A New Blow-Up Criterion for the DGH Equation

Abstract

We investigate the DGH equation. Analogous to the Camassa-Holm equation, this equation possesses the blow-up phenomenon. We establish a new blow up criterion on the initial data to guarantee the formulation of singularities in finite time.

1. Introduction

In 2001, Dullin et al. [1] derived the following equation to describe the unidirectional propagation of surface waves in a shallow water regime:𝑦_𝑡+𝑐₀𝑢_𝑥+𝑢𝑦_𝑥+2𝑦𝑢_𝑥+𝛾𝑢_𝑥𝑥𝑥=0, (1.1) where 𝑦=𝑢−𝛼²𝑢_𝑥𝑥, the constants 𝛼² and 𝛾/𝑐₀ are squares of length scales, and the constant 𝑐₀=𝑔ℎ>0 is the critical shallow water speed for undisturbed water at rest at spatial infinity, where ℎ is the mean fluid depth and 𝑔 is the gravitational constant, 𝑔=9.8m/s².

In [2], local well-posedness of strong solutions to (1.1) was established by applying Katos theory [3] and some sufficient conditions were found to guarantee the finite blowup of the corresponding solutions for the spatially nonperiodic case. In [4], the author finds best constants for two convolution problems on the unit circle via a variational method. Then, by using the best constants on the DGH equation, sufficient conditions on the initial data are given to guarantee finite time singularity formation for the corresponding solutions.

When 𝛾=0 and 𝛼=1, system (1.1) reduces to the Camassa-Holm equation, which was derived physically by Camassa and Holm in [5] (found earlier by Fokas and Fuchssteiner [6] as a bi-Hamiltonian generalization of the KdV equation) by approximating directly the Hamiltonian for Eulers equation in the shallow water region with 𝑢(𝑥,𝑡) representing the free surface above a flat bottom. For the hydrodynamic derivation one should also refer to the discussion in the papers [7–9]. The Camassa-Holm equation is completely integrable and has infinitely many conservation laws. The local well-posedness for Camassa-Holm equation in the Sobolev space 𝐻^𝑠 with 𝑠>3/2 was proved in [10, 11]. One of the remarkable features of Camassa-Holm equation is the presence of breaking waves. Wave breaking for a large class of initial data has been established in [10–16]. Here we would like to mention that in [16] we give a new and direct proof for McKeans theorem. The wave breaking and propagation speed for more general family of one-dimensional shallow water equations are studied in [17]. In [18] Guo and Zhu established sufficient conditions on the initial data to guarantee blow-up phenomenon for the modified two component Camassa-Holm (MCH2) system. The existence of global solutions was also explored in [10, 12]. In addition to [10, 12], it is worth pointing out that the solution can be continued uniquely after wave breaking either as a conservative global solution or as a dissipative global solution, see the discussions in the papers [19, 20]. The solitary waves of Camassa-Holm equation are peaked solitons. The peaked solitons replicate a feature that is characteristic for the waves of great height-waves of largest amplitude which are exact solutions of the governing equations for water waves compare the papers [21–23]. The orbital stability of the peakons was shown by Constantin and Strauss in [24], here the orbital stability means that the shape of the waves is stable under perturbations, so that these patterns can be detected. It is worthy of being mentioned here the property of propagation speed of solutions to the Camassa-Holm equation; the first results in this direction were provided in the papers [25–27], which was also presented by Zhou and his collaborators in [28]. In [29], it is proved that strong solution to the Camassa-Holm equation decays algebraically with the same exponent as that of the initial datum.

In this paper, we will establish a new blow-up criterion on the profile of the initial data. Now, we give our main result as follows.

Theorem 1.1. Suppose that 𝑢₀∈𝐻^𝑠(ℝ), 𝑠≥3/2, 𝑦₀=𝑢₀−𝛼²𝑢_0𝑥𝑥 satisfies 𝑦₀(𝑥₀)+(1/2)(𝑐₀+𝛾/𝛼²)=0, ^𝑥0_−∞𝑒^𝜉/𝛼𝑦₀(𝜉)+12𝑐₀+𝛾𝛼²𝑑𝜉>0,^∞_𝑥0𝑒^−𝜉/𝛼𝑦₀(𝜉)+12𝑐₀+𝛾𝛼²𝑑𝜉<0, (1.2)for some point 𝑥₀∈ℝ. Then the solution 𝑢(𝑥,𝑡) to (1.1) with initial datum 𝑢₀(𝑥) blows up in finite time.

