., prove Theorem 1.1 and 1.2.The properties of the Besov spaces and the Littlewood-Paley theory can be found in [39,40].

Local Well-Posedness to Equ. (1.1) in the Besov Spaces B s l,r
At the beginning we introduce the following definition.Definition 2.1 For T > 0, s ∈ ℝ and 1 ≤ l ≤ +∞ and s ≠ 2 + 1 l , we define and E s l,r ≐ ∩ T>0 E s l,r (T).
First, we get the uniqueness and continuity for the solution to the Equ.(1.1) with respect to the initial data, and we denote the generic constant C > 0 is only depend- ing on l, r, s, p, q, |a|, |b|. (1.17) Journal of Nonlinear Mathematical Physics (2023) 30:1153-1189
Proof The hypothesis of this theorem implies that ) , and (u 12 , v 12 ) and m 12 , n 12 solves the transport equations According to Lemma 2.2 (i) in [40], we have Since the property (1 − 2 x ) ∈ OP(S 2 ) , by Proposition 2.2 (7) in [40], for all s ∈ ℝ , we obtain that [40] and B s−2 l,r being an algebra, we arrive at For the case s > 3 + 1 l , the inequality (2.3) also holds true since that B s−3 l,r is an algebra.Thus, The second component v can be treat by the similar way, and get the following inequality Therefore Using Gronwall's lemma, we obtain (i).
Next, we apply the interpolation method to deal with the critical case: ) ∈ (0, 1) .According to Proposition 2.2(5) in [40] and the above inequality, we have

3
Journal of Nonlinear Mathematical Physics (2023) 30:1153-1189 which yields (ii).◻ Next, in order to prove the local existence theorem 1.1, we start establish the approximate solutions to Equ. (1.1) by the famous Friedrichs regularization approach.
and there is a lifespan T > 0 such that the sequence of smooth functions enjoying the following properties: Proof The fact S k+1 u 0 ∈ B ∞ l,r and Lemma 2.3 in [40] enables us to show that the equation (T k ) exists a global solution by induction, moveover, which belongs to is true for all k ∈ ℕ .Due to s > 2 + 1 l , we know that B s−2 l,r is an algebra and B s−2 l,r ↪ L ∞ .Thus, we have and also for ‖n k+1 ‖ B s−2 l,r . Thus, adding the two resulted inequalities yields where �  , and suppose by induction that for all t ∈ [0, T).Noticing and substituting (2.5) and the above inequality into (2.4), one obtain ) ) ) Journal of Nonlinear Mathematical Physics (2023) 30:1153-1189 which implies that the sequence ) .The linear equation (T k ) and the proofs of Lemma 2.1 implies that the sequence ) .In fact, from the equation (T k ) , for all k, j ∈ ℕ , we have, where Apparently, we have with For s > max{2 + 1 l , 3 − 1 l , 5 2 } and s ≠ 3 + 1 l , 4 + 1 l , using a similar argument in the proof of Lemma 2.1, on can arrive Proposition 2.1 in [40] gives By the same way, we can get Due to the sequence {u k , v k } k∈ℕ being uniformly bounded in the spaces E s l,r (T) , we can get a constant C T > 0 independent of k, i and verifying Arguing by the induction procedure, we have when k → ∞ , we get the desired result.The interpolation method leads to the criti- cal case s = 4 + 1 l , which yields the desired result.◻ Therefore, we can finish the proof of the existence and uniqueness for the solution of Equ.(1.1) in the nonhomogeneous Besov space.
Proof of Theorem 1.1 Let us first show that the limit (u, v) ∈ E s l,r (T) × E s l,r (T) and sat- isfies system (1.1).Proposition 2.2(6)and Lemma 2.2 in [40] means that 1 3 Journal of Nonlinear Mathematical Physics (2023) 30:1153-1189 Combining an interpolation argument with Lemma 2.2 gets Taking limit in the equation ) and satisfy the Cauchy problem (1.1) for all s ′ < s .Note the fact B s l,r is an algebra as s > 2 + 1 l , and applying the Lemma 2.2 and Lemma 2.3 in [40] produces (u, v) ∈ E s l,r (T) × E s l,r (T).At the end, the continuity on the initial data in the spaces can be proved through the use of Lemma 2.1 and a interpolation argument.While the continuity in the spaces when r < ∞ can be obtained by a sequence of viscosity approximation solutions (u  , v  ) >0 for the initial-value problem (1.1) which converges uniformly in these spaces.The proof of Theorem 1.1 is completed. .Then there exists a sequence

