Wave Breaking and Propagation Speed for a Class of One-Dimensional Shallow Water Equations

Abstract

We investigate a more general family of one-dimensional shallow water equations. Analogous to the Camassa-Holm equation, these new equations admit blow-up phenomenon and infinite propagation speed. First, we establish blow-up results for this family of equations under various classes of initial data. It turns out that it is the shape instead of the size and smoothness of the initial data which influences breakdown in finite time. Then, infinite propagation speed for the shallow water equations is proved in the following sense: the corresponding solution 𝑢(𝑡,𝑥) with compactly supported initial datum 𝑢₀(𝑥) does not have compact 𝑥-support any longer in its lifespan.

1. Introduction

In this paper we consider the equation𝑢_𝑡−𝑢_𝑥𝑥𝑡+(𝑎+𝑏)𝑢𝑢_𝑥=𝑎𝑢_𝑥𝑢_𝑥𝑥+𝑏𝑢𝑢_𝑥𝑥𝑥,(1.1) where 𝑎>0 and 𝑏>0 are real constants.

Set Λ=(1−𝜕²_𝑥)^1/2, then we can rewrite (1.1) as𝑢_𝑡+𝑏𝑢𝑢_𝑥+𝜕_𝑥Λ⁻²𝑎2𝑢²+3𝑏−𝑎2𝑢²_𝑥=0.(1.2) Let 𝑦(𝑡,𝑥)=𝑢(𝑡,𝑥)−𝑢_𝑥𝑥(𝑡,𝑥), then (1.1) can be reformulated in terms of 𝑦(𝑡,𝑥):𝑦_𝑡+𝑎𝑢_𝑥𝑦+𝑏𝑢𝑦_𝑥=0.(1.3)

By comparison with the Camassa-Holm equation [1], the Degasperis-Procesi equation, [2] and the Holm-Staley 𝑏-family of equations [3], it is easy to find that (1.1) is more general. The Camassa-Holm equation, the Degasperis-Procesi equation, and the Holm-Staley b-family of equations are the special cases with 𝑎=2,𝑏=1; 𝑎=3,𝑏=1, and 𝑏=1, respectively. The equation of type (1.1) arises in the modeling of shallow water waves; compare with the discussion in the papers [4, 5].

When 𝑎=2𝑏, use the scaling 𝑢(𝑡,𝑥)=𝑏𝑢(𝑡,𝑥), then (1.1) can be reformulated into𝑢_𝑡−𝑢_𝑥𝑥𝑡+3𝑢𝑢_𝑥=2𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥,(1.4) which is the well-known Camassa-Holm equation. The Camassa-Holm equation was first written explicitly and derived physically as a water wave equation by Camassa and Holm [1], who also studied its solutions. It has infinitely many conserved integrals including the 𝐻¹-norm, but wave breaking also happens in finite time, which coincides with physical phenomenon. Some satisfactory results have been obtained recently [6–8] for strong solutions. Moreover, wave breaking for a large class of initial data has been established in [9–11], and recently, a new and direct proof for McKean's theorem is given in [12]. In [13] some new criterion on blowup is established for the Camassa-Holm equation with weakly dissipative term. In [14] (see also [15]), Xin and Zhang showed global existence and uniqueness for weak solutions with 𝑢₀(𝑥)∈𝐻^𝑠. The solitary waves of the Camassa-Holm equation are peaked solitons and are orbitally stable [16] (see also [17, 18]). It is worthwhile to point out that the peakons replicate a feature that is characteristic for the waves of great height waves of largest amplitude that are exact solutions of the governing equations for water waves; compare with the papers [19–21]. Similarly, if 𝑎=3𝑏, use the some scaling, then (1.1) can be reformulated into the Degasperis-Procesi equation, the existence of global solutions, persistence properties and propagation speed of Degasperis-Procesi equation is given in [22] and the references therein. For the the weakly dissipative Degasperis-Procesi equation, [23] is concerned with some aspects of existence of global solutions, persistence properties, and propagation speed. So, in our following discussion, we will always exclude these two cases.

In general, the family of equations are not always completely integrable systems. However, one can find the following conservation laws: 𝐻₀=_ℝ𝑦𝑑𝑥,𝐻₁=_ℝ𝑦^𝑏/𝑎𝑑𝑥,𝐻₂=_ℝ𝑦²_𝑥𝑦^{−2−(𝑏/𝑎)}+𝑎²𝑏𝑦^−𝑏/𝑎𝑑𝑥,(1.5)

which are quite different from the invariants of the Camassa-Holm equation 𝐻₀=_ℝ𝑦𝑑𝑥,𝐻₁=_ℝ𝑢²+𝑢²_𝑥𝑑𝑥,𝐻₂=_ℝ𝑢³+𝑢𝑢²_𝑥𝑑𝑥.(1.6)

Due to the similarity of (1.2) and the Camassa-Holm equation, just by following the argument for the Camassa-Holm equation, it is easy to establish the following well-posedness theorem for (1.2).