2. Proof of Theorem 1.1

As direct calculation, we can rewrite (1.1) as𝑦_𝑡+𝑢𝑦_𝑥+2𝑦𝑢_𝑥−𝛾𝛼²𝑦_𝑥+𝑐₀+𝛾𝛼²𝑢_𝑥=0. (2.1)

Letting 𝜆=−𝛾/𝛼² and 𝜅=(𝑐₀+(𝛾/𝛼²))/2, the above equation can reduce to𝑦_𝑡+𝑢𝑦_𝑥+2𝑦𝑢_𝑥+𝜆𝑦_𝑥+2𝜅𝑢_𝑥=0. (2.2)Set Λ=(1−𝛼²𝜕²_𝑥)^1/2, then the operator Λ⁻² can be expressed by its associated Green’s function 𝐺=(1/2𝛼)𝑒^{−|𝑥|/𝛼} as Λ⁻²𝑓(𝑥)=𝐺∗𝑓(𝑥)=(1/2𝛼)∫_ℝ𝑒^{−|𝑥−𝑦|/𝛼}𝑓(𝑦)𝑑𝑦. Then, (2.2) is equivalent to the following equation:𝑢_𝑡+(𝑢+𝜆)𝑢_𝑥+𝜕_𝑥𝐺∗𝑢²+𝛼²2𝑢²_𝑥+2𝜅𝑢=0. (2.3)Motivated by Mckean’s deep observation for the Camassa-Holm equation [13], we can do similar particle trajectory as𝑞_𝑡𝑞()=𝑢𝑞,𝑡+𝜆,0<𝑡<𝑇,𝑥∈ℝ,𝑥,0=𝑥,𝑥∈ℝ, (2.4)where 𝑇 is the life span of the solution, then 𝑞 is a diffeomorphism of the line; here it is worthwhile pointing out that this corresponds to the Lagrangian viewpoint in hydrodynamics (see the discussion in [30]). Differentiating the first equation in (2.4) with respect to 𝑥, one has𝑑𝑞_𝑡𝑑𝑥=𝑞_𝑥𝑡=𝑢_𝑥𝑞,𝑡𝑞_𝑥,𝑡∈(0,𝑇). (2.5)Hence𝑞_𝑥(𝑥,𝑡)=exp^𝑡₀𝑢_𝑥𝑞,𝑠𝑑𝑠,𝑞_𝑥(𝑥,0)=1. (2.6)

From (2.2) and (2.5) we can obtain𝑑𝑦𝑞𝑞𝑑𝑡+𝜅²_𝑥=𝑦_𝑡𝑞+𝑢𝑞,𝑡+𝜆𝑦_𝑥𝑞+2𝑢_𝑥𝑞,𝑡𝑦𝑞+𝑘𝑞²_𝑥=0, (2.7)