Local
) satisfy the linear Cauchy problem (T k ) (see, Lemma 2.1).Furthermore, the solutions (u k , v k ) enjoying the following two properties: Proof Firstly, we claim that the sequence 2,1 ) for all s < 5  2 .This argument is very similar to the proof of Lemma 2.2, we omit the details here for concise.
The stability of the solution to Equ. (1.1) was obtain by the following lemma: , then there exists a neighborhood V correspond to z 0 in B ≤ .Indeed,we already get that if we fix a time T > 0 satisfy that T < , form the proof of the local well-posedness, then + .Here, one can actually choose some suitable constant C verify that and M = 2 1∕ (‖z 0 ‖ 2,1 ).On the other hand, we claim that the map ) be the solution of the initial- value problem (1.1) correspond to the initial data z k (0) .Through the above proce- dure we may obtain ≤ e 2 CM T .

3
Journal of Nonlinear Mathematical Physics (2023) 30:1153-1189 Apparently, proving 2,1 ) as k → ∞.Let us recall that (u k , v k ) solves the linear transport equation: Applying the Kato theory [13], we decompose the solution and Using properties of Besov spaces(cf.[13]), it is easily see that the sequence (m k , n k ) k∈ℕ are uniformly bounded in the spaces C([0, T];B ≤ M, for any n ∈ ℕ, t ∈ T. (2.8) (2.9) . (2.11) By the argument in the first step, we can get that (u k , v k ) n∈ℕ≐ℕ∪{∞} is uniformly bounded in the spaces C([0, T];B 2,1 ) is now completed.Using the operator t to original equations (1.1), then repeating the above procedure to the obtaining system in views of ( t u, t v) , we obtain that the map Φ in the spaces 2,1 ) , Lemma 2.5 further show that the sequence {u n , v n } n∈ℕ tends to the limit 2,∞ and converges to some limit function 2,∞ , and 2,1 is indeed a solution of (1.1).Furthermore, which is convergence in the spaces C([0, T], B s 1 2,1 ), s 1 < 5∕2 through interpolation theorem.
Journal of Nonlinear Mathematical Physics (2023) 30:1153-1189 On the other hands, Lemma 2.4 also obtain that the map 2,1 ) is continu- ous.Let us pass the limit in the system (T k ) (see, Lemma 2.1), one can easy get that the pair (u, v) is a solution to Equ. (1.1) and verifies The Lemma 2.3 implies that the continuity with the initial data 2,1 ) .Now, we only need to prove the unique- ness and stability of strong solutions to (1.1).Assume that ) , then get If we set (r) = r(1 − ln r) which satisfies the condition ∫ By virtue of Osgood lemma (cf.Lemma 3.4.in [3]) with (t) = W(t) C , we verify that which leads to Next, we apply the interpolation argument ensures that where = 1 2 − s � ∈ (0, 1] .The above inequality implies the uniqueness.Conse- quently, we prove the theorem 1.2.◻

Blow-Up Criterion
In present section, we shall build up a blow-up criteria for Equ.(1.1).We first recall two useful lemmas as follows.
Lemma 3.1 (See [40]) If the Sobolev index r > 0 , then H r ∩ L ∞ is an algebra, and where the constant c depend only on r.
Lemma 3.2 (See [40]) If Sobolev index r > 0 , then where the constant c depend only on r.