Theorem 1.1. Given 𝑢₀(𝑥)∈𝐻^𝑠(ℝ), 𝑠>3/2, then there exist a 𝑇 and a unique solution 𝑢 to (1.1) such that 𝑢(𝑡,𝑥)∈𝐶[0,𝑇);𝐻^𝑠(ℝ)∩𝐶¹[0,𝑇);𝐻^𝑠−1(ℝ).(1.7)

To make the paper concise we would like to omit the detailed proof, since one can find similar ones for these types of equations in [7, 24].

When we study the Camassa-Holm equation, the most frequently (crucially) used conservation law is that of the 𝐻¹-norm of the solution. However, if 𝑢 is a strong solution (decays rapidly at infinity) to (1.1), direct computation yields𝑑𝑑𝑡_ℝ𝑢²+𝑢²_𝑥𝑑𝑥=(2𝑏−𝑎)_ℝ𝑢³_𝑥𝑑𝑥.(1.8) Hence, the 𝐻¹-norm of the solution is not conserved at all except for 𝑎=2𝑏. However, we also have a clear blow-up scenario as follows: the solution blows up if and only if the first-order derivative blows up that is, wave breaking occurs. More precisely, assume that 𝑇 is the lifespan of the corresponding solution, thenlimsup_𝑡→𝑇𝑢_𝑥(𝑡,𝑥)_𝐿∞(ℝ)=+∞.(1.9)

Before giving our main results, we list some notations that will be used in our paper. (⋅,⋅) denotes the inner product in the Hilbert space 𝐿²(ℝ), (𝑓,𝑔)_𝑟≐(Λ^𝑟𝑓,Λ^𝑟𝑔); use ‖⋅‖_𝐿^𝑠_(ℝ) for the corresponding 𝐿^𝑠(ℝ) norm and ‖⋅‖_𝐻^𝑠_(ℝ) to denote the corresponding 𝐻^𝑠(ℝ) norm.

The remainder of the paper will be organized as follows. In the next section, we establish blow-up results for this family of equations under various classes of initial data. In the last section, infinite propagation speed for the shallow water equations is proved.

2. Blow-Up Phenomenon

After local well posedness of strong solutions is established, the next question is whether this local solution can exist globally. If the solution exists only for a finite time, how about the behavior of the solution when it blows up? What induces the blowup? On the other hand, to find sufficient conditions to guarantee the finite time blowup or global existence is of great interest, particularly for sufficient conditions added on the initial data.

Set 𝑝(𝑡,𝑥) as the characteristic evolved by the solution; that is, it satisfies𝑝_𝑡()(),𝑝()𝑡,𝑥=𝑏𝑢𝑡,𝑝𝑡,𝑥0,𝑥=𝑥.(2.1) Taking derivative with respect to 𝑥 in (2.1), we obtain 𝑑𝑝_𝑥𝑑𝑡=𝑏𝑢_𝑥𝑝,𝑡𝑝_𝑥.(2.2)

Hence 𝑝_𝑥(𝑡,𝑥)=𝑒^{𝑏∫𝑡0𝑢𝑥(𝑝(𝜏,𝑥),𝜏)𝑑𝜏},(2.3)

which is always positive before the blow-up time. Therefore, the function 𝑝(𝑡,𝑥) is an increasing diffeomorphism of the line before blowup. In addition, from (1.1), the following identity is proved:𝑦𝑡,𝑝(𝑡,𝑥)𝑝_𝑥^𝑎/𝑏(𝑡,𝑥)=𝑦₀(𝑥).(2.4)

The following theorem is proved in [25] it is one of sufficient conditions to guarantee the finite time blowup added on the initial data.

Theorem 2.1. Let 𝑎>0,𝑏>0,𝑎>𝑏,3𝑏−𝑎>0, 𝑢₀∈𝐻^𝑠(ℝ) with 𝑠>3/2 being odd and 𝑢^′₀(0)<0. Then the corresponding solution of (1.1) blows up in finite time.

From the above blow-up result we can obtain the following.

Theorem 2.2. Let 𝑎>0,𝑏>0,𝑎>𝑏,3𝑏−𝑎>0, 𝑢₀∈𝐻³(ℝ). Suppose that 𝑦₀ is odd and ∫^∞₀𝑒^−𝜉𝑦₀(𝜉)≤0. Then the corresponding solution of (1.1) blows up in finite time.

Proof. If 𝑦₀(𝑥) is odd, then 𝑢₀(𝑥) is also odd. In fact 𝑢₀(𝑥)=12_ℝ𝑒^{−|𝑥−𝑧|}𝑦₀(𝑧)=1𝑑𝑧2_ℝ𝑒^{−|𝑥−𝑧|}−𝑦₀()1−𝑧𝑑𝑧=−2_ℝ𝑒^{−|𝑥−𝑧|}𝑦₀(𝑧)𝑑𝑧=−𝑢₀(−𝑥).(2.5) On the other hand 𝑢^′₀(0)1=−2⁰_−∞𝑒^𝜉𝑦₀(𝜉)1𝑑𝜉+2⁰_−∞𝑒^−𝜉𝑦₀(𝜉)=𝑑𝜉_∞0𝑒^−𝜉𝑦₀(𝜉)𝑑𝜉<0.(2.6) So Theorem 2.2, follows from Theorem 2.1.

