On the Cauchy Problem for a Class of Weakly Dissipative One-Dimensional Shallow Water Equations

Abstract

We investigate a more general family of one-dimensional shallow water equations with a weakly dissipative term. First, we establish blow-up criteria for this family of equations. Then, global existence of the solution is also proved. Finally, we discuss the infinite propagation speed of this family of equations.

1. Introduction

Recently, in [1], the following one-dimensional shallow water equations were studied: 𝑦_𝑡+𝑎𝑢_𝑥𝑦+𝑏𝑢𝑦_𝑥=0,𝑡>0,𝑥∈ℝ, (1) where 𝑢(𝑥,𝑡)∈ℝ and 𝑦(𝑥,𝑡)=(1−𝜕²_𝑥)𝑢(𝑥,𝑡). A detailed description of the corresponding strong solution with the initial data 𝑢₀ was also given by them in [1].

When 𝑎=𝜃−1,  𝑏=𝜃, and 𝜆=0, (1) reduces to 𝜃-equation which is studied by Ni and Zhou in [2].

When 𝑎=2, 𝑏=1, and 𝜆=0, (1) reduces to the Camassa-Holm equation, which was derived physically by Camassa and Holm in [3] (found earlier by Fuchssteiner and Fokas [4] as a bi-Hamiltonian generalization of the KdV equation) by approximating directly the Hamiltonian for Euler’s equations in the shallow water region with 𝑢(𝑥,𝑡) representing the free surface above a flat bottom. The Camassa-Holm equation is completely integrable and has infinite conservation laws. Local well-posedness for the initial datum 𝑢₀(𝑥)∈𝐻^𝑠 with 𝑠>3/2 was proved in [5, 6]. One of the remarkable features of Camassa-Holm equation is the presence of breaking waves and global solutions. Necessary and sufficient condition for wave breaking was established by Mckean [7] in 1998. A new and direct proof was also given in [8]. The solitary waves of Camassa-Holm equation are peaked solitons. The orbital stability of the peakons was shown by Constantin and Strauss in [9] (see also [10]). The property of propagation speed of solutions to the Camassa-Holm equation, which was presented by Himonas and his collaborators in their work is worthy of being mentioned here [11].

The Degasperis-Procesi equation [12] and b-family equation [13] are the special cases with 𝑎=3, 𝑏=1, and 𝑏=1, respectively. There have been extensive studies on the two equations, (cf. [14, 15]).

In this paper, we consider the following weakly dissipative one-dimensional shallow water equation: 𝑦_𝑡+𝑎𝑢_𝑥𝑦+𝑏𝑢𝑦_𝑥+𝜆𝑦=0,𝑡>0,𝑥∈ℝ, (2) where 𝜆𝑦=𝜆(𝑢−𝑢_𝑥𝑥) is the weakly dissipative term.

It is worth pointing out that many works have been done for related equations which have a weakly dissipative term (cf. [16–19]).

The paper is organized as follows. In Section 2, we establish the local well-posedness of the initial-value problem associated with (2) and present the precise blow-up scenario. Some blow-up results are given in Section 3. In Section 4, we establish a sufficient condition added on the initial data to guarantee global existence. We will consider the infinite propagation speed in Section 5.

2. Local Well-Posedness and Blow-Up Scenario

In this section, we first establish the local well-posedness of (2) by using Kato’s theory. Then, we provide the precise blow-up scenario for solutions to (2).

System (2) is equivalent to the following system: 𝑢_𝑡+𝑏𝑢𝑢_𝑥+𝜕_𝑥𝑎𝐺∗2𝑢²+3𝑏−𝑎2𝑢²_𝑥+𝜆𝑢=0, (3) where 𝐺(𝑥)=(1/2)𝑒^−|𝑥|, ∗ means doing convolution.

Theorem 1. Given 𝑢₀∈𝐻^𝑠(ℝ), 𝑠>3/2, then there exist a 𝑇 and a unique solution 𝑢 to (2) such that 𝑢(𝑥,𝑡)∈𝐶[0,𝑇);𝐻^𝑠(ℝ)∩𝐶¹[0,𝑇);𝐻^𝑠−1(ℝ). (4)

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 [5].