Proof of Theorem 1.3
This theorem can be proved by an inductive method with respect to the Sobolev index s.The proof consist by the following three steps.
Step 1.For the cases s ∈ 1 2 , 1 , Using the Theorem 3.2 in [23] to the equations (1.1), one gets for all t ∈ (0, T * m 0 ,n 0 2 e −|x| and * stands for the convolution on ℝ .Then As per the Young inequality, we have Journal (3.3) Step 2. For the cases s ∈ [1, 2), we differentiate the system (1.1) with respect to x yields By Theorem 3.2 in [23], we have According to the Moser-type estimates in [40]  Step 3. Let us assume that Theorem 1.3 holds for the cases k − 1 ≤ s < k and 2 ≤ k ∈ ℕ .By the mathematical induction, we shall claim that it is true for k ≤ s < k + 1 as well.Differentiating the system (1.1) k times with respect to the space variant x leads to (3.8) (3.9) Journal of Nonlinear Mathematical Physics (2023) 30:1153-1189 According to Lemma 2.2 in [40], we get By Sobolev embedding inequality and Moser-type estimate, we derive where we applied with 0 ∈ 0, 1  4 and H (3.13) (3.14) ) If there exist a maximal existence time T * m 0 ,n 0 < ∞ varify that then by the solution uniqueness in Theorem 1.1, we know that ‖n‖ ) .As per the mathematical induction assumption, we obtained a contradiction that Therefore, we complete the proof of Theorem 1.
Morover, if there exists a positive constant M satisfies that then the solution z(t, ⋅) with the H s × H s -norm does not blow up on [0, T).
Proof The local well-posedness was guaranteed by Theorem 1.1.
Using the operator D s to the system (3.18),multiplying the result system by D s u and D s v , respectively.Then integrating the obtained system over ℝ , we may arrive at (3.17) Journal of Nonlinear Mathematical Physics (2023) 30:1153-1189 with Now, we estimate the right-hand side of (3.19), In the above inequality, we applied Lemma 3.2 with r = s is used.By Lemma 3.1 and the mathematical induction, we have 1 2 If p > 2a (or q > 2b ) and the slope of the function v p (or u q ) is lower bounded or if p < 2a (or q < 2b ) and the slope of the function v p (or u q ) is upper bounded on [0, T) × ℝ , then there exists a positive constant M > 0 verify that In view of the Gronwall's inequality, we obtain This inequality and Theorem 3.1 imply that the solution of Equ.(1.1) does not blow up in a finite time.
On the other hand, combing Theorem 3.1 and Sobolev's imbedding theorem give that if the slope of the functions v p , u q becomes unbounded either lower or upper in a finite time, then the solution will blow up in a finite time.This complete the proof of Theorem.◻ Next, let us consider the following Cauchy problem: where u, v denote the solution to the problem (1.1).Adopting classical ordinary differential equations theory leads to the results on , , which are key to the blow-up scenarios. (3.21) Journal of Nonlinear Mathematical Physics (2023) 30:1153-1189 Lemma 3.3 Let T > 0 be the lifespan of the solution to Equ. (1.1)with corre- spond to u 0 , v 0 ∈ H s (s > 5∕2 ).Then the system (3.22)exists a unique solution , ∈ C 1 ([0, T), ℝ) .and the map (t, ⋅), (t, ⋅) is an increasing diffeomorphism over ℝ , where for all (t, x) ∈ [0, T) × ℝ. (u q )  (s, (s, x))ds > 0, Journal of Nonlinear Mathematical Physics (2023) In present section, in order to prove Theorem 1.9, we construct some appropriate sequences of peakon solutions by the method of undetermined coefficients.First, let us show that the peakon formulas (1.14-1.15)and multi-peakon formulas (1.16-1.17)define some weak solutions to Equ. (1.1) both on a circle and on a line, respectively.

