WAVE BREAKING AND PERSISTENT DECAY OF SOLUTION TO A SHALLOW WATER WAVE EQUATION

. As we all know, wave breaking of the water wave is important and interesting to physicist and mathematician. In the article, we devote to the study of blow-up phenomena, the decay of solution and traveling wave solution to a shallow water wave equation. First, based on the blow-up scenario, some new blow-up phenomena is derived. By virtue of a weighted function, the persistent decay of solution is established. Finally, we explore the analytic solutions and traveling wave solutions.

Let p = 1 12 and ς = u, Eq.(12) is equivalent to Eq.(1.1). One of the most prominent relatives is the Camassa-Holm (CH) equation [2] which derived by Camassa and Holm y t + uy x + 2u x y = 0, y = u − u xx (1.4) from an asymptotic approximation to the Hamiltonian for the Green-Naghdi equations in shallow water theory, which approximates to the incompressible Euler equation at the next order beyond the KdV equation [9]. It has a bi-Hamiltonian structure [17], a Lax pair based on a linear spectral problem of second order, and is completely integrable [5]. Moreover, Eq.(1.4) not only has wave breaking and global existence of strong solution [5,6,7,10], but also exists peakon solitions [3], which are orbital stable [11].
The CH equation is not the only integrable PDE of its kind, being a shallow water equation whose dispersionless version has weak solitons. Degasperis and Procesi [14] used an asymptotic integrability approach to isolate integrable third order equations, and discovered the Degasperis-Procesi (DP) equation The DP equation can be regarded as a model for nonlinear shallow water dynamics [20]. Degasperis, Holm and Hone [13] prove the formal integrability of Eq.(1.5) by constructing a Lax pair. They also show [13] that it has a bi-Hamiltonian structure and an infinite sequence of conserved quantities, and admits exact peakon solutions. Despite the form of DP is similar to the CH equation, it should be emphasized that these two equations are truly different, such as the conservation laws of Eq.(1.5) are weaker than them of Eq.(1.4) [15]. One of the important features of Eq. (1.5) is that it has not only peakon solitons [13], i.e. solutions at the form u(t, x) = ce −|x−ct| and periodic peakon solutions [35], but also has shock peakons [4,24] which are given by and periodic shock peakons [16] Compared with the CH and DP equations, by the Hamiltonian structure, Eq.(1.1) has the following infinite sequence of conserved quantities and admits solitary travelling wave solutions decaying at infinity [18]. The orbital stability of it has been recently obtained by Mutlubas and Geyer in [25] using the Hamiltonian structure of Eq.(1.1) as = 4, µ = 12. The well-posedness of Eq.(1.1) in space H s , s > 3 2 or B s p,r , s > max{ 3 2 , 1 + 1 p } is the same as it of the CH and DP equations [9,12,26,34]. Moreover, Eq.(1.1) itself is not symmetrical, i.e. (u, x) (−u, −x), some results of the equation are truly different with the CH and DP equations [9].
In this paper, based on the conservation law (1.6) and blow-up criterion, In particular, the blow-up rate of blow-up solution is 4 7 . Next, in view of a weighted function which is defined in [1], the persistence decay of solution is established. The analyticity of solutions for water wave equation has been studied extensively [8,19,21,28,30]. We finally explore the existence and uniqueness of analytic solutions and traveling wave solutions.
The remainder of the paper is organized as follows. In Section 2, note that the conservation law (1.6) is equivalent to the H 1 -norm of solution, we derive the blowup scenario of solution by Theorem 2.1 and Theorem 2.2. Then some new results of wave breaking phenomena are investigated. In Section 3, by the weight function, we explore the persistent decay of solution in a weight L p -spaces. As the argument which was used in [23,30], in Section 4, the existence and uniqueness of analytic solutions to Eq.(1.1) with the analytic initial data is obtained, we also prove that Eq.(1.1) has a family traveling wave solutions.
2. Global existence and wave breaking. In this subsection, based on a conservation law of strong solution, we will derive the blow-up scenario and establish the singularity of strong solution to Eq.(1.1).
Lemma 2.1. [22] Assume that s > 0. Then we have where c is constant depending only on s, and f, g x ∈ L ∞ ∩ H s−1 .
On the one hand, the first term of RHS in (2.2) can be dealt with as follows where we have used Lemma 2.1 in the second inequality.
On the other hand, note that is a conservation law of Eq.(1.1), then it follows that Therefore, we estimate the second term of RHS in (2.2) to yield By virtue of Gronwall's inequality to (2.6) is given by If there exist a M > 0 such that lim t↑T u x (t) L ∞ ≤ M , then by the Gronwall inequality to (2.6) obtains that Assume lim t↑T u x (t) L ∞ = ∞, by Sobolevs embedding theorem, we deduce that the solution u(t, x) will blow up in finite time T . This completes the proof of Theorem 2.1.
Proof. If the slope of the solution u satisfies (2.7) or (2.8) in finite time, then by Theorem 2.1 and Sobolev's embedding theorem, the solution u will blow up in space H s (R) in finite time T . Differentiating Eq.(2.1) with respect to x variable. Let the Green function p =