3. Blow-Up Phenomenon

In this section, we will give some conditions to guarantee the finite time blowup. Motivated by Mckean’s deep observation for the Camassa-Holm equation [7], we can consider the similar particle trajectory as 𝑞_𝑡𝑞()=𝑏𝑢𝑞,𝑡,0<𝑡<𝑇,𝑥∈ℝ,𝑥,0=𝑥,𝑥∈ℝ, (5) where 𝑇 is the lifespan of the solution: then 𝑞 is a diffeomorphism of the line. Taking derivative (5) with respect to 𝑥, we obtain 𝑑𝑞_𝑡𝑑𝑥=𝑞_𝑡𝑥=𝑏𝑢_𝑥𝑞,𝑡𝑞_𝑥,𝑡∈(0,𝑇). (6) Therefore 𝑞_𝑥(𝑥,𝑡)=exp𝑏^𝑡₀𝑢_𝑥𝑞,𝑠𝑑𝑠,𝑞_𝑥(𝑥,0)=1. (7) Hence, from (2), the following identity can be proved: 𝑦𝑞𝑞_𝑥^𝑎/𝑏=𝑦₀(𝑥)𝑒^−𝜆𝑡. (8) In fact, direct calculation yields 𝑑𝑦𝑞𝑞𝑑𝑡^{𝑥𝑎/𝑏}=𝑦_𝑡𝑞+𝑏𝑢𝑞,𝑡𝑦_𝑥𝑞+𝑎𝑢_𝑥𝑞,𝑡𝑦𝑞𝑞_𝑥^𝑎/𝑏=−𝜆𝑦𝑞_𝑥^𝑎/𝑏. (9)

Motivated by [19], we give the following theorem.

Theorem 2. Let 𝑎−2𝑏>0, 𝑏>0: suppose that 𝑢₀∈𝐻²(ℝ), and there exists a 𝑥₀∈ℝ such that 𝑦₀(𝑥₀)=(1−𝜕²_𝑥)𝑢₀(𝑥₀)=0, 𝑦₀≥0(≢0),𝑓𝑜𝑟𝑥∈−∞,𝑥₀,𝑦₀≤0(≢0),𝑓𝑜𝑟𝑥∈𝑥₀,𝑒,∞_{−𝑥0𝑥0−∞}𝑒^𝜉𝑦₀(𝜉)𝑑𝜉>2𝜆𝑏,𝑒^𝑥0∞_𝑥0𝑒^−𝜉𝑦₀(𝜉)𝑑𝜉<−2𝜆𝑏. (10)(11) Then the corresponding solution 𝑢(𝑥,𝑡) to (2) with 𝑢₀ as the initial datum blows up in finite time.