then it follows that𝑦𝑞+𝜅𝑞²_𝑥=𝑦₀(𝑥)+𝜅. (2.8)In the case of the CH-equation (with 𝜅=0), the invariance of (2.8) has a geometric interpretation, compare [31, 32]. Differentiating (2.2), it follows from the definition of 𝑞(𝑥,𝑡) that𝑑𝑢𝑑𝑡^𝑥𝑞𝑥₀,𝑡,𝑡=𝑢_𝑡𝑥𝑞𝑥₀,𝑡,𝑡+𝑢_𝑥𝑥𝑞𝑥₀,𝑡,𝑡𝑢𝑞𝑥₀1,𝑡,𝑡+𝜆=−2𝑢²_𝑥𝑞𝑥₀+1,𝑡,𝑡𝛼²𝑢²𝑞𝑥₀+1,𝑡,𝑡𝛼²2𝜅𝑢𝑞𝑥₀−1,𝑡,𝑡𝛼²𝐺∗𝑢²+𝛼²2𝑢²_𝑥+2𝜅𝑢𝑞𝑥₀1,𝑡,𝑡=−2𝑢²_𝑥𝑞𝑥₀+1,𝑡,𝑡𝛼²(𝑢+𝜅)²𝑞𝑥₀−1,𝑡,𝑡𝛼²𝐺∗(𝑢+𝜅)²+𝛼²2𝑢²_𝑥𝑞𝑥₀1,𝑡,𝑡≤−2𝑢²_𝑥𝑞𝑥₀+1,𝑡,𝑡2𝛼²(𝑢+𝜅)²𝑞𝑥₀,𝑡,𝑡, (2.9)here we have used the inequality𝐺∗(𝑢+𝜅)²+𝛼²2𝑢²_𝑥≥12(𝑢+𝜅)². (2.10)In fact,𝐺∗(𝑢+𝜅)²+𝛼²2𝑢²_𝑥=12𝛼_ℝ𝑒^{−|𝑥−𝜉|/𝛼}(𝑢+𝜅)²+𝛼²2𝑢²_𝜉)=1(𝜉,𝑡𝑑𝜉𝑒2𝛼^{−𝑥/𝛼𝑥}_−∞𝑒^𝜉/𝛼(𝑢+𝜅)²+𝛼²2𝑢²_𝜉)+1(𝜉,𝑡𝑑𝜉𝑒2𝛼^𝑥/𝛼∞_𝑥𝑒^−𝜉/𝛼(𝑢+𝜅)²+𝛼²2𝑢²_𝜉(𝜉,𝑡)𝑑𝜉. (2.11)From^𝑥_−∞𝑒^𝜉/𝛼(𝑢+𝜅)²+𝛼²𝑢²_𝜉(𝜉,𝑡)𝑑𝜉≥2𝛼^𝑥_−∞𝑒^𝜉/𝛼(𝑢+𝜅)𝑢_𝜉𝑑𝜉=𝛼^𝑥_−∞𝑒^𝜉/𝛼(𝑢+𝜅)²_𝜉𝑑𝜉=𝛼𝑒^𝑥/𝛼(𝑢+𝜅)²−^𝑥_−∞𝑒^𝜉/𝛼(𝑢+𝜅)²𝑑𝜉, (2.12)we can deduce^𝑥_−∞𝑒^𝜉/𝛼(𝑢+𝜅)²+𝛼²2𝑢²_𝜉)𝛼(𝜉,𝑡𝑑𝜉≥2𝑒^𝑥/𝛼(𝑢+𝜅)². (2.13)Similarly we can obtain^∞_𝑥𝑒^−𝜉/𝛼(𝑢+𝜅)²+𝛼²2𝑢²_𝜉)𝛼(𝜉,𝑡𝑑𝜉≥2𝑒^−𝑥/𝛼(𝑢+𝜅)². (2.14)Combining (2.11), (2.13), and (2.14) we deduce our inequality (2.10).

