Well-Posedness, Blow-Up Phenomena, and Asymptotic Profile for a Weakly Dissipative Modified Two-Component Camassa-Holm Equation

Abstract

We study the Cauchy problem of a weakly dissipative modified two-component Camassa-Holm equation. We firstly establish the local well-posedness result. Then we present a precise blow-up scenario. Moreover, we obtain several blow-up results and the blow-up rate of strong solutions. Finally, we consider the asymptotic behavior of solutions.

1. Introduction

In this paper, we consider the Cauchy problem of the following weakly dissipative modified two-component Camassa-Holm system: 𝑚_𝑡+𝑢𝑚_𝑥+2𝑚𝑢_𝑥+𝜌𝜌_𝑥𝜌+𝜆𝑚=0,𝑡>0,𝑥∈ℝ,_𝑡+𝜌𝑢_𝑥𝑚()+𝜆𝜌=0,𝑡>0,𝑥∈ℝ,0,𝑥=𝑚₀(𝑥)𝜌(),𝑥∈ℝ,0,𝑥=𝜌₀(𝑥),𝑥∈ℝ, (1) where 𝑚=(1−𝜕²_𝑥)𝑢,𝜌=(1−𝜕²_𝑥)(𝜌−𝜌₀), and 𝜆 is a nonnegative dissipative parameter.

The Camassa-Holm equation [1] has been recently extended to a two-component integrable system (CH2) 𝑚_𝑡+𝑢𝑚_𝑥+2𝑚𝑢_𝑥=𝜌𝜌_𝑥𝜌,𝑡>0,𝑥∈ℝ,_𝑡+𝜌𝑢_𝑥=0,𝑡>0,𝑥∈ℝ, (2) with 𝑚=𝑢−𝑢_𝑥𝑥, which is a model for wave motion on shallow water, where 𝑢(𝑡,𝑥) describes the horizontal velocity of the fluid and 𝜌(𝑡,𝑥) is in connection with the horizontal deviation of the surface from equilibrium, all measured in dimensionless units. Moreover, 𝑢 and 𝜌 satisfy the boundary conditions: 𝑢→0 and 𝜌→1 as |𝑥|→∞. The system can be identified with the first negative flow of the AKNS hierarchy and possesses the interesting peakon and multikink solutions [2]. Moreover, it is connected with the time-dependent Schrödinger spectral problem [2]. Popowicz [3] observes that the system is related to the bosonic sector of an 𝑁=2 supersymmetric extension of the classical Camassa-Holm equation. Equation (2) with 𝜌≡0 becomes the Camassa-Holm equation, which has global conservative solutions [4] and dissipative solutions [5]. For other methods to handle the problems relating to various dynamic properties of the Camassa-Holm equation and other shallow water equations, the reader is referred to [6–8] and the references therein.

Since the system was derived physically by Constantin and Ivanov [9] in the context of shallow water theory (also by Chen et al. in [2] and Falqui in [10]), many researchers have paid extensive attention to it. In [11], Escher et al. establish the local well-posedness and present the precise blow-up scenarios and several blow-up results of strong solutions to (2) on the line. In [9], Constantin and Ivanov investigate the global existence and blow-up phenomena of strong solutions of (2) on the line. Later, Guan and Yin [12] obtain a new global existence result for strong solutions to (2) and get several blow-up results, which improve the recent results in [9]. Recently, they study the global existence of weak solutions to (2) [13]. In [14], Henry studies the infinite propagation speed for (2). Gui and Liu [15] establish the local well-posedness for (2) in a range of the Besov spaces; they also derive a wave-breaking mechanism for strong solutions. Mustafa [16] gives a simple proof of existence for the smooth travelling waves for (2). Hu and Yin [17, 18] study the blow-up phenomena and the global existence of (2) on the circle.

Recently, the CH2 system was generalized into the following modified two-component Camassa-Holm (MCH2) system: 𝑚_𝑡+𝑢𝑚_𝑥+2𝑚𝑢_𝑥=−𝑔𝜌𝜌_𝑥𝜌,𝑡>0,𝑥∈ℝ,_𝑡+𝜌𝑢_𝑥=0,𝑡>0,𝑥∈ℝ, (3) where 𝑚=(1−𝜕²_𝑥)𝑢,𝜌=(1−𝜕²_𝑥)(𝜌−𝜌₀),𝑢 denotes the velocity field, 𝜌₀ is taken to be a constant, and 𝑔 is the downward constant acceleration of gravity in applications to shallow water waves. This MCH2 system does admit peaked solutions in the velocity and average density; we refer this to [19] for details. There, the authors analytically identified the steepening mechanism that allows the singular solutions to emerge from smooth spatially confined initial data. They found that wave breaking in the fluid velocity does not imply singularity in the pointwise density 𝜌 at the point of vertical slope. Some other recent works can be found in [20–31]. We find that the MCH2 system is expressed in terms of an averaged or filtered density 𝜌 in analogy to the relation between momentum and velocity by setting 𝜌=(1−𝜕²_𝑥)(𝜌−𝜌₀), but it may not be integrable unlike the CH2 system. The important point here is that MCH2 has the following conservation law ∫_𝑅(𝑢²+𝑢²_𝑥+𝜌²+𝜌²_𝑥)𝑑𝑥, which plays a crucial role in the study of (3), noting that, for the CH2 system, we cannot obtain the conservation of 𝐻¹ norm.