Before giving our main result, we first recall the following three lemmas that will be used in our proof.

Lemma 2.3. If 𝑠>0, then 𝐻^𝑠(ℝ)∩𝐿^∞(ℝ) is algebra. Moreover 𝑓𝑔_{𝐻𝑠(ℝ)}≤𝐶𝑓_𝐿∞(ℝ)𝑔_{𝐻𝑠(ℝ)}+𝑔_𝐿∞(ℝ)𝑓_{𝐻𝑠(ℝ)}.(2.7)

Lemma 2.4. If 𝑠>0, then Λ^𝑠,𝑓𝑔_𝐿2(ℝ)≤𝐶𝜕_𝑥𝑓_𝐿∞(ℝ)Λ^𝑠−1𝑔_𝐿2(ℝ)+𝑔_𝐿∞(ℝ)Λ^𝑠𝑓_𝐿2(ℝ).(2.8) Here [𝐴,𝐵]=𝐴𝐵−𝐵𝐴 denotes the commutator of the linear operators 𝐴 and 𝐵.

Lemma 2.5 (see [26]). Suppose that Ψ(𝑡) is twice continuously differentiable satisfying Ψ^′′(𝑡)≥𝐶₀Ψ^′(𝑡)Ψ(𝑡),𝑡>0,𝐶₀Ψ(0)>0,>0,Ψ^′(0)>0.(2.9) Then Ψ(𝑡) blows up in finite time. Moreover the blow-up time 𝑇 can be estimated in terms of the initial datum as 2𝑇≤max𝐶₀Ψ(0),Ψ(0)Ψ^′(0).(2.10)

Now, we are ready to give our main results. Firstly, we present the result which improves Theorem 2.1.

Theorem 2.6. Let 𝑢₀∈𝐻^𝑟(ℝ) with 𝑟>3/2, and assume that 𝑇 is the existence time of the corresponding solution with the initial data 𝑢₀. If there exists 𝑀>0 such that 𝑢_𝑥(𝑡,𝑥)_𝐿∞(ℝ)≤𝑀,𝑡∈[0,𝑇),(2.11) then the 𝐻^𝑟(ℝ) norm of 𝑢(𝑡,𝑥) does not blow up on [0,𝑇).

Proof. Let 𝑢 be the solution to (1.1) with initial data 𝑢₀∈𝐻^𝑟(ℝ) and let 𝑇 be the maximal existence time of the solution 𝑢. Applying the operator Λ^𝑟 to (1.2), multiplying by Λ^𝑟𝑢, and integrating, we obtain 𝑑‖𝑢‖𝑑𝑡^{2𝐻𝑟(ℝ)}=−2𝑏𝑢,𝑢𝑢_𝑥𝑟+2𝑢,𝑓(𝑢)_𝑟+2𝑢,𝑔(𝑢)_𝑟,(2.12) where 𝑓(𝑢)=−𝑎(1−𝜕²_𝑥)⁻¹(𝑢𝑢_𝑥) and 𝑔(𝑢)=−𝜕_𝑥(1−𝜕²_𝑥)⁻¹(((3𝑏−𝑎)/2)𝑢²_𝑥). We have 𝑢𝑢_𝑥,𝑢_𝑟=Λ^𝑟𝑢𝑢_𝑥,Λ^𝑟𝑢=Λ^𝑟,𝑢𝜕_𝑥𝑢,Λ^𝑟𝑢+𝑢Λ^𝑟𝜕_𝑥𝑢,Λ^𝑟𝑢=Λ^𝑟,𝑢𝜕_𝑥𝑢_𝐿2(ℝ)Λ^𝑟𝑢_𝐿2(ℝ)+12𝑢_𝑥Λ^𝑟𝑢,Λ^𝑟𝑢≤𝐶𝑢_{𝑥𝐿∞(ℝ)}‖𝑢‖²_𝐻^𝑟_(ℝ),(2.13) where we applied Lemma 2.4 with 𝑠=𝑟. Again applying Lemma 2.4 with 𝑠=𝑟−1, we obtain 𝑓(𝑢),𝑢_𝑟=−𝑎1−𝜕²_𝑥⁻¹𝑢𝑢_𝑥,𝑢_𝑟=|𝑎|Λ^𝑟−1𝑢𝑢_𝑥,Λ^𝑟−1𝑢=|𝑎|Λ^𝑟−1,𝑢𝜕_𝑥𝑢,Λ^𝑟−1𝑢+𝑢Λ^𝑟−1𝜕_𝑥𝑢,Λ^𝑟−1𝑢≤𝐶𝑢_{𝑥𝐿∞(ℝ)}‖𝑢‖²_𝐻^𝑟_(ℝ),(2.14) and, from Lemma 2.3, we have 𝑔(𝑢),𝑢_𝑟≤𝑔(𝑢)_{𝐻𝑟(ℝ)}‖𝑢‖_𝐻^𝑟_(ℝ)=−𝜕_𝑥1−𝜕²_𝑥⁻¹3𝑏−𝑎2𝑢²_{𝑥𝐻𝑟(ℝ)}‖𝑢‖_𝐻^𝑟_(ℝ)≤𝐶𝑢_{𝑥𝐿∞(ℝ)}‖𝑢‖²_𝐻^𝑟_(ℝ).(2.15) Combining the above estimates, we obtain 𝑑‖𝑢‖𝑑𝑡^{2𝐻𝑟(ℝ)}≤𝐶𝑀‖𝑢‖²_𝐻^𝑟_(ℝ).(2.16) From Gronwall inequality, we obtain ‖𝑢(𝑡)‖²_𝐻^𝑟_(ℝ)≤𝑒^𝐶𝑀𝑡‖𝑢(0)‖²_𝐻^𝑟_(ℝ).(2.17) This completes the proof of the theorem.

