On the Cauchy Problem for the b-Family Equations with a Strong Dispersive Term

Abstract

In this paper, we consider b-family equations with a strong dispersive term. First, we present a criterion on blow-up. Then global existence and persistence property of the solution are also established. Finally, we discuss infinite propagation speed of this equation.

1. Introduction

Recently, Holm and Staley [1] studied the exchange of stability in the dynamics of solitary wave solutions under changes in the nonlinear balance in a 1+1 evolutionary partial differential equation related both to shallow water waves and to turbulence. They derived the following equations (the b-family equations):𝑦_𝑡+𝑢𝑦_𝑥+𝑏𝑢_𝑥𝑦=0,𝑡>0,𝑥∈ℝ, (1.1) where 𝑢(𝑥) denotes the velocity field and 𝑦(𝑥,𝑡)=𝑢−𝑢_𝑥𝑥.

Detailed description of the corresponding strong solutions to (1.1) with 𝑢₀ being its initial data was given by Zhou [2]. He established a sufficient condition in profile on the initial data for blow-up in finite time. The necessary and sufficient condition for blow-up is still a challenging problem for us at present. More precious, Theorem 3.1 in [2] means that no matter what the profile of the compactly supported initial datum 𝑢₀(𝑥) is (no matter whether it is positive or negative), for any 𝑡>0 in its lifespan, the solution 𝑢(𝑥,𝑡) is positive at infinity and negative at negative infinity; it is really a very nice property for the b-family equations.

The famous Camassa-Holm equation [3] and Degasperis-Procesi equation [4] are the special cases with 𝑏=2 and 𝑏=3, respectively. Many papers [5–12] are devoted to their study.

In this paper, we consider the following b-family equations with a strong dispersive term:𝑦_𝑡+𝑢𝑦_𝑥+𝑏𝑢_𝑥𝑦+𝜆𝑦_𝑥=0,𝑡>0,𝑥∈ℝ, (1.2) where 𝑦=𝑢−u_𝑥𝑥, 𝜆>0, and 𝜆𝑦_𝑥=𝜆(𝑢_𝑥−𝑢_𝑥𝑥𝑥) is the strong dispersive term.

Let Λ=(1−𝜕²_𝑥)^1/2; then, the operator Λ⁻² can be expressed by its associated Green's function 𝐺=(1/2)𝑒^−|𝑥| as Λ⁻²𝑓(𝑥)=𝐺∗𝑓(𝑥)=(1/2)∫_ℝ𝑒^{−|𝑥−𝑦|}𝑓(𝑦)𝑑𝑦. So (1.2) is equivalent to the following equation:𝑢_𝑡+𝑢𝑢_𝑥+𝜕_𝑥𝑏𝐺∗2𝑢²+3−𝑏2𝑢²_𝑥+𝜆𝑢_𝑥=0. (1.3)

Similar to the Camassa-Holm equation [5], it is easy to establish the following local well-posedness theorem for (1.2).

Theorem 1.1. Given 𝑢₀∈𝐻^𝑠(ℝ), 𝑠>3/2, then there exist a 𝑇 and a unique solution 𝑢 to (1.3) such that 𝑢(𝑥,𝑡)∈𝐶[0,𝑇);𝐻^𝑠(ℝ)∩𝐶¹[0,𝑇);𝐻^𝑠−1(ℝ). (1.4)

To make the paper concise, we would like to omit the detailed proof.

The paper is organized as follows. In Section 2, we get a criterion on blow-up. A condition for global existence is found in Section 3. Persistence property is considered in Section 4. In Section 5, the infinite propagation speed will be established analogous to the b-family equation.

2. Blow-Up

The maximum value of 𝑇 in Theorem 1.1 is called the lifespan of the solution, in general. If 𝑇<∞, that is, lim_𝑡→𝑇⁻‖𝑢(⋅,𝑡)‖_𝐻^𝑠=∞, we say the solution blows up in finite time.

The following lemma tells us that the solution blows up if and only if the first-order derivative blows up.

Lemma 2.1. Assume that 𝑢₀∈𝐻^𝑠(ℝ), 𝑠>2. If 𝑏=1/2, then the solution of (1.2) will exist globally in time. If 𝑏>1/2, then the solution blows up if and only if 𝑢_𝑥 becomes unbounded from below in finite time. If 𝑏<1/2, the solution blows up in finite time if and only if 𝑢_𝑥 becomes unbounded from above in finite time.