Claim
𝑢_𝑥(𝑞(𝑥₀),𝑡)<0 is strictly decreasing and (𝑢+𝜅)²(𝑞(𝑥₀,𝑡),𝑡)<𝛼²𝑢²_𝑥(𝑞(𝑥₀,𝑡),𝑡) for all 𝑡≥0.
Suppose there exists a 𝑡₀ such that (𝑢+𝜅)²(𝑞(𝑥₀,𝑡),𝑡)>𝛼²𝑢²_𝑥(𝑞(𝑥₀,𝑡),𝑡) on [0,𝑡₀) and (𝑢+𝜅)²(𝑞(𝑥₀,𝑡₀),𝑡₀)=𝛼²𝑢²_𝑥(𝑞(𝑥₀,𝑡₀),𝑡₀). From the expression of 𝑢(𝑥,𝑡) in terms of 𝑦(𝑥,𝑡), we can rewrite 𝑢(𝑥,𝑡)+𝜅 and 𝑢_𝑥(𝑥,𝑡) as follows: 𝑢()=1𝑥,𝑡+𝜅𝑒2𝛼^{−𝑥/𝛼𝑥}_−∞𝑒^𝜉/𝛼𝑦()1𝜉,𝑡+𝜅𝑑𝜉+𝑒2𝛼^𝑥/𝛼∞_𝑥𝑒^−𝜉/𝛼𝑦()𝑢𝜉,𝑡+𝜅𝑑𝜉,_𝑥()1𝑥,𝑡=−2𝛼²𝑒^{−𝑥/𝛼𝑥}_−∞𝑒^𝜉/𝛼𝑦()1𝜉,𝑡+𝜅𝑑𝜉+2𝛼²𝑒^𝑥/𝛼∞_𝑥𝑒^−𝜉/𝛼𝑦(𝜉,𝑡)+𝜅𝑑𝜉. (2.15)Now, let 𝐼(𝑡)∶=𝑒^{−𝑞(𝑥0,𝑡)/𝛼𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼𝑦()(𝑡)𝜉,𝑡+𝜅𝑑𝜉,𝐼𝐼∶=𝑒^{𝑞(𝑥0,𝑡)/𝛼∞}_{𝑞𝑥0,𝑡}𝑒^−𝜉/𝛼𝑦(𝜉,𝑡)+𝜅𝑑𝜉. (2.16)Then, 𝑑𝐼(𝑡)1𝑑𝑡=−𝛼𝑢𝑞𝑥₀,𝑡+𝜆𝑒^{−𝑞(𝑥0,𝑡)/𝛼𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼𝑦(𝜉,𝑡)+𝜅𝑑𝜉+𝑒^{−𝑞(𝑥0,𝑡)/𝛼𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼𝑦_𝑡(𝜉,𝑡)𝑑𝜉. (2.17)From (2.2), the equation for 𝑦(𝑥,𝑡) can be written as 𝑦_𝑡+𝑦+𝜅𝑢_𝑥+12𝑢²−𝛼²𝑢²_𝑥𝑥+𝜆𝑦+𝜅_𝑥+𝜅𝑢_𝑥=0. (2.18)Putting (2.18) into the second term on the right-hand side of (2.17) and using (2.8), we have 𝑒^{−𝑞(𝑥0,𝑡)/𝛼𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼𝑦_𝑡(𝜉,𝑡)𝑑𝜉=−𝑒^{−𝑞(𝑥0,𝑡)/𝛼𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼𝑦+𝜅𝑢_𝑥+12𝑢²−𝛼²𝑢²_𝑥𝑥+𝜆𝑦+𝜅_𝑥+𝜅𝑢_𝑥()=1𝜉,𝑡𝑑𝜉2𝛼²𝑢²_𝑥−𝑢²𝑞𝑥₀+1,𝑡,𝑡−𝜅𝑢𝛼𝑒^{−𝑞(𝑥0,𝑡)/𝛼𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼1𝑦+𝜅𝑢+2𝑢²−𝛼²𝑢²_𝑥()=1+𝜆𝑦+𝜅+𝜅𝑢𝜉,𝑡𝑑𝜉2𝛼²𝑢²_𝑥−𝑢²𝑞𝑥₀+1,𝑡,𝑡−𝜅𝑢𝛼𝑒^{−𝑞(𝑥0,𝑡)/𝛼𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼32𝑢²−𝛼²2𝑢²_𝑥−𝛼²𝑢𝑢_𝑥𝑥()=𝛼+𝜆𝑦+𝜅+2𝜅𝑢𝜉,𝑡𝑑𝜉₂2𝑢²_𝑥−𝛼𝑢𝑢_𝑥+1−𝜅𝑢𝛼𝑒^{−𝑞(𝑥0,𝑡)/𝛼𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼𝑢²+𝛼²2𝑢²_𝑥+𝜆𝑦+𝜅+2𝜅𝑢(𝜉,𝑡)𝑑𝜉. (2.19) Here we have used −𝛼𝑒^{−𝑞(𝑥0,𝑡)/𝛼𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼𝑢𝑢_𝑥𝑥(𝜉,𝑡)𝑑𝜉=−𝛼𝑢𝑢_𝑥𝑞𝑥₀+1,𝑡,𝑡2𝑢²𝑞𝑥₀,𝑡,𝑡+𝛼𝑒^{−𝑞(𝑥0,𝑡)/𝛼𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼𝑢²_𝑥()−1𝜉,𝑡2𝛼²𝑢²(𝜉,𝑡)𝑑𝜉. (2.20)On the other hand, after integration by parts, the first term on the right-hand side of (2.17) yields −1𝛼𝑢𝑞𝑥₀,𝑡𝑒^{−𝑞(𝑥0,𝑡)/𝛼𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼𝑦(𝜉,𝑡)+𝜅𝑑𝜉=𝛼𝑢𝑢_𝑥𝑞𝑥₀,𝑡,𝑡−𝑢²𝑞𝑥₀,𝑡,𝑡−𝜅𝑢. (2.21)Hence after combining the above terms and inequalities together, (2.17) reads as 𝑑𝐼(𝑡)=𝛼𝑑𝑡2²𝑢²_𝑥−𝑢²−2𝜅𝑢𝑞𝑥₀+1,𝑡,𝑡𝛼𝑒^{−𝑞(𝑥0,𝑡)/𝛼𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼𝑢²𝛼+2𝜅𝑢+²2𝑢²_𝑥()=𝛼𝜉,𝑡𝑑𝜉₂2𝑢²_𝑥−(𝑢+𝜅)²𝑞𝑥₀+1,𝑡,𝑡𝛼𝑒^{−𝑞(𝑥0,𝑡)/𝛼𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼(𝑢+𝜅)²+𝛼²2𝑢²_𝑥()≥1𝜉,𝑡𝑑𝜉2𝛼²𝑢²_𝑥−(𝑢+𝜅)²𝑞𝑥₀,𝑡,𝑡>0,on0,𝑡₀, (2.22)here (2.13) is used again. From the continuity property we have 𝑒^{−𝑞(𝑥0,𝑡0)/𝛼𝑞(𝑥0,𝑡0)}_−∞𝑒^𝜉/𝛼𝑦𝜉,𝑡₀+𝜅𝑑𝜉>𝑒^{−𝑥0/𝛼𝑥0}_−∞𝑒^𝜉/𝛼𝑦₀(𝜉)+𝜅𝑑𝜉>0. (2.23)Similarly, 𝑑𝐼𝐼(𝑡)≤1𝑑𝑡2(𝑢+𝜅)²−𝛼²𝑢²_𝑥𝑞𝑥₀,𝑡,𝑡<0,on0,𝑡₀. (2.24)Thus by continuity property 𝑒^{𝑞(𝑥0,𝑡0)/𝛼∞}_{𝑞𝑥0,𝑡0}𝑒^−𝜉/𝛼𝑦𝜉,𝑡₀+𝜅𝑑𝜉<𝑒^{𝑥0/𝛼∞}_𝑥0𝑒^−𝜉/𝛼𝑦₀(𝜉)+𝜅𝑑𝜉<0. (2.25)Summarizing (2.23) and (2.25), we obtain 𝛼²𝑢²_𝑥𝑞𝑥₀,𝑡₀,𝑡₀−(𝑢+𝜅)²𝑞𝑥₀,𝑡₀,𝑡₀1=−𝛼^{2𝑞(𝑥0,𝑡0)}_−∞𝑒^𝜉/𝛼𝑦𝜉,𝑡₀+𝜅𝑑𝜉^∞_{𝑞𝑥0,𝑡0}𝑒^−𝜉/𝛼𝑦𝜉,𝑡₀1+𝜅𝑑𝜉>−𝛼^2𝑥0_−∞𝑒^𝜉/𝛼𝑦₀(𝜉)+𝜅𝑑𝜉^∞_𝑥0𝑒^−𝜉/𝛼𝑦₀(𝜉)+𝜅𝑑𝜉=𝛼²𝑢²_0𝑥𝑥₀−𝑢₀+𝜅²𝑥₀>0. (2.26)This is a contradiction. So the claim is true.