Proof. Suppose that the solution exists globally. From (8) and initial condition (10), we have 𝑦(𝑞(𝑥₀,𝑡),𝑡)=0 and 𝑦𝑞(𝑥,𝑡),𝑡≥0(≢0),for𝑥∈−∞,𝑞𝑥₀,𝑦𝑞()()𝑞𝑥,𝑡𝑥,𝑡,𝑡≤0≢0,for𝑥∈₀,𝑡,∞, (12) for all 𝑡≥0. Due to 𝑢(𝑥,𝑡)=𝐺∗𝑦(𝑥,𝑡), we can write 𝑢(𝑥,𝑡) and 𝑢_𝑥(𝑥,𝑡) as 𝑢()=1𝑥,𝑡2𝑒^−𝑥𝑥_−∞𝑒^𝜉𝑦()1𝜉,𝑡𝑑𝜉+2𝑒^𝑥∞_𝑥𝑒^−𝜉𝑦()𝑢𝜉,𝑡𝑑𝜉,_𝑥()1𝑥,𝑡=−2𝑒^−𝑥𝑥_−∞𝑒^𝜉𝑦()1𝜉,𝑡𝑑𝜉+2𝑒^𝑥∞_𝑥𝑒^−𝜉𝑦(𝜉,𝑡)𝑑𝜉. (13) Therefore, 𝑢²_𝑥(𝑥,𝑡)−𝑢²(𝑥,𝑡)=−^𝑥_−∞𝑒^𝜉𝑦(𝜉,𝑡)𝑑𝜉^∞_𝑥𝑒^−𝜉𝑦(𝜉,𝑡)𝑑𝜉, (14) for all 𝑡>0.
By direct calculation, for 𝑥≤𝑞(𝑥₀,𝑡), we have 𝑢²_𝑥(𝑥,𝑡)−𝑢²(𝑥,𝑡)=−^𝑥_−∞𝑒^𝜉𝑦(𝜉,𝑡)𝑑𝜉^∞_𝑥𝑒^−𝜉𝑦(𝜉,𝑡)𝑑𝜉=−^{𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉𝑦(𝜉,𝑡)𝑑𝜉−^𝑞(𝑥0_𝑥^,𝑡)𝑒^𝜉𝑦()×𝜉,𝑡𝑑𝜉_{∞𝑞𝑥0,𝑡}𝑒^−𝜉𝑦(𝜉,𝑡)𝑑𝜉+^𝑞(𝑥0_𝑥^,𝑡)𝑒^−𝜉𝑦(𝜉,𝑡)𝑑𝜉=𝑢²_𝑥𝑞𝑥₀,𝑡,𝑡−𝑢²𝑞𝑥₀−,𝑡,𝑡_𝑥−∞𝑒^𝜉𝑦(𝜉,𝑡)𝑑𝜉^𝑞(𝑥0_𝑥^,𝑡)𝑒^−𝜉𝑦()+𝜉,𝑡𝑑𝜉_{𝑞(𝑥0𝑥,𝑡)}𝑒^𝜉𝑦(𝜉,𝑡)𝑑𝜉^∞_{𝑞𝑥0,𝑡}𝑒^−𝜉𝑦(𝜉,𝑡)𝑑𝜉≤𝑢²_𝑥𝑞𝑥₀,𝑡,𝑡−𝑢²𝑞𝑥₀,𝑡,𝑡. (15) Similarly, for 𝑥≥𝑞(𝑥₀,𝑡), we have 𝑢²_𝑥(𝑥,𝑡)−𝑢²(𝑥,𝑡)≤𝑢²_𝑥𝑞𝑥₀,𝑡,𝑡−𝑢²𝑞𝑥₀,𝑡,𝑡. (16) So for any fixed 𝑡, combination of (15) and (16), we obtain 𝑢²_𝑥(𝑥,𝑡)−𝑢²(𝑥,𝑡)≤𝑢²_𝑥𝑞𝑥₀,𝑡,𝑡−𝑢²𝑞𝑥₀,𝑡,𝑡, (17) for all 𝑥∈ℝ.
From the expression of 𝑢_𝑥(𝑥,𝑡) in terms of 𝑦(𝑥,𝑡), differentiating 𝑢_𝑥(𝑞(𝑥₀,𝑡),𝑡) with respect to 𝑡, we have 𝜕_𝑡𝑢_𝑥𝑞𝑥₀,𝑡,𝑡=𝑢_𝑥𝑡𝑞𝑥₀,𝑡,𝑡+𝑢_𝑥𝑥𝑞𝑥₀,𝑡,𝑡𝑞_𝑡𝑞𝑥₀=𝑎,𝑡,𝑡2𝑢²𝑞𝑥₀,𝑡,𝑡+𝑎−𝑏2𝑢²_𝑥𝑞𝑥₀,𝑡,𝑡−𝜆𝑢_𝑥𝑞𝑥₀𝑎,𝑡,𝑡−𝐺∗2𝑢²(𝑥,𝑡)+3𝑏−𝑎2𝑢²_𝑥()𝑎𝑥,𝑡=𝐺∗2𝑢²𝑞𝑥₀,𝑡,𝑡+𝑎−𝑏2𝑢²_𝑥𝑞𝑥₀−𝑎,𝑡,𝑡2𝑢²(𝑥,𝑡)−3𝑏−𝑎2𝑢²_𝑥(𝑥,𝑡)−𝜆𝑢_𝑥𝑞𝑥₀,𝑡,𝑡=𝐺∗𝑎−2𝑏2𝑢²𝑞𝑥₀,𝑡,𝑡−𝑢²_𝑥𝑞𝑥₀,𝑡,𝑡−𝑢²(𝑥,𝑡)+𝑢²_𝑥(𝑥,𝑡)+𝐺∗𝑏𝑢²𝑞𝑥₀−𝑏,𝑡,𝑡2𝑢²_𝑥𝑞𝑥₀,𝑡,𝑡−𝑏𝑢²()−𝑏𝑥,𝑡2𝑢²_𝑥(𝑥,𝑡)−𝜆𝑢_𝑥𝑞𝑥₀≤𝑏,𝑡,𝑡2𝑢²𝑞𝑥₀−𝑏,𝑡,𝑡2𝑢²_𝑥𝑞𝑥₀,𝑡,𝑡−𝜆𝑢_𝑥𝑞𝑥₀,𝑡,𝑡, (18) where we have used (17), and the inequality 𝐺∗(𝑢²(𝑥,𝑡)+(1/2)𝑢²_𝑥(𝑥,𝑡))≥(1/2)𝑢². In addition, we also used the equation 𝑢_𝑡𝑥+𝑢𝑢_𝑥𝑥−(𝑎/2)𝑢²−((𝑏−𝑎)/2)𝑢²_𝑥+𝐺∗((𝑎/2)𝑢²+((3𝑏−𝑎)/2)𝑢²_𝑥)+𝜆𝑢_𝑥=0, which is obtained by differentiating equation (3).