Proof. By direct computation, one has 𝑦²_𝐿2=_ℝ𝑢−𝑢_𝑥𝑥²𝑑𝑥=_ℝ𝑢²+2𝑢²_𝑥+𝑢²_𝑥𝑥𝑑𝑥. (2.1) Hence, ‖𝑢‖²_𝐻2≤𝑦²_𝐿2≤2‖𝑢‖²_𝐻2. (2.2) Applying 𝑦 on (1.2) and integration by parts, we obtain 𝑑𝑑𝑡_ℝ𝑦²𝑑𝑥=_ℝ2𝑦𝑦_𝑡𝑑𝑥=−2_ℝ𝑦𝑢𝑦_𝑥+𝑏𝑢_𝑥𝑦+𝜆𝑦_𝑥𝑑𝑥=(1−2𝑏)_ℝ𝑢_𝑥𝑦²𝑑𝑥. (2.3)
If 𝑏=1/2, then (𝑑/𝑑𝑡)∫_ℝ𝑦²𝑑𝑥=0. Hence, ‖𝑢‖²_𝐻2≤𝑦²_𝐿2=𝑦₀²_𝐿2. (2.4) Equation (2.4) implies the corresponding solution exists globally.
If 𝑏>1/2, due to the Gronwall inequality, it is clear that, from (2.3), 𝑢_𝑥 is bounded from below on [0,𝑇) and then the 𝐻₂-norm of the solution is also bounded on [0,𝑇). On the other hand, 𝑢(𝑥,𝑡)=1−𝜕²_𝑥⁻¹𝑦(𝑥,𝑡)=_ℝ𝐺(𝑥−𝜉)𝑦(𝜉)𝑑𝜉. (2.5) Therefore 𝑢_𝑥𝐿∞≤_ℝ𝐺_𝑥(𝑥−𝜉)𝑦(𝜉)𝑑𝜉≤𝐺_𝑥𝐿2𝑦_𝐿2=12𝑦_𝐿2≤‖𝑢‖_𝐻², (2.6) where we use (2.2). Hence, (2.6) tells us if 𝐻₂-norm of the solution is bounded, then the 𝐿_∞-norm of the first derivative is bounded.
By the same argument, we can get the similar result for 𝑏<1/2.
This completes the proof.

Motivated by Mckean's deep observation for the Camassa-Holm equation [7], we can do the similar particle trajectory as𝑞_𝑡𝑞()=𝑢𝑞,𝑡+𝜆,0<𝑡<𝑇,𝑥∈ℝ,𝑥,0=𝑥,𝑥∈ℝ, (2.7) where 𝑇 is the lifespan of the solution; then, 𝑞 is a diffeomorphism of the line. Differentiating the first equation in (2.7) with respect to 𝑥, one has 𝑑𝑞_𝑡𝑑𝑥=𝑞_𝑥𝑡=𝑢_𝑥𝑞,𝑡,𝑡∈(0,𝑇). (2.8) Hence 𝑞_𝑥(𝑥,𝑡)=exp^𝑡₀𝑢_𝑥𝑞,𝑠𝑑𝑠,𝑞_𝑥(𝑥,0)=1. (2.9)

Since 𝑑𝑦𝑞𝑞𝑑𝑡^𝑏𝑥=𝑦_𝑡𝑞+𝑢𝑞,𝑡+𝜆𝑦_𝑥𝑞−𝑏𝑢_𝑥𝑞,𝑡𝑦𝑞𝑞^𝑏_𝑥=0, (2.10) it follows that𝑦𝑞𝑞^𝑏_𝑥=𝑦₀(𝑥). (2.11)

Then we establish sufficient condition on the initial data to guarantee blow-up for (1.2).

Theorem 2.2. Let 𝑏≥2. Suppose that 𝑢₀∈𝐻²(ℝ) and there exists an 𝑥₀∈ℝ such that 𝑦₀(𝑥₀)=(1−𝜕²_𝑥)𝑢₀(𝑥₀)=0, 𝑦₀≥0(≢0)for𝑥∈−∞,𝑥₀,𝑦₀≤0(≢0)for𝑥∈𝑥₀,∞. (2.12) Then the corresponding solution 𝑢(𝑥,𝑡) to (1.2) with 𝑢₀ as the initial datum blows up in finite time.