Moreover, due to (2.22) and (2.24), we have the following for (𝛼²𝑢²_𝑥−(𝑢+𝜅)²)(𝑞(𝑥₀,𝑡),𝑡): 𝑑𝛼𝑑𝑡²𝑢²_𝑥−(𝑢+𝜅)²𝑞𝑥₀1,𝑡,𝑡=−𝛼²𝑑𝑑𝑡^{𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉𝑦+𝜅(𝜉,𝑡)𝑑𝜉^∞_{𝑞𝑥0,𝑡}𝑒^−𝜉()1𝑦+𝜅𝜉,𝑡𝑑𝜉=−𝛼²𝑑𝑒𝑑𝑡^{−𝑞(𝑥0,𝑡)𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉𝑦+𝜅(𝜉,𝑡)𝑑𝜉𝑒^{𝑞(𝑥0,𝑡)∞}_{𝑞𝑥0,𝑡}𝑒^−𝜉()−1𝑦+𝜅𝜉,𝑡𝑑𝜉𝛼²𝑒^{−𝑞(𝑥0,𝑡)𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉()𝑑𝑦+𝜅𝜉,𝑡𝑑𝜉𝑒𝑑𝑡^{𝑞(𝑥0,𝑡)∞}_{𝑞𝑥0,𝑡}𝑒^−𝜉()1𝑦+𝜅𝜉,𝑡𝑑𝜉≥−2𝛼²𝛼²𝑢²_𝑥−(𝑢+𝜅)²𝑞𝑥₀,𝑡,𝑡𝑒^{𝑞(𝑥0,𝑡)∞}_{𝑞𝑥0,𝑡}𝑒^−𝜉)+1𝑦+𝜅(𝜉,𝑡𝑑𝜉2𝛼²𝛼²𝑢²_𝑥−(𝑢+𝜅)²𝑞𝑥₀,𝑡,𝑡𝑒^{−𝑞(𝑥0,𝑡)𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉𝑦+𝜅(𝜉,𝑡)𝑑𝜉=−𝑢_𝑥𝑞𝑥₀,𝑡,𝑡𝛼²𝑢²_𝑥−(𝑢+𝜅)²𝑞𝑥₀,𝑡,𝑡. (2.27) Now, substituting (2.9) into (2.27), it yields𝑑𝛼𝑑𝑡²𝑢²_𝑥−(𝑢+𝜅)²𝑞𝑥₀≥1,𝑡,𝑡2𝛼²𝛼²𝑢²_𝑥−(𝑢+𝜅)²𝑞𝑥₀×,𝑡,𝑡_𝑡0𝛼²𝑢²_𝑥−(𝑢+𝜅)²𝑞𝑥₀,𝜏,𝜏𝑑𝜏−2𝛼²𝑢_0𝑥𝑥₀. (2.28) Before completing the proof, we need the following technical lemma.