For (11), we know that 𝑢_0𝑥𝑥₀+𝜆𝑏²−𝑢₀𝑥₀+𝜆𝑏²=−𝑒^{−𝑥0𝑥0}_−∞𝑒^𝜉𝑦₀(𝜉)𝑑𝜉×𝑒^𝑥0∞_𝑥0𝑒^−𝜉𝑦₀(𝜉)𝑑𝜉+2𝜆𝑏𝑢>0,_0𝑥𝑥₀+𝜆𝑏²−𝑢₀𝑥₀−𝜆𝑏²=−𝑒^𝑥0∞_𝑥0𝑒^−𝜉𝑦₀(𝜉)𝑑𝜉×𝑒^{−𝑥0𝑥0}_−∞𝑒^𝜉𝑦₀(𝜉)𝑑𝜉−2𝜆𝑏>0. (19)
Claim. 𝑢_𝑥(𝑞(𝑥₀,𝑡),𝑡)<0 is decreasing. (𝑢(𝑞(𝑥₀,𝑡),𝑡)+𝜆/𝑏)²<(𝑢_𝑥(𝑞(𝑥₀,𝑡),𝑡)+𝜆/𝑏)² and (𝑢(𝑞(𝑥₀,𝑡),𝑡)−𝜆/𝑏)²<(𝑢_𝑥(𝑞(𝑥₀,𝑡),𝑡)+𝜆/𝑏)², for all 𝑡≥0.
Suppose that there exists a 𝑡₀ such that (𝑢(𝑞(𝑥₀,𝑡),𝑡)+𝜆/𝑏)²<(𝑢_𝑥(𝑞(𝑥₀,𝑡),𝑡)+𝜆/𝑏)² and (𝑢(𝑞(𝑥₀,𝑡),𝑡)−𝜆/𝑏)²<(𝑢_𝑥(𝑞(𝑥₀,𝑡),𝑡)+𝜆/𝑏)² on [0,𝑡₀); then (𝑢(𝑞(𝑥₀,𝑡₀),𝑡₀)+𝜆/𝑏)²=(𝑢_𝑥(𝑞(𝑥₀,𝑡₀),𝑡₀)+𝜆/𝑏)² or (𝑢(𝑞(𝑥₀,𝑡₀),𝑡₀)−𝜆/𝑏)²=(𝑢_𝑥(𝑞(𝑥₀,𝑡₀),𝑡₀)+𝜆/𝑏)².
Now, let 𝐼(𝑡)1∶=2𝑒^{−𝑞(𝑥0,𝑡)𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉𝑦()(𝑡)1𝜉,𝑡𝑑𝜉,𝐼𝐼∶=2𝑒^{𝑞(𝑥0,𝑡)∞}_{𝑞𝑥0,𝑡}𝑒^−𝜉𝑦(𝜉,𝑡)𝑑𝜉. (20) Firstly, differentiating 𝐼(𝑡), we have 𝑑𝐼(𝑡)𝑏𝑑𝑡=−2𝑢𝑞𝑥₀,𝑡,𝑡𝑒^{−𝑞(𝑥0,𝑡)𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉𝑦()+1𝜉,𝑡𝑑𝜉2𝑒^{−𝑞(𝑥0,𝑡)𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉𝑦_𝑡()=𝑏𝜉,𝑡𝑑𝜉2𝑢𝑢_𝑥−𝑢𝑞𝑥₀−1,𝑡,𝑡2𝑒^{−𝑞(𝑥0,𝑡)}×^{𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉𝑏𝑢𝑦_𝑥+2𝑏𝑢_𝑥𝑦+𝑎−2𝑏2𝑢²−𝑢²_𝑥𝑥≥𝑏+𝜆𝑦𝑑𝜉2𝑢𝑢_𝑥−𝑢𝑞𝑥₀+𝑏,𝑡,𝑡4𝑢²+𝑢²_𝑥−2𝑢𝑢_𝑥×𝑞𝑥₀−𝜆,𝑡,𝑡2𝑢−𝑢_𝑥𝑞𝑥₀=𝑏,𝑡,𝑡4𝑢²_𝑥−𝑢²𝑞𝑥₀−𝜆,𝑡,𝑡2𝑢−𝑢_𝑥𝑞𝑥₀=𝑏,𝑡,𝑡4𝑢_𝑥𝑞𝑥₀+𝜆,𝑡,𝑡𝑏²−𝑏4𝑢𝑞𝑥₀+𝜆,𝑡,𝑡𝑏²>0,on0,𝑡₀. (21) Secondly, by the same argument, we obtain 𝑑𝐼𝐼(𝑡)=𝑏𝑑𝑡2𝑢𝑞𝑥₀,𝑡,𝑡𝑒^{𝑞(𝑥0,𝑡)∞}_{𝑞𝑥0,𝑡}𝑒^−𝜉𝑦()+1𝜉,𝑡𝑑𝜉2𝑒^{𝑞(𝑥0,𝑡)∞}_{𝑞𝑥0,𝑡}𝑒^−𝜉𝑦_𝑡()=𝑏𝜉,𝑡𝑑𝜉2𝑢𝑢_𝑥+𝑢𝑞𝑥₀−1,𝑡,𝑡2𝑒^{𝑞(𝑥0,𝑡)}×^∞_{𝑞𝑥0,𝑡}𝑒^−𝜉𝑏𝑢𝑦_𝑥+2𝑏𝑢_𝑥𝑦+𝑎−2𝑏2𝑢²−𝑢²_𝑥𝑥≤𝑏+𝜆𝑦𝑑𝜉2𝑢𝑢_𝑥+𝑢𝑞𝑥₀−𝑏,𝑡,𝑡4𝑢²+𝑢²_𝑥+2𝑢𝑢_𝑥𝑞𝑥₀−𝜆,𝑡,𝑡2𝑢_𝑥+𝑢𝑞𝑥₀𝑏,𝑡,𝑡=−4𝑢²_𝑥−𝑢²𝑞𝑥₀−𝜆,𝑡,𝑡2𝑢_𝑥+𝑢𝑞𝑥₀𝑏,𝑡,𝑡=−4𝑢_𝑥𝑞𝑥₀+𝜆,𝑡,𝑡𝑏²+𝑏4𝑢𝑞𝑥₀−𝜆,𝑡,𝑡𝑏²<0,on0,𝑡₀. (22) Therefore, it