In order to help the readers to understand the following theorem, we would like to give an intuitive explanation for this result. Neglecting lower-order source terms, (1.2) reduces to the generalized Hunter-Saxton equation:𝑢_𝑡+𝑏𝑢𝑢_𝑥=3𝑎−𝑏2^𝑥_−∞−^∞_𝑥𝑢²_𝑥𝑑𝑦.(2.18) Differentiating with respect to 𝑥, one obtains𝑢_𝑥𝑡+𝑏𝑢𝑢_𝑥𝑥=(2𝑏−𝑎)𝑢²_𝑥.(2.19) By looking at the evolution of 𝑢_𝑥 along characteristics, it is clear that one can have 𝑢_𝑥→−∞ or 𝑢_𝑥→+∞ depending on the sign of 2𝑏−𝑎.

Theorem 2.7. Let 𝑢₀∈𝐻^𝑟(ℝ) with 𝑟>3/2. If 𝑎=𝑏/2, then every solution will exist globally in time. If 𝑎>𝑏/2, then the solution blows up in finite time if and only if the slope of the solution becomes unbounded from below in finite time. If 𝑎<𝑏/2, then the solution blows up in finite time if and only if the slope of the solution becomes unbounded from above in finite time.

Proof. It suffices to consider the case 𝑟=3. Let 𝑇>0 be the maximal time of existence of the solution of (1.1). From [25] we know that 𝑢∈𝐶([0,𝑇);𝐻³(ℝ))∩𝐶¹([0,𝑇);𝐻²(ℝ)). Multiplying (1.2) by 𝑦=𝑢−𝑢_𝑥𝑥 and integrating by parts, we get 12𝑑𝑑𝑡_ℝ𝑦²𝑑𝑥=−𝑎_ℝ𝑦²𝑢_𝑥𝑑𝑥−𝑏_ℝ𝑢𝑦𝑦_𝑥=𝑏𝑑𝑥−𝑎+2_ℝ𝑦²𝑢_𝑥𝑑𝑥.(2.20) Note that ‖𝑢‖_𝐻²_(ℝ)≤𝐶₁𝑦_𝐿2(ℝ)≤𝐶₂‖𝑢‖_𝐻²_(ℝ).(2.21) From (2.20), for 𝑎=𝑏/2, we have 𝑢_{𝑥𝐿∞(ℝ)}≤‖𝑢‖_𝐻²_(ℝ)≤𝐶₁𝑦(𝑡)_𝐿2(ℝ)=𝐶₁𝑦(0)_𝐿2(ℝ).(2.22) From Theorem 2.6, for 𝑎=𝑏/2 the solution exists globally in time. If 𝑎>𝑏/2 and the slope of the solution is bounded from below or if 𝑎<𝑏/2 and the slope of the solution is bounded from above on [0,𝑇), then there exists 𝑀>0 such that 𝑑𝑑𝑡_ℝ𝑦²𝑑𝑥≤𝑀_ℝ𝑦²𝑑𝑥.(2.23) By Gronwall inequality, we have 𝑦(𝑡)_𝐿2(ℝ)≤𝑒^𝑀𝑡𝑦(0)_𝐿2(ℝ).(2.24)

There is no doubt that it is the shape of the initial data but not their smoothness or size that influences the lifespan. However, we try to understand what induces wave breaking. Another sufficient condition is given in the following theorem. The approach used to prove the following theorem is rooted in the considerations made in the paper [27].

Theorem 2.8. Let 𝑎−2𝑏>0, 𝑏>0. Suppose that 𝑢₀∈𝐻²(ℝ) and there exists 𝑥₀∈ℝ such that 𝑦₀(𝑥₀)=(1−𝜕²_𝑥)𝑢₀(𝑥₀)=0, and 𝑦₀(𝑥)≥0for𝑥∈−∞,𝑥₀,𝑦₀(𝑥)≤0for𝑥∈𝑥₀,∞.(2.25) Then the corresponding solution 𝑢(𝑡,𝑥) of (1.1) blows up in finite time with the lifespan 𝑇≤max−2𝑏𝑢₀(𝑥),−2𝑢₀(𝑥)𝑏𝑢²_0𝑥(𝑥)−𝑢²₀(𝑥).(2.26)