Proof. Suppose that the solution exists globally. Due to (2.11) and the initial condition (2.12), we have 𝑦(𝑞(𝑥₀,𝑡),𝑡)=0, and 𝑦𝑞(𝑥,𝑡),𝑡≥0(≢0),for𝑥∈−∞,𝑥₀,𝑦𝑞(𝑥,𝑡),𝑡≤0(≢0),for𝑥∈𝑥₀,∞, (2.13) for all 𝑡≥0. Since 𝑢(𝑥,𝑡)=𝐺∗𝑦(𝑥,𝑡), one 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. (2.14) Consequently, 𝑢²_𝑥(𝑥,𝑡)−𝑢²(𝑥,𝑡)=−^𝑥_−∞𝑒^𝜉𝑦(𝜉,𝑡)𝑑𝜉^∞_𝑥𝑒^−𝜉𝑦(𝜉,𝑡)𝑑𝜉, (2.15) for all 𝑡>0.
For any fixed 𝑡, if 𝑥≤𝑞(𝑥₀,𝑡), then 𝑢²_𝑥(𝑥,𝑡)−𝑢²(𝑥,𝑡)=−^{𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉𝑦(𝜉,𝑡)𝑑𝜉−^𝑞(𝑥0_𝑥^,𝑡)𝑒^𝜉𝑦()×𝜉,𝑡𝑑𝜉_{∞𝑞(𝑥0,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^∞_{𝑞(𝑥0,𝑡)}𝑒^−𝜉𝑦(𝜉,𝑡)𝑑𝜉≤𝑢²_𝑥𝑞𝑥₀,𝑡,𝑡−𝑢²𝑞𝑥₀,𝑡,𝑡. (2.16) Similarly, for 𝑥≥𝑞(𝑥₀,𝑡), we also have 𝑢²_𝑥(𝑥,𝑡)−𝑢²(𝑥,𝑡)≤𝑢²_𝑥𝑞𝑥₀,𝑡,𝑡−𝑢²𝑞𝑥₀,𝑡,𝑡. (2.17) Combining (2.16) and (2.17) together, we get that for any fixed 𝑡, 𝑢²_𝑥(𝑥,𝑡)−𝑢²(𝑥,𝑡)≤𝑢²_𝑥𝑞𝑥₀,𝑡,𝑡−𝑢²𝑞𝑥₀,𝑡,𝑡, (2.18) for all 𝑥∈ℝ.
Differentiating (1.3), we get 𝑢_𝑡𝑥+𝑢𝑢_𝑥𝑥−𝑏2𝑢²−1−𝑏2𝑢²_𝑥𝑏+𝐺∗2𝑢²+3−𝑏2𝑢²_𝑥+𝜆𝑢_𝑥𝑥=0. (2.19) Differentiating 𝑢_𝑥(𝑞(𝑥₀,𝑡),𝑡) with respect to 𝑡, where 𝑞 is the diffeomorphism defined in (2.7), 𝜕_𝑡𝑢_𝑥𝑞𝑥₀,𝑡,𝑡=𝑢_𝑥𝑡𝑞𝑥₀,𝑡,𝑡+𝑢_𝑥𝑥𝑞𝑥₀,𝑡,𝑡𝑞_𝑡𝑞𝑥₀=𝑏,𝑡,𝑡2𝑢²𝑞𝑥₀,𝑡,𝑡+1−𝑏2𝑢²_𝑥𝑞𝑥₀𝑏,𝑡,𝑡−𝐺∗2𝑢²(𝑥,𝑡)+3−𝑏2𝑢²_𝑥()𝑏𝑥,𝑡=𝐺∗2𝑢²𝑞𝑥₀,𝑡,𝑡+1−𝑏2𝑢²_𝑥𝑞𝑥₀−𝑏,𝑡,𝑡2𝑢²(𝑥,𝑡)−3−𝑏2𝑢²_𝑥(𝑥,𝑡)=𝐺∗𝑏−22𝑢²𝑞𝑥₀,𝑡,𝑡−𝑢²_𝑥𝑞𝑥₀,𝑡,𝑡−𝑢²(𝑥,𝑡)+𝑢²_𝑥()𝑢𝑥,𝑡+𝐺∗²𝑞𝑥₀−1,𝑡,𝑡2𝑢²_𝑥𝑞𝑥₀,𝑡,𝑡−𝑢²()−1𝑥,𝑡2𝑢²_𝑥()≤1𝑥,𝑡2𝑢²𝑞𝑥₀−1,𝑡,𝑡2𝑢²_𝑥𝑞𝑥₀,𝑡,𝑡, (2.20) where we use (2.18) and the following inequality: 𝐺∗(𝑢²(𝑥,𝑡)+(1/2)𝑢²_𝑥(𝑥,𝑡))≥(1/2)𝑢².
Claim 1. 𝑢_𝑥(𝑞(𝑥₀),𝑡)<0 is decreasing and 𝑢²(𝑞(𝑥₀,𝑡),𝑡)<𝑢²_𝑥(𝑞(𝑥₀,𝑡),𝑡) for all 𝑡≥0.
Suppose not; that is, there exists a 𝑡₀ such that 𝑢²(𝑞(𝑥₀,𝑡),𝑡)<𝑢²_𝑥(𝑞(𝑥₀,𝑡),𝑡) on [0,𝑡) and 𝑢²(𝑞(𝑥₀,𝑡),𝑡)=𝑢²_𝑥(𝑞(𝑥₀,𝑡),𝑡). Now, let 𝐼(𝑡)1∶=2𝑒^{−𝑞(𝑥0,𝑡)𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉𝑦()(𝑡)1𝜉,𝑡𝑑𝜉,𝐼𝐼∶=2𝑒^{𝑞(𝑥0,𝑡)∞}_{𝑞(𝑥0,𝑡)}𝑒^−𝜉𝑦(𝜉,𝑡)𝑑𝜉. (2.21) Firstly, differentiating 𝐼(𝑡), we have 