follows from (21), (22), and the continuity property of ODEs that 𝑢_𝑥𝑞𝑥₀+𝜆,𝑡,𝑡𝑏²−𝑢𝑞𝑥₀+𝜆,𝑡,𝑡𝑏²(𝑡(𝑡)+𝜆=−4𝐼)𝐼𝐼𝑏(0(0)+𝜆>−4𝐼)𝐼𝐼𝑏𝑢>0,_𝑥𝑞𝑥₀+𝜆,𝑡,𝑡𝑏²−𝑢𝑞𝑥₀−𝜆,𝑡,𝑡𝑏²𝐼(𝑡)−𝜆=−4𝑏(𝑡)𝐼(0)−𝜆𝐼𝐼>−4𝑏𝐼𝐼(0)>0, (23) for all 𝑡>0. This implies that 𝑡₀ can be extended to the infinity.
Moreover, using (21) and (22) again, we have the following equation for [2(𝑢_𝑥+𝜆/𝑏)²−(𝑢+𝜆/𝑏)²−(𝑢−𝜆/𝑏)²](𝑞(𝑥₀,𝑡),𝑡): 𝑑2𝑢𝑑𝑡^𝑥+𝜆𝑏²−𝜆𝑢+𝑏²−𝜆𝑢−𝑏²𝑞𝑥₀𝑑,𝑡,𝑡=−4𝐼(𝑡)(𝑡)+𝜆𝑑𝑡𝐼𝐼𝑏𝑑−4𝐼(𝑡)−𝜆𝑑𝑡𝑏(𝑡𝑢𝐼𝐼)≥−𝑏_𝑥+𝜆𝑏²−𝜆𝑢+𝑏²𝑞𝑥₀(𝑡)+𝜆,𝑡,𝑡𝐼𝐼𝑏+𝑏𝑢_𝑥+𝜆𝑏²−𝜆𝑢−𝑏²𝑞𝑥₀𝐼(𝑡)𝑢,𝑡,𝑡−𝑏_𝑥+𝜆𝑏²−𝜆𝑢+𝑏²𝑞𝑥₀(𝑡)𝑢,𝑡,𝑡𝐼𝐼+𝑏_𝑥+𝜆𝑏²−𝜆𝑢−𝑏²𝑞𝑥₀𝐼(𝑡)−𝜆,𝑡,𝑡𝑏−𝜆=𝑏𝑏2𝑢_𝑥+𝜆𝑏²−𝜆𝑢+𝑏²−𝜆𝑢−𝑏²×𝑞𝑥₀,𝑡,𝑡−𝑢_𝑥𝑞𝑥₀×2𝑢,𝑡,𝑡_𝑥𝑞𝑥₀+𝜆,𝑡,𝑡𝑏²𝜆+2𝑢+𝑏²(𝑡)𝜆𝐼𝐼−2𝑢−𝑏²𝐼(𝑡)=𝑏−𝑢_𝑥𝑞𝑥₀+𝜆,𝑡,𝑡𝑏×2𝑢_𝑥+𝜆𝑏²−𝜆𝑢+𝑏²−𝜆𝑢−𝑏²×𝑞𝑥₀,𝑡,𝑡−𝑢_𝑥𝑞𝑥₀×𝜆,𝑡,𝑡𝑢+𝑏²+𝜆𝑢−𝑏²𝑞𝑥₀𝜆,𝑡,𝑡+2𝑢+𝑏²(𝑡)𝜆𝐼𝐼−2𝑢−𝑏²𝐼(𝑡)≥−𝑏𝑢_𝑥𝑞𝑥₀+𝜆,𝑡,𝑡𝑏×2𝑢_𝑥+𝜆𝑏²−𝜆𝑢+𝑏²−𝜆𝑢−𝑏²𝑞𝑥₀,𝑡,𝑡, (24) where we use 𝑢_𝑥(𝑞(𝑥₀,𝑡),𝑡)=−𝐼(𝑡)+𝐼𝐼(𝑡).
Now, recalling (18), we have 𝜕_𝑡𝑢_𝑥𝑞𝑥₀,𝑡,𝑡𝑞𝑥₀≤𝑏,𝑡,𝑡2𝑢²𝑞𝑥₀−𝑏,𝑡,𝑡2𝑢²_𝑥𝑞𝑥₀,𝑡,𝑡−𝜆𝑢_𝑥=𝑏4𝜆𝑢+𝑏²+𝜆𝑢−𝑏²−2𝑢_𝑥+𝜆𝑏²×𝑞𝑥₀,𝑡,𝑡. (25)
Putting (25) into (24), it yields 𝑑2𝑢𝑑𝑡^𝑥+𝜆𝑏²−𝜆𝑢+𝑏²−𝜆𝑢−𝑏²𝑞𝑥₀≥𝑏,𝑡,𝑡₂42𝑢_𝑥+𝜆𝑏²−𝜆𝑢+𝑏²−𝜆𝑢−𝑏²×𝑞𝑥₀×,𝑡,𝑡_𝑡02𝑢_𝑥+𝜆𝑏²−𝜆𝑢+𝑏²−𝜆𝑢−𝑏²×𝑞𝑥₀,𝜏,𝜏𝑑𝜏−4𝑢_0𝑥𝑥₀−4𝜆𝑏. (26) Before finishing the proof, we need the following technical lemma.
Lemma  3 (see [15]). Suppose that Ψ(𝑡) is twice continuously differential satisfyingΨ^′′(𝑡)≥𝐶₀Ψ^′(𝑡)Ψ(𝑡),𝑡>0,𝐶₀Ψ(𝑡)>0,>0,Ψ^′(𝑡)>0. (27)Then 𝜓(𝑡) blows up in finite time. Moreover the blow-up time can be estimated in terms of the initial datum as2𝑇≤max𝐶₀Ψ(0),Ψ(0)Ψ^′(0). (28)
Let Ψ(𝑡)=∫^𝑡₀[2(𝑢_𝑥+𝜆/𝑏)²−(𝑢+𝜆/𝑏)²−(𝑢−𝜆/𝑏)²](𝑞(𝑥₀,𝜏),𝜏)𝑑𝜏−4𝑢_0𝑥(𝑥₀)−4𝜆/𝑏; then (26) is an equation of type (27) with 𝐶₀=𝑏²/4. The proof is complete by applying Lemma 3.