Proof. Suppose that the solution exists globally. Due to (2.4) and the initial condition, we have 𝑦₀(𝑡,𝑝(𝑡,𝑥₀))=0, and 𝑦𝑡,𝑝(𝑡,𝑥)≥0for𝑥∈−∞,𝑥₀,𝑦𝑡,𝑝(𝑡,𝑥)≤0for𝑥∈𝑥₀,∞(2.27) for all 𝑡.
Since 𝑢(𝑥,𝑡)=𝐺∗𝑦(𝑡,𝑥),𝑥∈ℝ,𝑡≥0, one can write 𝑢(𝑡,𝑥) and 𝑢_𝑥(𝑡,𝑥) as 𝑢()=1𝑡,𝑥2𝑒^−𝑥𝑥_−∞𝑒^𝜉𝑦()1𝑡,𝜉𝑑𝜉+2𝑒^𝑥∞_𝑥𝑒^−𝜉𝑦()𝑢𝑡,𝜉𝑑𝜉,_𝑥()1𝑡,𝑥=−2𝑒^−𝑥𝑥_−∞𝑒^𝜉𝑦()1𝑡,𝜉𝑑𝜉+2𝑒^𝑥∞_𝑥𝑒^−𝜉𝑦(𝑡,𝜉)𝑑𝜉.(2.28)
Consequently, 𝑢²_𝑥(𝑡,𝑥)−𝑢²(𝑡,𝑥)=−^𝑥_−∞𝑒^𝜉𝑦(𝑡,𝜉)𝑑𝜉^∞_𝑥𝑒^−𝜉𝑦(𝑡,𝜉)𝑑𝜉.(2.29)
From the expression of 𝑢_𝑥(𝑡,𝑥) in terms of 𝑦(𝑡,𝑥), 𝑑𝑢𝑑𝑡^𝑥𝑡,𝑝𝑡,𝑥₀=𝑏𝑢²𝑡,𝑝𝑡,𝑥₀−12𝑒^{−𝑝(𝑡,𝑥0)𝑝(𝑡,𝑥0)}_−∞𝑒^𝜉𝑦_𝑡()1𝑡,𝜉𝑑𝜉+2𝑒^{𝑝(𝑡,𝑥0)∞}_{𝑝𝑡,𝑥0}𝑒^𝜉𝑦_𝑡(𝑡,𝜉)𝑑𝜉.(2.30)
Rewrite (1.1) as 𝑦_𝑡+𝑏𝑢𝑦_𝑥+2𝑏𝑢_𝑥𝑦+𝑎−2𝑏2𝑢²−𝑢²_𝑥𝑥=0.(2.31) Using this identity, we can obtain 𝑑𝑢𝑑𝑡^𝑥𝑡,𝑝𝑡,𝑥₀=𝑏𝑢²𝑡,𝑝𝑡,𝑥₀+𝑏2𝑒^{−𝑝(𝑡,𝑥0)𝑝(𝑡,𝑥0)}_−∞𝑒^𝜉𝑢𝑦_𝜉+2𝑢_𝜉𝑦𝑏𝑑𝜉−2𝑒^{𝑝(𝑡,𝑥0)∞}_{𝑝𝑡,𝑥0}𝑒^−𝜉𝑢𝑦_𝜉+2𝑢_𝜉𝑦+𝑑𝜉𝑎−2𝑏4𝑒^{−𝑝(𝑡,𝑥0)𝑝(𝑡,𝑥0)}_−∞𝑒^𝜉𝑢²−𝑢²_𝜉𝜉𝑑𝜉−𝑎−2𝑏4𝑒^{𝑝(𝑡,𝑥0)∞}_{𝑝𝑡,𝑥0}𝑒^−𝜉𝑢²−𝑢²_𝜉𝜉𝑑𝜉.(2.32)
By direct calculation, we have ^{𝑝(𝑡,𝑥0)}_−∞𝑒^𝜉𝑢𝑦_𝜉+2𝑦𝑢_𝜉()=𝑡,𝜉𝑑𝜉_{𝑝(𝑡,𝑥0)−∞}𝑒^𝜉𝑢(𝑡,𝜉)𝑦(𝑡,𝜉)_𝜉𝑑𝜉+^{𝑝(𝑡,𝑥0)}_−∞𝑒^𝜉𝑦(𝑡,𝜉)𝑢_𝜉(𝑡,𝜉)𝑑𝜉=−^{𝑝(𝑡,𝑥0)}_−∞𝑒^𝜉𝑢()𝑦()1𝑡,𝜉𝑡,𝜉𝑑𝜉+2^{𝑝(𝑡,𝑥0)}_−∞𝑒^𝜉𝑢²(𝑡,𝜉)−𝑢²_𝜉(𝑡,𝜉)_𝜉𝑑𝜉=−^{𝑝(𝑡,𝑥0)}_−∞𝑒^𝜉𝑢²()+1𝑡,𝜉2𝑢²_𝑥(𝑡,𝜉)𝑑𝜉+𝑒^𝜉𝑢(𝑡,𝜉)𝑢_𝑥()−1𝑡,𝜉2𝑢²_𝑥(𝑡,𝜉)_{𝜉=𝑝(𝑡,𝑥0)}.(2.33) Using the inequality ∫^𝑥_−∞𝑒^𝜉[𝑢²(𝑡,𝜉)+1/2𝑢²_𝑥(𝑡,𝜉)]𝑑𝜉≥𝑒^𝑥(𝑢²(𝑡,𝑥)/2) (see [26]), we obtain 𝑏2𝑒^{−𝑝(𝑡,𝑥0)𝑝(𝑡,𝑥0)}_−∞𝑒^𝜉𝑢𝑦_𝑥+2𝑦𝑢_𝑥()𝑏𝑡,𝜉𝑑𝜉≤−4𝑢²𝑡,𝑝𝑡,𝑥₀−𝑏4𝑢²_𝑥𝑡,𝑝𝑡,𝑥₀+𝑏2𝑢𝑡,𝑝𝑡,𝑥₀𝑢_𝑥𝑡,𝑝𝑡,𝑥₀.