XUE YANG AND XINGLONG WU
By Young's inequality and Hölder's inequality, one has that 2u + 5 2 On the other hand, if < 0, let M (t) = u x (t, ζ(t)) = sup x∈R u x (t, x). Since u xx (t, ζ(t)) = 0, in view of (2.9) and (2.10), for a.e. t ∈ [0, T ), we deduce that i.e. the slope of solution is bounded from above. Thanks to Theorem 2.1, the solution u blows up in finite time T if and only if which derives the result of Theorem 2.2. Based on the method which comes from [29], we now present some results of wave breaking of Eq.(2.1).
2 and > 0. Assume u(t, x) be corresponding solution of Eq.(2.1) with the initial datum u 0 . If for n ∈ N + , the slope of u 0 satisfies then there exists the lifespan T < ∞ such that the solution u blows up in finite time T . Moreover, the above bound of lifespan T is estimated by dx and the constant K satisfies (2.16).
Proof. In view of (2.9), for all n ∈ N/{0}, it follows that (2.13) Thanks to the conservation law, there exists M such that where we have used p L 1 = 1, p L ∞ = (3/µ) 1 2 , the second inequality comes from Gagliardo-Nirenberg's inequality, the last inequality is guaranteed by the Young's inequality with ε.
Let ε = ( 21 n(2n−1) 2µ(n−1)(2n+1) ) n−1 n , combining (2.13) with (2.14) to deduce where the constant K satisfies By virtue of the following interpolation inequality (2.18) By virtue of the assumption of Theorem 2.3 , then h (t) > 0 and h(t) is a increasing function. If the solution u globally exists, then there is t 1 > 0 such that Since h(t 1 ) > 0, let t ≥ t 1 large enough, the above inequality will lead a contradiction. There exists lifespan T < ∞ such that lim t↑T h(t) = ∞. Due to Thanks to Theorem 2.2, it follows that lim sup By solving (2.18), one can easily check that the lifespan T satisfies Since lim t↑T h(t) = ∞, the above inequality can be changed into , then there exists the lifespan T < ∞ such that the solution u blows up in finite time T . In particular, the above bound of lifespan T is estimated by dx and the constant K satisfies (2.16).
Proof. Proceeding exactly as the estimate of (2.15), one has that using inequality (2.17), due to < 0, we immediately deduce that In view of the assumption of Theorem 2.4 Next, we will derive another blow-up phenomena to Eq.(2.1).
Theorem 2.5. Given u 0 ∈ H s (R), s > 3 2 . Assume u(t, x) be corresponding solution of Eq.(2.1) with the initial datum u 0 . If there exists a point x 0 ∈ R such that then the solution u blows up in finite time. Moreover the lifespan T of solution satisfies where M (0) = sup x∈R u 0 (x), m(0) = inf x∈R u 0 (x) and Since u xx (t, ζ(t)) = 0, by virtue of (2.9) is given by By Young's inequality and Hölder's inequality   (2.27) and lim t↑T M (t) = ∞, i.e. the solution blows up in finite time T and Similarly, by the RHS of inequality (2.26), it easily check that the lower bound of lifespan T satisfies If < 0, define m(t) = u x (t, ξ(t)) = inf x∈R u x (t, x). Since u xx (t, ξ(t)) = 0, we also can get which concludes the proof of Theorem 2.5.
If the solution u to Eq.(2.1) with the initial datum u 0 blows up in finite time T , which is guaranteed by Theorem 2.3-2.5, then the blow-up rate of solution satisfies while the L ∞ -norm of solution remains bounded.
Proof. By the assumption T < ∞ and Theorem 2.2, it follows that lim sup 3. The persistent decay of solution. By a weight function, our aim is to investigate the persistent decay of solution in a weight L p -spaces, At first, we recall some definitions which comes from [1].
then, for all t ∈ [0, T ], the following estimate of solution holds where the constant depends on T, v, ϕ and u x L ∞ .
Proof. Let φ(x) = min{N, ϕ(x)} for any N ∈ Z + . Then the function φ is uniformly v-moderate with respect to N . Multiplying Eq.(2.1) by the admissible weight function φ, after applying p|uφ| p−2 uφ, integration by parts to variable x, Note that in view of Hölder's inequality to inequality (3.2). Let G = (3/µ) On the other hand, we deal with (2.9) as the above process Thus, one can easily check that Remark 3.2. If we choose ϕ = ϕ abcd as in (3.1) and satisfies then ϕ abcd is the admissible weight function of Eq.(1.1). Let ϕ = ϕ 00c0 with c > 0 and p = ∞. Then Theorem 3.1 deduces that the uniform algebra decay of solution.

4.
Analytic solutions and traveling wave solutions.
In view of the Fourier transform, the norm of H k,ρ is equivalent to It is not difficult to check the following properties.
It is easy to check the conditions (i) and (ii) is ture to Eq.(4.2). In order to obtain the result of Theorem 4.1, it suffice to derive the following result.
Proof. Thanks to the equivalent norm of H 1,ρ (4.1), for any 0 < ρ 1 < ρ, in view of simple calculation, one has that In view of above relations and Proposition 4.1, it follows that which completes the proof of Lemma 4.1.
Next, we will prove that Eq.(1.1) have a family traveling wave solutions. First, it is necessary to present two definitions. for almost every x ∈ R. We say that b(t) is the symmetric axis of u(t, x). Remark 4.1. Since C ∞ 0 (R + × R) is dense in C 1 0 (R + , C 3 0 (R)). By the density argument, we only need to consider the test functions belong to C 1 0 (R + , C 3 0 (R)). As the method proving Theorem 4.1 in [31], we can derive Theorem 4.2. Let u(t, x) be x-symmetric. If u(t, x) be a unique weak solution of Eq.(1.1), then u(t, x) is a traveling wave.