Remark 4. When 𝑏=1, Theorem 2 reduces to the result in [19].

Theorem 5. Let 𝑎=2𝑏>0. Suppose that 𝑢₀∈𝐻²(ℝ) and there exists a 𝑥₀∈ℝ such that 𝑦₀(𝑥₀)=(1−𝜕²_𝑥)𝑢₀(𝑥₀)=0, 𝑒^{−𝑥0𝑥0}_−∞𝑒^𝜉𝑦₀(𝜉)𝑑𝜉>2𝜆𝑏,𝑒^𝑥0∞_𝑥0𝑒^−𝜉𝑦₀(𝜉)𝑑𝜉<−2𝜆𝑏. (29) Then the corresponding solution 𝑢(𝑥,𝑡) to (2) with 𝑢₀ as the initial datum blows up in finite time.

Proof. We easily obtain 𝑢_𝑡𝑥+𝑏𝑢𝑢_𝑥𝑥−𝑎2𝑢²+𝑏2𝑢²_𝑥𝑎+𝐺∗2𝑢²+𝑏2𝑢²_𝑥+𝜆𝑢_𝑥=0. (30) Differentiating 𝑢_𝑥 at the point (𝑞(𝑥₀,𝑡),𝑡) with respect to 𝑡, we get 𝑑𝑢𝑑𝑡^𝑥𝑞𝑥₀≤𝑏,𝑡,𝑡2𝑢²𝑞𝑥₀−𝑏,𝑡,𝑡2𝑢²_𝑥𝑞𝑥₀,𝑡,𝑡−𝜆𝑢_𝑥𝑞𝑥₀,𝑡,𝑡. (31)
Process of the proof is similar to Theorem 2. Thus to be concise, we omit the detailed proof.