(2.34) Similarly, we have −𝑏2𝑒^{𝑝(𝑡,𝑥0)𝑝(𝑡,𝑥0)}_−∞𝑒^−𝜉𝑢𝑦_𝑥+2𝑦𝑢_𝑥()𝑏𝑡,𝜉𝑑𝜉≤−4𝑢²𝑡,𝑝𝑡,𝑥₀−𝑏4𝑢²_𝑥𝑡,𝑝𝑡,𝑥₀−𝑏2𝑢𝑡,𝑝𝑡,𝑥₀𝑢_𝑥𝑡,𝑝𝑡,𝑥₀.(2.35) Now using the inequality (see [26]) 𝑢²_𝑥(𝑡,𝑥)−𝑢²(𝑡,𝑥)≤𝑢²_𝑥−𝑢²𝑡,𝑝𝑡,𝑥₀,(2.36) we obtain 𝑎−2𝑏4𝑒^{−𝑝(𝑡,𝑥0)𝑝(𝑡,𝑥0)}_−∞𝑒^𝜉𝑢²−𝑢²_𝜉𝜉(𝑡,𝜉)𝑑𝜉≤0.(2.37) Similarly, we have 𝑎−2𝑏4𝑒^{𝑝(𝑡,𝑥0)∞}_{𝑝𝑡,𝑥0}𝑒^−𝜉𝑢²−𝑢²_𝑥(𝑡,𝜉)𝑑𝜉≥0.(2.38) Combining all the above terms together, we have 𝑑𝑢𝑑𝑡^𝑥𝑡,𝑝𝑡,𝑥₀≤𝑏2𝑢²𝑡,𝑝𝑡,𝑥₀−𝑏2𝑢²_𝑥𝑡,𝑝𝑡,𝑥₀.(2.39)Claim. 𝑢_𝑥(𝑡,𝑝(𝑡,𝑥₀))<0 is decreasing and 𝑢²(𝑡,𝑝(𝑡,𝑥₀))<𝑢²_𝑥(𝑡,𝑝(𝑡,𝑥₀)) for all 𝑡≥0.
Suppose not that there exists a 𝑡₀ such that 𝑢²(𝑡,𝑝(𝑡,𝑥₀))<𝑢²_𝑥(𝑡,𝑝(𝑡,𝑥₀)) on [0,𝑡₀) and 𝑢²(𝑡₀,𝑝(𝑡₀,𝑥₀))=𝑢²_𝑥(𝑡₀,𝑝(𝑡₀,𝑥₀)). Now, let 𝐼(𝑡)=12𝑒^{−𝑝(𝑡,𝑥0)𝑝(𝑡,𝑥0)}_−∞𝑒^𝜉𝑦()(𝑡)=1𝑡,𝜉𝑑𝜉>0,𝐼𝐼2𝑒^{𝑝(𝑡,𝑥0)∞}_{𝑝𝑡,𝑥0}𝑒^−𝜉𝑦(𝑡,𝜉)𝑑𝜉<0.(2.40) Firstly, by the same trick as above, we obtain 𝑑𝐼(𝑡)≥𝑏𝑑𝑡4𝑢²_𝑥−𝑢²𝑡,𝑝𝑡,𝑥₀(𝑡)>0,𝑑𝐼𝐼𝑏𝑑𝑡≤−4𝑢²_𝑥−𝑢²𝑡,𝑝𝑡,𝑥₀𝑢<0,_2𝑥−𝑢²𝑡,𝑝𝑡,𝑥₀=−4𝐼(𝑡)𝐼𝐼(𝑡)≥−4𝐼(0)𝐼𝐼(0)>0.(2.41) This implies that 𝑡₀ can be extended to infinity.
Moreover, due to the above inequality, we have 𝑑𝑢𝑑𝑡^2𝑥−𝑢²𝑡,𝑝𝑡,𝑥₀𝑑=4𝐼(𝑡)⋅((𝑡(𝑡)⋅𝑑𝑑𝑡−𝐼𝐼))+4𝐼((𝑡𝑢𝑑𝑡−𝐼𝐼))≥𝑏²_𝑥−𝑢²𝑡,𝑝𝑡,𝑥₀[𝐼(𝑡)−𝐼𝐼(𝑡)]=−𝑏𝑢_𝑥𝑡,𝑝𝑡,𝑥₀𝑢²_𝑥−𝑢²𝑡,𝑝𝑡,𝑥₀.(2.42) Now substituting (2.39) in (2.42), we get 𝑑𝑢𝑑𝑡^2𝑥−𝑢²𝑡,𝑝𝑡,𝑥₀≥𝑏²2𝑢²_𝑥−𝑢²𝑡,𝑝𝑡,𝑥₀^𝑡₀𝑢²_𝑥−𝑢²𝜏,𝑝𝜏,𝑥₀2𝑑𝜏−𝑏𝑢_0𝑥𝑥₀.(2.43) Now the theorem follows from Lemma 2.5 with Ψ(𝑡)=∫^𝑡₀(𝑢²_𝑥−𝑢²)(𝜏,𝑝(𝜏,𝑥₀))𝑑𝜏−(2/𝑏)𝑢_0𝑥(𝑥₀) and 𝐶₀=𝑏²/2. Then, we complete our proof.