Proof of Theorem 1.7
The non-periodic peakon solution in the form of (1.14).Without loss of generality, we set x 0 = 0 .First, Rewriting the model (1.1) as where Noticing that then we have A simple computation reveals For the case x < ct , we derive For the case x > ct , we deduce Consequently, we obtain and ∞ ct e −(p+2)y+x+(p+1)ct dy = p −e (p+1)(x−ct) + e x−ct .Combining Equs.(4.2-4.4) with the assumption = c 1∕q , = c 1∕p , we get that the first equation of the system (4.1)holds on the line in the sense of distribution.
The periodic peakon solution in the forms of (1.15).We claim that Equ.(4.1) is equivalent to Equ. (1.1), let us start from the original system(1.1).Let f ∈ L 1 loc (X) , and the open set X ⊂ ℝ .Assume that f � ∈ L 1 loc (X) and is continuous except at a sin- gle point x 0 ∈ X ; then the right-handed and left-handed limits f (x ± 0 ) exist, moreover, where T f is the distribution associated to the func- tion f and x 0 is the Dirac delta distribution centered at where ct is the periodic Dirac delta distribution centered at x = ct mod 2 , we have u − u xx = 2 sinh( ) ct and Employing the hyperbolic identity cosh 2 x = 1 + sinh 2 x yields Then, we find Similarly, we have and Therefore, we have x (cosh p+1 K + p cosh p−1 K sinh 2 K − 2 sinh( ) cosh p ( ) ct ) = x ((p + 1) cosh p+1 K − p cosh p−1 K − 2 sinh( ) cosh p ( ) ct ) = p [(p + 1) 2 cosh p K sinh K − p(p − 1) cosh p−2 K sinh K − 2 sinh( ) cosh p ( ) � ct ].
u f i ( ġi − v p (g i )) x (q i ) + ( ̇fi + a − p p (v p ) x (g i )f i )(g i ) .

Lemma 2 . 3
Well-Posedness for Equ.(1.1) in Critical Besov SpaceIn present section, local well-posedness of the solution for Equ.(1.1) in critical Besov spaces was established.Inspired by the argument of local existence about CH type equations[13], by the famous Friedrichs regularization methold, one can construct the approximate solutions of Equ.(1.1).Given (u 0 , v 0 ) = 0 and the initial data (u 0 , v 0 ) ∈ B

1 1 ) 1 , 2 2, 1 ) 2
and a positive time T satisfy that every solution z of the initial-value problem (1.1) is continuous.Proof At the beginning, we will claim that the map Φ in C([0, T];B is continuous.Fix  > 0 and z 0 ∈ B we prove that there exists two positive constants T and M verify that the solution z = Φ(z) ∈ C([0, T];B 5 and satisfies ‖z‖ Substitute this uniform bounds into Lemma 2.4 yieldsHence Φ is Hölder continuous from B

1 2 2, 1 )
. Moreover, In light of the product law in the Besov spaces and Lemma 4.3 in[13] to equation (

1 ) 1 ) 1 )
, and which tends to the limit (u ∞ , v ∞ ) in the spaces C([0, T];B as k → ∞ .Therefore, adopt- ing Proposition 3 in[13] reveals that ( k , k ) tends to (m ∞ , n ∞ ) in the spaces C([0, T];B .Therefore, adding this convergence result into the esti- mates (2.7) and (2.11), for large enough n ∈ ℤ + leads to By the Gronwall's inequality, we have where the constant C depends only on the constants M and T. Continuity of the map Φ in the spaces C([0, T];B and (3.2), we obtain and Plugging Eqs.(3.10) and (3.11) into Equ.(3.9) gives By a similar argument to the second equation in (3.8) produces Considering the estimate for (3.6) and the fact one may see that (3.6) holds for all s ∈ [1, 2) .Repeating the above procedure as shown in Step 1, thus, Theorem 1.3 holds for all s ∈ [1, 2).

Theorem 3 . 1
3. ◻ To prove Theorem 1.4, let us rewrite the initial-value problem of the transport equation (1.1) as follows with the functions Let us first provide the sufficient conditions lead to global existence for the solutions to Equ. (1.1).Assume that T be the maximal time of the solution z = (u, v) to the Cauchy problem (1.1) with the initial data way, from (3.20) we can get the estimate for ||v|| 2 H s .So, we arrive at Adopting the assumption of the theorem and the Gronwall's inequality imply which completes the proof of Theorem 3.1.◻ Now, we use Theorem 3.1 to show the blow-up scenario for Equ.(1.1).

av p− 1
v x u + (p − a)v p−1 v x u xx = p p sinh K cosh p K.In a similar way, for n(t) we obtain So, the periodic peaked function(1.15) is a solution to the equation (1.1) iff = c 1∕q cosh( ) , = c 1∕p cosh( ) .
− g j )f j e −|g −g j | M ∑ j=1 f j e −|g −g j | p−1