𝑑𝐼(𝑡)1𝑑𝑡=−2𝑢𝑞𝑥₀,𝑡,𝑡+𝜆𝑒^{−𝑞(𝑥0,𝑡)𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉𝑦()+1𝜉,𝑡𝑑𝜉2𝑒^{−𝑞(𝑥0,𝑡)𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉𝑦_𝑡()=1𝜉,𝑡𝑑𝜉2(𝑢+𝜆)𝑢_𝑥−𝑢𝑞𝑥₀−1,𝑡,𝑡2𝑒^{−𝑞(𝑥0,𝑡)}×^{𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉𝑢𝑦_𝑥+2𝑢_𝑥𝑦+𝑏−22𝑢²−𝑢²_𝑥𝑥𝑞𝑥₀,𝑡,𝑡+𝜆𝑦_𝑥≥1𝑑𝜉2(𝑢+𝜆)𝑢_𝑥−𝑢𝑞𝑥₀+1,𝑡,𝑡4𝑢²+𝑢²_𝑥−2𝑢𝑢_𝑥𝑞𝑥₀−𝜆,𝑡,𝑡2𝑢_𝑥−𝑢𝑞𝑥₀=1,𝑡,𝑡4𝑢²_𝑥−𝑢²𝑞𝑥₀,𝑡,𝑡>0,on0,𝑡₀. (2.22) Secondly, by the same argument, we get 𝑑𝐼𝐼(𝑡)=1𝑑𝑡2𝑢𝑞𝑥₀,𝑡,𝑡+𝜆𝑒^{𝑞(𝑥0,𝑡)∞}_{𝑞(𝑥0,𝑡)}𝑒^−𝜉𝑦()+1𝜉,𝑡𝑑𝜉2𝑒^{𝑞(𝑥0,𝑡)∞}_{𝑞(𝑥0,𝑡)}𝑒^−𝜉𝑦_𝑡()=1𝜉,𝑡𝑑𝜉2(𝑢+𝜆)𝑢_𝑥+𝑢𝑞𝑥₀−1,𝑡,𝑡2𝑒^{𝑞(𝑥0,𝑡)}×^∞_{𝑞𝑥0,𝑡}𝑒^−𝜉𝑢𝑦_𝑥+2𝑢_𝑥𝑦+𝑏−22𝑢²−𝑢²_𝑥𝑥𝑞𝑥₀,𝑡,𝑡+𝜆𝑦_𝑥≤1𝑑𝜉2(𝑢+𝜆)𝑢_𝑥+𝑢𝑞𝑥₀−1,𝑡,𝑡4𝑢²+𝑢²_𝑥+2𝑢𝑢_𝑥𝑞𝑥₀−𝜆,𝑡,𝑡2𝑢_𝑥+𝑢𝑞𝑥₀1,𝑡,𝑡=−4𝑢²_𝑥−𝑢²𝑞𝑥₀,𝑡,𝑡<0,on0,𝑡₀. (2.23) Hence, it follows from (2.22), (2.23), and the continuity property of ODEs that 𝑢²_𝑥−𝑢²𝑞𝑥₀,𝑡,𝑡=−4𝐼(𝑡)𝐼𝐼(𝑡)>−4𝐼(0)𝐼𝐼(0)>0. (2.24) Moreover, due to (2.22) and (2.23) again, we have the following equation for (𝑢²_𝑥−𝑢²)(𝑞(𝑥₀,𝑡),𝑡): 𝑑𝑢𝑑𝑡^2𝑥−𝑢²𝑞𝑥₀𝑑,𝑡,𝑡=−𝑑𝑡^{𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉𝑦(𝜉,𝑡)𝑑𝜉^∞_{𝑞(𝑥0,𝑡)}𝑒^−𝜉𝑦()𝑑𝜉,𝑡𝑑𝜉=−𝑒𝑑𝑡^{−𝑞(𝑥0,𝑡)𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉𝑦(𝜉,𝑡)𝑑𝜉𝑒^{𝑞(𝑥0,𝑡)∞}_{𝑞(𝑥0,𝑡)}𝑒^−𝜉𝑦(𝜉,𝑡)𝑑𝜉−𝑒^{−𝑞(𝑥0,𝑡)𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉𝑦()𝑑𝜉,𝑡𝑑𝜉𝑒𝑑𝑡^{𝑞(𝑥0,𝑡)∞}_{𝑞(𝑥0,𝑡)}𝑒^−𝜉𝑦()1𝜉,𝑡𝑑𝜉≥−2𝑢²_𝑥−𝑢²𝑞𝑥₀,𝑡,𝑡𝑒^{𝑞(𝑥0,𝑡)∞}_{𝑞(𝑥0,𝑡)}𝑒^−𝜉𝑦()+1𝜉,𝑡𝑑𝜉2𝑢²_𝑥−𝑢²𝑞𝑥₀,𝑡,𝑡𝑒^{−𝑞(𝑥0,𝑡)𝑞(𝑥0,𝑡)}_−∞𝑒^𝜉𝑦(𝜉,𝑡)𝑑𝜉=−𝑢_𝑥𝑞𝑥₀,𝑡,𝑡𝑢²_𝑥−𝑢²𝑞𝑥₀,𝑡,𝑡. (2.25) Now, substituting (2.20) into (2.25), it yields 𝑑𝑢𝑑𝑡^2𝑥−𝑢²𝑞𝑥₀≥1,𝑡,𝑡2𝑢²_𝑥−𝑢²𝑞𝑥₀,𝑡,𝑡^𝑡₀𝑢²_𝑥−𝑢²𝑞𝑥₀,𝜏,𝜏𝑑𝜏−2𝑢_0𝑥𝑥₀. (2.26) Before completing the proof, we need the following technical lemma.
Lemma 2.3. Suppose that Ψ(𝑡) is twice continuously differential satisfying Ψ^′′(𝑡)≥𝐶₀Ψ^′(𝑡)Ψ(𝑡),𝑡>0,𝐶₀Ψ(𝑡)>0,>0,Ψ^′(𝑡)>0. (2.27) 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). (2.28)
Let Ψ(𝑡)=∫^𝑡₀(𝑢²_𝑥−𝑢²)(𝑞(𝑥₀,𝜏),𝜏)𝑑𝜏−2𝑢_0𝑥(𝑥₀); then, (2.26) is an equation of type (2.27) with 𝐶₀=1/2. The proof is complete by applying Lemma 2.3.