When 𝑎=2𝑏>0,𝜆=0, using 𝑢(𝑥,𝑡)=𝑏𝑢(𝑥,𝑡), (2) can be reformulated into 𝑢_𝑥+𝑢_𝑥𝑥𝑡+3𝑢𝑢_𝑥−2𝑢_𝑥𝑢_𝑥𝑥−𝑢𝑢_𝑥𝑥𝑥=0, (32) which is the well-known Camassa-Holm equation. Meanwhile, we also find that the condition in Theorem 5 can be reformulated into ^𝑥0_−∞𝑒^𝜉𝑦₀(𝜉)𝑑𝜉>0,^∞_𝑥0𝑒^−𝜉𝑦₀(𝜉)𝑑𝜉<0, (33) which is one of the sufficient conditions to guarantee blow-up add-on initial data for the Camassa-Holm equation.

So, we show the necessary and sufficient condition for the special case 𝑎=2𝑏>0 and 𝜆=0 in the following theorem.

Theorem 6. When 𝑎=2𝑏>0 and 𝜆=0, then the nonlinear wave equation (2) breaks if and only if some portion of the positive part of 𝑦₀(𝑥) lies to the left of some portion of its negative part.

Proof. As studied in [1], when 𝑎=2𝑏>0 and 𝜆=0, rewriting (2) yields 𝑢_𝑥+𝑢_𝑥𝑥𝑡+3𝑏𝑢𝑢_𝑥−2𝑏𝑢_𝑥𝑢_𝑥𝑥−𝑏𝑢𝑢_𝑥𝑥𝑥=0. (34)
Recalling Mckean’s theorem in [7], (32) breaks if and only if some portion of the positive part of 𝑦₀(𝑥,𝑡)=(1−𝜕²_𝑥)𝑢₀ lies to the left of some portion of its negative part.
So (34) breaks if and only if some portion of the positive part of 𝑏𝑦₀(𝑥,𝑡)=𝑏(1−𝜕²_𝑥)𝑢₀ lies to the left of some portion of its negative part.
This completes the proof.

Remark 7. Mckean’s theorem [7] is for the special case 𝑎=2, 𝑏=1. Condition 𝑎=2𝑏 here is more general. However, the necessary and sufficient condition for (2) is still a challenging problem for us at present.

4. Global Existence

Now, let us try to find a condition for global existence. Unfortunately, When 𝑎≠2𝑏, like the Degasperis-Procesi equation [12], only the following easy one can be proved at present.

Theorem 8. Suppose that 𝑢₀∈𝐻³(ℝ), and 𝑦₀=(1−𝜕²_𝑥)𝑢₀ is one sign. Then the corresponding solution to (2) exists globally.

Proof. We can assume that 𝑦₀≥0. It is sufficient to prove that 𝑢_𝑥(𝑥,𝑡) has a lower and upper bound for all 𝑡. In fact, 𝑢_𝑥()1𝑥,𝑡=−2𝑒^−𝑥𝑥_−∞𝑒^𝜉𝑦()1𝜉,𝑡𝑑𝜉+2𝑒^𝑥∞_𝑥𝑒^−𝜉𝑦(𝜉,𝑡)𝑑𝜉. (35) Therefore, we have 𝑢_𝑥()1𝑥,𝑡≥−2𝑒^−𝑥𝑥_−∞𝑒^𝜉𝑦()1𝜉,𝑡𝑑𝜉≥−2^𝑥_−∞𝑦()1𝜉,𝑡𝑑𝜉≥−2^∞_−∞𝑦()1𝜉,𝑡𝑑𝜉=−2^∞_−∞𝑦₀()𝑢𝜉,𝑡𝑑𝜉,_𝑥()≤1𝑥,𝑡2𝑒^𝑥∞_𝑥𝑒^−𝜉𝑦()1𝜉,𝑡𝑑𝜉≤2^∞_𝑥𝑦()≤1𝜉,𝑡𝑑𝜉2^∞_−∞𝑦()1𝜉,𝑡𝑑𝜉=2^∞_−∞𝑦₀(𝜉,𝑡)𝑑𝜉. (36) This completes the proof.

5. Infinite Propagation Speed

In this section, we will give a more detailed description on the corresponding strong solution 𝑢(𝑥,𝑡) to (2) in its life span with initial data 𝑢₀(𝑥) being compactly supported. The main theorem reads as follows.

Theorem 9. Let 0<𝑎≤3𝑏. Assume that for some 𝑇≥0 and 𝑠≥5/2, 𝑢∈𝐶([0,𝑇);𝐻^𝑠(ℝ)) is a strong solution of (2). If 𝑢₀(𝑥)=𝑢(𝑥,0) has compact support [a,c], then for 𝑡∈(0,𝑇), one has 𝑢()=𝐿(𝑡)𝑒𝑥,𝑡^−𝑥(),𝑙(𝑡)𝑒,𝑓𝑜𝑟𝑥>𝑞𝑐,𝑡_−𝑥,𝑓𝑜𝑟𝑥<𝑞(𝑎,𝑡), (37) where 𝐿(𝑡) and 𝑙(𝑡) denote continuous nonvanishing functions, with 𝐿(𝑡)>0 and 𝑙(𝑡)<0 for 𝑡∈(0,𝑇). Furthermore, 𝐿(𝑡) is strictly increasing function, while 𝑙(𝑡) is strictly decreasing function.