The final goal for us is to establish a necessary and sufficient condition for guaranteeing singularity formation in finite time, which is in the opposite direction to global existence. So if the necessary and sufficient condition can be found, then the problem can be solved completely.

Now, let us try to find a condition for global existence. Unfortunately, when 𝑎≠2𝑏 like the Degasperis-Procesi equation [22] and the Holm-Staley 𝑏-family of equations [26], only the following easy form can be proved at present.

Theorem 2.9. Suppose that 𝑢₀∈𝐻³(ℝ), 𝑦₀(𝑥)=(1−𝜕²_𝑥)𝑢₀ does not change sign. Then the corresponding solution to (1.1) exists globally.

Proof. Without loss of generality we assume 𝑦₀(𝑥)≥0. It is sufficient to prove that 𝑢_𝑥(𝑡,𝑥) is bounded for all 𝑡.
In fact, 𝑢_𝑥()1𝑡,𝑥=−2𝑒^−𝑥𝑥_−∞𝑒^𝜉𝑦()1𝑡,𝜉𝑑𝜉+2𝑒^𝑥∞_𝑥𝑒^−𝜉𝑦()1𝑡,𝜉𝑑𝜉≥−2𝑒^−𝑥𝑥_−∞𝑒^𝜉𝑦()1𝑡,𝜉𝑑𝜉≥−2^𝑥_−∞𝑦()1𝑡,𝜉𝑑𝜉≥−2^+∞_−∞𝑦()1𝑡,𝜉𝑑𝜉=−2^+∞_−∞𝑦₀(𝜉)𝑑𝜉,(2.44) while we can also obtain 𝑢_𝑥()1𝑡,𝑥=−2𝑒^−𝑥𝑥_−∞𝑒^𝜉𝑦()1𝑡,𝜉𝑑𝜉+2𝑒^𝑥∞_𝑥𝑒^−𝜉𝑦()≤1𝑡,𝜉𝑑𝜉2𝑒^𝑥∞_𝑥𝑒^−𝜉𝑦()≤1𝑡,𝜉𝑑𝜉2^+∞_−∞𝑦₀(𝜉)𝑑𝜉.(2.45) Combining the above terms together, we have 𝑢_𝑥(≤1𝑡,𝑥)2^+∞_−∞𝑦₀(𝜉)𝑑𝜉.(2.46) Then, we complete our proof.

3. Propagation Speed

The purpose of this section is to give a detailed description of the corresponding strong solution 𝑢(𝑡,𝑥) in its lifespan with 𝑢₀ being compactly supported. The analogous results were obtained in [6] for the Camassa-Holm equation and in the papers [28, 29] for the Degasperis-Procesi equation. The main theorem reads as follows.

Theorem 3.1. Let 𝑎>0 and 3𝑏−𝑎>0. Assume that the initial datum 𝑢₀∈𝐻³(ℝ) is compactly supported in [𝑐,𝑑]. Then, the corresponding solution of (1.1) has the following property: for 0<𝑡<𝑇𝑢(𝑡,𝑥)=𝐿(𝑡)𝑒^−𝑥𝑎𝑠𝑥>𝑝(𝑡,𝑑),𝑢(𝑡,𝑥)=𝑙(𝑡)𝑒^𝑥𝑎𝑠𝑥<𝑝(𝑡,𝑐),(3.1) with 𝐿(𝑡)>0 and 𝑙(𝑡)<0, respectively, where 𝑝(𝑡,𝑥) is defined by (2.1) and 𝑇 is its lifespan. Furthermore, 𝐿(𝑡) and 𝑙(𝑡) denote continuous nonvanishing functions, with 𝐿(𝑡)>0 and 𝑙(𝑡)<0 for 𝑡∈(0,𝑇]. And 𝐿(𝑡) is a strictly increasing function, while 𝑙(𝑡) is a strictly decreasing function.