Remark 2.4. Mckean got the necessary and sufficient condition for the Camassa-Holm equation in [7]. It is worth pointing out that Zhou and his collaborators [13] gave a new proof to Mckean's theorem. However, the necessary and sufficient condition for (1.2) is still a challenging problem for us at present.

3. Global Existence

In this section, a global existence result is proved.

Theorem 3.1. Supposing that 𝑢₀∈𝐻³, 𝑦₀=(1−𝜕²_𝑥)𝑢₀ is one sign. Then the corresponding solution to (1.2) exists globally.

Proof. We can assume that 𝑦₀≥0. It is sufficient to prove 𝑢_𝑥(𝑥,𝑡) has a lower and supper bound for all 𝑡. In fact, 𝑢_𝑥()1𝑥,𝑡=−2𝑒^−𝑥𝑥_−∞𝑒^𝜉𝑦()1𝜉,𝑡𝑑𝜉+2𝑒^𝑥∞_𝑥𝑒^−𝜉𝑦(𝜉,𝑡)𝑑𝜉, (3.1) so 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. (3.2) This completes the proof.

4. Persistence Property

Now, we will investigate the following property for the strong solutions to (1.2) in 𝐿^∞-space which asymptotically exponentially decay at infinity as their initial profiles. The main idea comes from a recent work of Zhou and his collaborators [6] for the standard Camassa-Holm equation (for slower decay rate, we refer to [14]).

Theorem 4.1. Assume that for some 𝑇>0 and 𝑠>5/2, 𝑢∈𝐶([0,𝑇];𝐻^𝑠(ℝ)) is a strong solution of (1.2) and that 𝑢₀(𝑥)=𝑢(𝑥,0) satisfies that for some 𝜃∈(0,1), 𝑢₀(𝑥),𝑢_0𝑥(𝑥)∼𝑂𝑒^−𝜃𝑥. (4.1) Then |𝑢(𝑥,𝑡)|,𝑢_𝑥(𝑥,𝑡)∼𝑂𝑒^−𝜃𝑥 (4.2) uniformly in the time interval [0,𝑇].