Lemma 2.1 (see [33]). Suppose that Ψ(𝑡) is twice continuously differential satisfying Ψ^′′(𝑡)≥𝐶₀Ψ^′(𝑡)Ψ(𝑡),𝑡>0,𝐶₀Ψ(𝑡)(𝑡)>0,>0,Ψ′>0. (2.29)Then Ψ(𝑡) blows up in finite time. Moreover, the blow-up time can be estimated in terms of the initial datum as 2𝑇≤max𝐶₀Ψ(0),Ψ(0)Ψ^′(0). (2.30)

Let Ψ(𝑡)=(∫^𝑡₀(𝛼²𝑢²_𝑥−(𝑢+𝜅)²)(𝑞(𝑥₀,𝜏),𝜏)𝑑𝜏−2𝛼²𝑢_0𝑥(𝑥₀)), then (2.28) is an equation of type (2.29) with 𝐶₀=1/2𝛼². The proof is complete by applying Lemma 2.1.

Remark 2.2. The special case 𝑐₀+𝛾/𝛼²=0 was studied in [4]. We can regard the special case as the CH(𝜅=0) equation with a strong dispersive term 𝑐₀𝑦_𝑥. Result here without the restriction is an improvement.

Remark 2.3. Recall, (2.2), we can easily find that the DGH equation is the CH(𝜅≠0) equation with a strong dispersive term.

Acknowledgments

This work is partially supported by Zhejiang Innovation Project (T200905), ZJNSF (Grant no. R6090109), and NSFC (Grant no. 11101376).