Proof. Since 𝑢₀ has compact support in 𝑥 in [𝑎,𝑐], from (8), so does 𝑦(,𝑡) has compact support in 𝑥 in [𝑞(𝑎,𝑡),𝑞(𝑐,𝑡)] in its lifespan. Hence the following functions are well-defined:𝐸(𝑡)=_ℝ𝑒^𝑥𝑦(𝑥,𝑡)𝑑𝑥,𝐹(𝑡)=_ℝ𝑒^−𝑥𝑦(𝑥,𝑡)𝑑𝑥, (38) with 𝐸₀=_ℝ𝑒^𝑥𝑦₀(𝑥)𝑑𝑥=0,𝐹₀=_ℝ𝑒^−𝑥𝑦₀(𝑥)𝑑𝑥=0. (39) Thus, for 𝑥>𝑞(𝑐,𝑡), we obtain 𝑢()=1𝑥,𝑡2𝑒^−|𝑥|()=1∗𝑦𝑥,𝑡2𝑒^−𝑥_𝑞()^{𝑞(𝑏,𝑡)}_𝑎,𝑡𝑒^𝜏𝑦()1𝜏,𝑡𝑑𝜏=2𝑒^−𝑥𝐸(𝑡). (40) Similarly, for 𝑥<𝑞(𝑎,𝑡), we have 𝑢()=1𝑥,𝑡2𝑒^−|𝑥|()=1∗𝑦𝑥,𝑡2𝑒^𝑥_𝑞()^{𝑞(𝑏,𝑡)}_𝑎,𝑡𝑒^−𝜏𝑦()=1𝜏,𝑡𝑑𝜏2𝑒^𝑥𝐹(𝑡). (41) Hence, as consequences of (40) and (41), we get 𝑢(𝑥,𝑡)=−𝑢_𝑥(𝑥,𝑡)=𝑢_𝑥𝑥()=1𝑥,𝑡2𝑒^−𝑥𝐸(𝑡),(),𝑢()as𝑥>𝑞𝑐,𝑡𝑥,𝑡=𝑢_𝑥(𝑥,𝑡)=𝑢_𝑥𝑥()=1𝑥,𝑡2𝑒^𝑥𝐹(𝑡),as𝑥<𝑞(𝑎,𝑡). (42) On the other hand, 𝑑𝐸(𝑡)=𝑑𝑡_ℝ𝑒^𝑥𝑦_𝑡(𝑥,𝑡)𝑑𝑥. (43) It is easy to get 𝑦_𝑡=−𝑏𝑢𝑢_𝑥+𝑏𝑢𝑢_𝑥𝑥𝑥−𝜕_𝑥𝑎2𝑢²+3𝑏−𝑎2𝑢²_𝑥−𝜆𝑢+𝜆𝑢_𝑥𝑥. (44) Putting the identity (44) into 𝑑𝐸(𝑡)/𝑑𝑡, we have 𝑑𝐸(𝑡)=𝑑𝑡_ℝ𝑒^𝑥−𝑏𝑢𝑢_𝑥+𝑏𝑢𝑢_𝑥𝑥𝑥−𝜕_𝑥𝑎2𝑢²+3𝑏−𝑎2𝑢²_𝑥+𝑑𝑥_ℝ𝑒^𝑥−𝜆𝑢+𝜆𝑢_𝑥𝑥=𝑑𝑥_ℝ𝑒^𝑥𝑎2𝑢²+3𝑏−𝑎2𝑢²_𝑥𝑑𝑥, (45) where we have used (42).
Therefore, in the lifespan of the solution, we get 𝐸(𝑡)=^𝑡_0ℝ𝑒^𝑥𝑎2𝑢²+3𝑏−𝑎2𝑢²_𝑥(𝑥,𝜏)𝑑𝑥𝑑𝜏>0. (46) By the same argument, one can check that the following identity for 𝐹(𝑡) is true: 𝐹(𝑡)=−^𝑡_0ℝ𝑒^−𝑥𝑎2𝑢²+3𝑏−𝑎2𝑢²_𝑥(𝑥,𝜏)𝑑𝑥𝑑𝜏<0. (47)
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 (T200905), ZJNSF (Grant no. R6090109), and NSFC (Grant no. 10971197 and 11101376).