Proof. First, we will introduce the weight function to get the desired result. This function 𝜑_𝑁(𝑥) with 𝑁∈ℤ⁺ is independent on 𝑡 as follows: 𝜑_𝑁(𝑥)=𝑒1,𝑥≤0,_𝜃𝑥(),𝑒,𝑥∈0,𝑁_𝜃𝑁,𝑥≥𝑁, (4.3) which implies that 0≤𝜑^′_𝑁(𝑥)≤𝜑_𝑁(𝑥). (4.4) From (1.3), we can get 𝜕_𝑡𝑢𝜑_𝑁+𝑢𝜑_𝑁𝑢_𝑥+𝜑_𝑁𝜕_𝑥𝑏𝐺∗2𝑢²+3−𝑏2𝑢²_𝑥+𝜆𝜑_𝑁𝑢_𝑥=0. (4.5) Multiplying (4.5) by (𝑢𝜑_𝑁)^2𝑝−1 with 𝑝∈ℤ⁺ and integrating the result in the 𝑥-variable, we get ^+∞_−∞𝜕_𝑡𝑢𝜑_𝑁𝑢𝜑_𝑁^2𝑝−1𝑑𝑥+^+∞_−∞𝑢𝜑_𝑁𝑢_𝑥𝑢𝜑_𝑁^2𝑝−1+𝑑𝑥_+∞−∞𝜑_𝑁𝜕_𝑥𝑏𝐺∗2𝑢²+3−𝑏2𝑢²_𝑥𝑢𝜑_𝑁^2𝑝−1+^+∞_−∞𝜆𝜑_𝑁𝑢_𝑥𝑢𝜑_𝑁^2𝑝−1𝑑𝑥=0, (4.6) from which we can deduce that 𝑑𝑑𝑡𝑢𝜑_{𝑁𝐿2𝑝}≤𝑢_𝑥𝐿∞𝑢𝜑_{𝑁𝐿2𝑝}+𝜑_𝑁𝜕_𝑥𝑏𝐺∗2𝑢²+3−𝑏2𝑢²_𝑥+𝜆𝑢_{𝑥𝐿2𝑝}. (4.7) Denoting 𝑀=sup_{𝑡∈[0,𝑇]}‖𝑢(𝑡)‖_𝐻^𝑠 and by Gronwall's inequality, we obtain 𝑢𝜑_{𝑁𝐿2𝑝}≤𝑢₀𝜑_{𝑁𝐿2𝑝}+^𝑡₀𝜑_𝑁𝜕_𝑥𝑏𝐺∗2𝑢²+3−𝑏2𝑢²_𝑥+𝜆𝑢_{𝑥𝐿2𝑝}𝑑𝜏𝑒^𝑀𝑡. (4.8) Taking the limits in (4.8), we get 𝑢𝜑_𝑁𝐿∞≤𝑢₀𝜑_𝑁𝐿∞+^𝑡₀𝜑_𝑁𝜕_𝑥𝑏𝐺∗2𝑢²+3−𝑏2𝑢²_𝑥+𝜆𝑢_𝑥𝐿∞𝑑𝜏𝑒^𝑀𝑡. (4.9) Next differentiating (1.3) in the 𝑥-variable produces the equation 𝑢_𝑡𝑥+𝑢𝑢_𝑥𝑥+𝑢²_𝑥+𝜕²_𝑥𝑏𝐺∗2𝑢²+3−𝑏2𝑢²_𝑥+𝜆𝑢_𝑥𝑥=0. (4.10) Using the weight function, we can rewrite (4.10) as 𝜕_𝑥𝑢_𝑥𝜑_𝑁+𝑢𝑢_𝑥𝑥𝜑_𝑁+𝑢_𝑥𝜑_𝑁𝑢_𝑥+𝜑_𝑁𝜕²_𝑥𝑏𝐺∗2𝑢²+3−𝑏2𝑢²_𝑥+𝜆𝜑_𝑁𝑢_𝑥𝑥=0. (4.11) Multiplying (4.11) by (𝑢_𝑥𝜑_𝑁)^2𝑝−1 with 𝑝∈ℤ⁺ and integrating the result in the 𝑥-variable, it follows that ^+∞_−∞𝜕_𝑡𝑢_𝑥𝜑_𝑁𝑢_𝑥𝜑_𝑁^2𝑝−1𝑑𝑥+^+∞_−∞𝑢𝑢_𝑥𝑥𝜑_𝑁𝑢_𝑥𝜑_𝑁^2𝑝−1+𝑑𝑥_+∞−∞𝑢_𝑥𝜑_𝑁𝑢_𝑥𝑢_𝑥𝜑_𝑁^2𝑝−1𝑑𝑥+^+∞_−∞𝜑_𝑁𝜕²_𝑥𝑏𝐺∗2𝑢²+3−𝑏2𝑢²_𝑥𝑢_𝑥𝜑_𝑁^2𝑝−1𝑑𝑥+^+∞_−∞𝜆𝜑_𝑁𝑢_𝑥𝑥𝑢_𝑥𝜑_𝑁^2𝑝−1𝑑𝑥=0. (4.12) For the second term on the right side of (4.12), we know ^+∞_−∞(𝑢+𝜆)𝑢_𝑥𝑥𝜑_𝑁𝑢_𝑥𝜑_𝑁^2𝑝−1𝑑𝑥≤2‖𝑢‖_𝐿^∞+𝑢_𝑥𝐿∞𝑢_𝑥𝜑_𝑁𝐿^2𝑝_2𝑝. (4.13) Using