Proof. Since 𝑢₀(𝑥) has a compact support, 𝑦₀(𝑥)=(1−𝜕²_𝑥)𝑢₀(𝑥) also does. From (2.4), it follows that 𝑦(𝑡,𝑥)=(1−𝜕²_𝑥)𝑢(𝑡,𝑥) is compactly supported in [𝑝(𝑡,𝑐),𝑝(𝑡,𝑑)] in its lifespan. Hence the following functions are well defined: 𝐸(𝑡)=_ℝ𝑒^𝑥𝑦(𝑡,𝑥)𝑑𝑥,𝐹(𝑡)=_ℝ𝑒^−𝑥𝑦(𝑡,𝑥)𝑑𝑥(3.2) with 𝐸(0)=0=𝐹(0). For 𝑥>𝑝(𝑡,𝑑), we have 𝑢()=1𝑡,𝑥2𝑒^−|𝑥|()=1∗𝑦𝑡,𝑥2𝑒^−𝑥_𝑝()^{𝑝(𝑡,𝑑)}_𝑡,𝑐𝑒^−𝜉𝑦()1𝑡,𝜉𝑑𝜉=2𝑒^−𝑥𝐸(𝑡),(3.3) and, for 𝑥<𝑝(𝑡,𝑐), we have 𝑢()=1𝑡,𝑥2𝑒^−|𝑥|()=1∗𝑦𝑡,𝑥2𝑒^𝑥_𝑝()^{𝑝(𝑡,𝑑)}_𝑡,𝑐𝑒^−𝜉𝑦()1𝑡,𝜉𝑑𝜉=2𝑒^𝑥𝐹(𝑡).(3.4) Hence, as consequence of (3.3) and (3.4), we have 𝑢(𝑡,𝑥)=−𝑢_𝑥(𝑡,𝑥)=𝑢_𝑥𝑥()=1𝑡,𝑥2𝑒^−𝑥𝐸(𝑡)(),𝑢(),for𝑥>𝑝𝑡,𝑑𝑡,𝑥=𝑢_𝑥(𝑡,𝑥)=𝑢_𝑥𝑥()=1𝑡,𝑥2𝑒^𝑥𝐹(𝑡),for𝑥<𝑝(𝑡,𝑐).(3.5) On the other hand, 𝑑𝐸(𝑡)=𝑑𝑡_ℝ𝑒^𝑥𝑦_𝑡(𝑡,𝑥)𝑑𝑥.(3.6) Differentiating (1.1) twice, we get 0=𝑢_𝑥𝑥𝑡+𝑏𝑢𝑢_𝑥𝑥𝑥+Λ⁻²𝜕³_𝑥𝑎2𝑢²+3𝑏−𝑎2𝑢²_𝑥=𝑢_𝑥𝑥𝑡+𝑏𝑢𝑢_𝑥𝑥𝑥+Λ⁻²𝜕_𝑥𝑎2𝑢²+3𝑏−𝑎2𝑢²_𝑥−Λ⁻²1−𝜕²_𝑥𝜕_𝑥𝑎2𝑢²+3𝑏−𝑎2𝑢²_𝑥=𝑢_𝑥𝑥𝑡+𝑏𝑢𝑢_𝑥𝑥𝑥+Λ⁻²𝜕_𝑥𝑎2𝑢²+3𝑏−𝑎2𝑢²_𝑥−𝜕_𝑥𝑎2𝑢²+3𝑏−𝑎2𝑢²_𝑥.(3.7) Combining (1.1) and (3.7), we obtain 𝑦_𝑡=−𝑏𝑢𝑢_𝑥+𝑏𝑢𝑢_𝑥𝑥𝑥−𝜕_𝑥𝑎2𝑢²+3𝑏−𝑎2𝑢²_𝑥.(3.8)
Substituting the identity (3.8) into 𝑑𝐸(𝑡)/𝑑𝑡 and using (3.5), we obtain 𝑑𝐸(𝑡)𝑑𝑡=−𝑏_ℝ𝑒^𝑥𝑢𝑢_𝑥𝑑𝑥+𝑏_ℝ𝑒^𝑥𝑢𝑢_𝑥𝑥𝑥𝑑𝑥−_ℝ𝑒^𝑥𝜕_𝑥𝑎2𝑢²+3𝑏−𝑎2𝑢²_𝑥=𝑑𝑥_ℝ𝑒^𝑥𝑎2𝑢²+3𝑏−𝑎2𝑢²_𝑥𝑑𝑥.(3.9) Therefore, in the lifespan of the solution, we have 𝐸(𝑡)=^𝑡_0ℝ𝑒^𝑥𝑎2𝑢²+3𝑏−𝑎2𝑢²_𝑥𝑑𝑥>0.(3.10) By the same argument, one can check that the following identity for 𝐹(𝑡) is true: 𝐹(𝑡)=−^𝑡_0ℝ𝑒^−𝑥𝑎2𝑢²+3𝑏−𝑎2𝑢²_𝑥𝑑𝑥<0.(3.11) In order to complete the proof, it is sufficient to let 𝐿(𝑡)=(1/2)𝐸(𝑡) and 𝑙(𝑡)=(1/2)𝐹(𝑡), respectively.

Acknowledgments

This work is partially supported by Zhejiang Innovation Project (Grant no. T200905) and NSFC (Grant no. 11101376). S. Hakkaev is also partially supported by Research Grant DVU_10_0338.