the above estimate and the Hölder inequality, we deduce that 𝑑𝑢𝑑𝑡^𝑥𝜑_{𝑁𝐿2𝑝}≤2𝑀𝑢_𝑥𝜑_{𝑁𝐿2𝑝}+𝜑_𝑁𝜕²_𝑥𝑏𝐺∗2𝑢²+3−𝑏2𝑢²_{𝑥𝐿2𝑝}. (4.14) Thanks to Gronwall's inequality, it holds that 𝑢_𝑥𝜑_{𝑁𝐿2𝑝}≤𝑢_0𝑥𝜑_{𝑁𝐿2𝑝}+𝜑_𝑁𝜕²_𝑥𝑏𝐺∗2𝑢²+3−𝑏2𝑢²_{𝑥𝐿2𝑝}𝑒^2𝑀𝑡. (4.15) Taking the limits in (4.15), we have 𝑢_𝑥𝜑_𝑁𝐿∞≤𝑢_0𝑥𝜑_𝑁𝐿∞+^𝑡₀𝜑_𝑁𝜕²_𝑥𝑏𝐺∗2𝑢²+3−𝑏2𝑢²_𝑥𝐿∞𝑑𝜏𝑒^2𝑀𝑡. (4.16) Combining (4.9) and (4.16) together, it follows that 𝑢𝜑_𝑁𝐿∞+𝑢_𝑥𝜑_𝑁𝐿∞≤𝑢₀𝜑_𝑁𝐿∞+𝑢_0𝑥𝜑_𝑁𝐿∞𝑒^2𝑀𝑡+𝑒^{2𝑀𝑡𝑡}₀𝜑_𝑁𝜕_𝑥𝑏𝐺∗2𝑢²+3−𝑏2𝑢²_𝑥+𝜆𝑢_𝑥𝐿∞+𝜑_𝑁𝜕²_𝑥𝑏𝐺∗2𝑢²+3−𝑏2𝑢²_𝑥𝐿∞𝑑𝜏. (4.17) A simple calculation shows that there exists 𝑐₀>0, depending only on 𝜃∈(0,1), such that for any 𝑁∈ℤ⁺, 𝜑_𝑁(𝑥)^∞_−∞𝑒^{−|𝑥−𝑦|}1𝜑_𝑁𝑦𝑑_𝑦≤𝑐₀=4.1−𝜃 (4.18) Thus, for any appropriate function 𝑓 and 𝑔, one sees that 𝜑_𝑁(𝑥)𝑔(𝑥=1𝐺∗𝑓)2𝜑_𝑁(𝑥)^∞_−∞𝑒^{−|𝑥−𝑦|}𝑓𝑦𝑔𝑦≤1𝑑𝑦2𝜑_𝑁(𝑥)^∞_−∞𝑒^{−|𝑥−𝑦|}1𝜑_𝑁𝑦𝜑_𝑁𝑦𝑓𝑦𝑔𝑦≤1𝑑𝑦2𝜑_𝑁(𝑥)^∞_−∞𝑒^{−|𝑥−𝑦|}1𝜑_𝑁𝑦𝑑𝑦𝜑_𝑁𝑓_𝐿∞𝑔_𝐿∞≤𝑐₀𝜑_𝑁𝑓_𝐿∞𝑔_𝐿∞. (4.19) Similarly, we can get 𝜑_𝑁𝜕_𝑥𝐺∗𝑓(𝑥)𝑔(𝑥)≤𝑐₀𝜑_𝑁𝑓_𝐿∞𝑔_𝐿∞,𝜑_𝑁𝜕²_𝑥𝐺∗𝑓(𝑥)𝑔(𝑥)≤𝑐₀𝜑_𝑁𝑓_𝐿∞𝑔_𝐿∞. (4.20) Thus, inserting the above estimates into (4.17), there exists a constant 𝑐̃=𝑐(̃𝑀,𝑇)≥0 such that 𝑢𝜑_𝑁𝐿∞+𝑢_𝑥𝜑_𝑁𝐿∞≤𝑐̃^𝑡₀‖𝑢‖_𝐿^∞+𝑢_𝑥𝐿∞+𝜆𝑢𝜑_𝑁𝐿∞+𝑢_𝑥𝜑_𝑁𝐿∞+𝑐̃𝑢𝑑𝜏₀𝜑_𝑁𝐿∞+𝑢_0𝑥𝜑_𝑁𝐿∞≤𝑐̃𝑢₀𝜑_𝑁𝐿∞+𝑢_0𝑥𝜑_𝑁𝐿∞+𝑐̃^𝑡₀𝑢𝜑_𝑁𝐿∞+𝑢_𝑥𝜑_𝑁𝐿∞𝑑𝜏. (4.21) Hence, for any 𝑡∈ℤ⁺ and any 𝑡∈[0,𝑇], we have 𝑢𝜑_𝑁𝐿∞+𝑢_𝑥𝜑_𝑁𝐿∞≤𝑐̃𝑢₀𝜑_𝑁𝐿∞+𝑢_0𝑥𝜑_𝑁𝐿∞≤𝑐̃𝑢₀max1,𝑒^𝜃𝑥_𝐿∞+𝑢_0𝑥max1,𝑒^𝜃𝑥_𝐿∞. (4.22) Finally, taking the limit as 𝑁 goes to infinity in (4.22), we find that for any 𝑡∈[0,𝑇]𝑢𝑒^𝜃𝑥_𝐿∞+𝑢_𝑥𝑒^𝜃𝑥_𝐿∞≤𝑐̃𝑢₀max1,𝑒^𝜃𝑥_𝐿∞+𝑢_0𝑥max1,𝑒^𝜃𝑥_𝐿∞, (4.23) which completes the proof of the theorem.

5. Infinite Propagation Speed

Recently, Himonas and his collaborators established infinite propagation speed for the Camassa-Holm equation in [6]. Later, Guo [15, 16] considered a similar problem on the weakly dissipative Camassa-Holm equation and the weakly dissipative Degasperis-Procesi equation. Recently, infinite propagation speed for a class of nonlocal dispersive 𝜃-equations was established in [17]. The purpose of this section is to give a more detailed description on the corresponding strong solution 𝑢(𝑥,𝑡) to (1.2) in its life span with initial data 𝑢₀(𝑥) being compactly supported. The main theorem is as follows.

Theorem 5.1. Let 0≤𝑏≤3. Assume that for some 𝑇≥0 and 𝑠≥5/2, 𝑢∈𝐶([0,𝑇];𝐻^𝑠(ℝ)) is a strong solution of (1.2). If 𝑢₀(𝑥)=𝑢(𝑥,0) has compact support [𝑎,𝑐], then for 𝑡∈(0,𝑇], one has 𝑢()=𝑓𝑥,𝑡⁺(𝑡)𝑒^−𝑥(),𝑓,for𝑥>𝑞𝑐,𝑡₋(𝑡)𝑒^−𝑥,for𝑥<𝑞(𝑎,𝑡), (5.1) where 𝑓₊(𝑡) and 𝑓₋(𝑡) denote continuous nonvanishing functions, with 𝑓₊(𝑡)>0 and 𝑓₋(𝑡)<0 for 𝑡∈(0,𝑇]. Furthermore, 𝑓₊(𝑡) is strictly increasing function, while 𝑓₋(𝑡) is strictly decreasing function.

Proof. Since 𝑢₀ has compact support in 𝑥 in the interval [𝑎,𝑐], from (2.11), so does 𝑦(𝑥,𝑡) in the interval [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, (5.2) with 𝐸₀=_ℝ𝑒^𝑥𝑦₀(𝑥)𝑑𝑥=0,𝐹₀=_ℝ𝑒^−𝑥𝑦₀(𝑥)𝑑𝑥=0. (5.3) Then for 𝑥>𝑞(𝑐,𝑡), we have 𝑢()=1𝑥,𝑡2𝑒^−|𝑥|()=1∗𝑦𝑥,𝑡2𝑒^{−𝑥𝑞(𝑏,𝑡)}_{𝑞(𝑎,𝑡)}𝑒^𝜏𝑦()1𝜏,𝑡𝑑𝜏=2𝑒^−𝑥𝐸(𝑡). (5.4) Similarly, when 𝑥<𝑞(𝑎,𝑡), we get 𝑢()=1𝑥,𝑡2𝑒^−|𝑥|()=1∗𝑦𝑥,𝑡2𝑒^{𝑥𝑞(𝑏,𝑡)}_{𝑞(𝑎,𝑡)}𝑒^−𝜏𝑦()1𝜏,𝑡𝑑𝜏=2𝑒^𝑥𝐹(𝑡). (5.5) Hence, as consequences of (5.4) and (5.5), we have 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). (5.6) On the other hand, 𝑑𝐸(𝑡)=𝑑𝑡_ℝ𝑒^𝑥𝑦_𝑡(𝑥,𝑡)𝑑𝑥. (5.7) It is easy to get 𝑦_𝑡=−𝑢𝑢_𝑥+𝑢𝑢_𝑥𝑥𝑥−𝜕_𝑥𝑏2𝑢²+3−𝑏2𝑢²_𝑥−𝜆𝑢_𝑥+𝜆𝑢_𝑥𝑥𝑥. (5.8) Substituting identity (5.8) into 𝑑𝐸(𝑡)/𝑑𝑡, we obtain 𝑑𝐸(𝑡)=𝑑𝑡_ℝ𝑒^𝑥−𝑢𝑢_𝑥+𝑢𝑢_𝑥𝑥𝑥−𝜕_𝑥𝑏2𝑢²+3−𝑏2𝑢²_𝑥+𝑑𝑥_ℝ𝑒^𝑥−𝜆𝑢_𝑥+𝜆𝑢_𝑥𝑥𝑥=𝑑𝑥_ℝ𝑒^𝑥𝑏2𝑢²+3−𝑏2𝑢²_𝑥𝑑𝑥, (5.9) where we use (5.6).
Therefore, in the lifespan of the solution, we have 𝐸(𝑡)=^𝑡_0ℝ𝑒^𝑥𝑏2𝑢²+3−𝑏2𝑢²_𝑥(𝑥,𝜏)𝑑𝑥𝑑𝜏>0. (5.10) By the same argument, one can check that the following identity for 𝐹(𝑡) is true: 𝐹(𝑡)=−^𝑡_0ℝ𝑒^−𝑥𝑏2𝑢²+3−𝑏2𝑢²_𝑥(𝑥,𝜏)𝑑𝑥𝑑𝜏<0. (5.11) In order to complete the proof, it is sufficient to let 𝑓₊(𝑡)=(1/2)𝐸(𝑡) and 𝑓₋(𝑡)=(1/2)𝐹(𝑡).

Remark 5.2. The main result in [18] is that any nontrivial classical solution of the b-family equation with dispersive term will not have compact support if its initial data has this property. But Theorem 4.1 means that no matter what the profile of the compactly supported initial datum 𝑢₀(𝑥) is (no matter whether it is positive or negative), for any 𝑡>0 in its lifespan, the solution 𝑢(𝑥,𝑡) is positive at infinity and negative at negative infinity. So Theorem 4.1 is an improvement of that in [18].

Acknowledgment

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