In general, it is quite difficult to avoid energy dissipation mechanisms in the real world. Ghidaglia [32] studies the long-time behavior of solutions to the weakly dissipative KdV equation as a finite-dimensional dynamical system. Recently, Hu and Yin [33] study the blow-up and blow-up rate of solutions to a weakly dissipative periodic rod equation. In [34, 35], Hu considered global existence and blow-up phenomena for a weakly dissipative two-component Camassa-Holm system on the circle and on the line. However, (1) on the line (nonperiodic case) has not been studied yet. The aim of this paper is to study the blow-up phenomena and asymptotic profile of the strong solutions to (1). We find that asymptotic profile of solutions to the weakly dissipative modified two-component periodic Camassa-Holm system (1) is similar to that of the modified two-component Camassa-Holm system (3), such as the local well-posedness and the blow-up scenario. In addition, we also find that the blow-up rate of (1) is not affected by the weakly dissipative term, but the occurrence of blow-up of (1) is affected by the dissipative parameter.

This paper is organized as follows. In Section 2, we present some notations and establish the local well-posedness for system (1) by applying Kato's semigroup approach to nonlinear hyperbolic evolution equations. In Section 3, we prove a precise blow-up scenario result. In Section 4, we present the blow-up results for strong solutions to (1) provided that the initial data satisfy appropriate conditions and we derive a blow-up rate estimate result. Finally, we consider the asymptotic behavior of solutions.

2. Local Well-Posedness

We now provide the framework in which we will reformulate system (1). With 𝑚=𝑢−𝑢_𝑥𝑥,𝜌=𝛾−𝛾_𝑥𝑥, and 𝛾=𝜌−𝜌₀, we can rewrite (1) as follows: 𝑚_𝑡+𝑢𝑚_𝑥+2𝑚𝑢_𝑥+𝜌𝛾_𝑥𝜌+𝜆𝑦=0,𝑡>0,𝑥∈ℝ,_𝑡+𝜌𝑢_𝑥𝑦()+𝜆𝜌=0,𝑡>0,𝑥∈ℝ,0,𝑥=𝑢₀(𝑥)−𝑢_0,𝑥𝑥(𝑥)𝜌(),𝑥∈ℝ,0,𝑥=𝛾₀(𝑥)−𝛾_0,𝑥𝑥(𝑥),𝑥∈ℝ. (4) Note that if 𝑝(𝑥)∶=(1/2)𝑒^−|𝑥|,𝑥∈𝑅, then (1−𝜕²_𝑥)⁻¹𝑓=𝑝∗𝑓 for all 𝑓∈𝐿²(𝑅),𝑝∗𝑦=𝑢, and 𝑝∗𝜌=𝛾. Here, we denote by ∗ the convolution. Using this identity, we can rewrite (5) as follows: 𝑢_𝑡+𝑢𝑢_𝑥=−𝜕_𝑥𝑝∗𝑢²+12𝑢²_𝑥+12𝛾²−12𝛾²_𝑥𝛾−𝜆𝑢,𝑡>0,𝑥∈ℝ,_𝑡+𝑢𝛾_𝑥=−𝑝∗𝑢_𝑥𝛾_𝑥𝑥+𝑢_𝑥𝛾𝑢()−𝜆𝛾,𝑡>0,𝑥∈ℝ,0,𝑥=𝑢₀(𝑥)𝛾(),𝑥∈ℝ,0,𝑥=𝛾₀(𝑥),𝑥∈ℝ, (5) or we can write it in the equivalent form 𝑢_𝑡+𝑢𝑢_𝑥=−𝜕_𝑥1−𝜕²_𝑥⁻¹𝑢²+12𝑢²_𝑥+12𝛾²−12𝛾²_𝑥𝛾−𝜆𝑢,𝑡>0,𝑥∈ℝ,_𝑡+𝑢𝛾_𝑥=−𝜕_𝑥1−𝜕²_𝑥⁻¹𝑢_𝑥𝛾_𝑥−1−𝜕²_𝑥⁻¹𝑢_𝑥𝑢()𝛾−𝜆𝛾,𝑡>0,𝑥∈ℝ,0,𝑥=𝑢₀(𝑥)𝛾(),𝑥∈ℝ,0,𝑥=𝛾₀(𝑥),𝑥∈ℝ. (6) The local well-posedness of the Cauchy problem (5) in Sobolev spaces 𝐻^𝑠 with 𝑠>5/2 can be obtained by applying Kato's theorem [23, 36]. As a result, we have the following well-posedness result.

Theorem 1. Given 𝑧₀=𝑧(𝑥,0)=(𝑢₀,𝛾₀)∈𝐻^𝑠×𝐻^𝑠, 𝑠>5/2, then there exist a maximal 𝑇=𝑇(‖𝑧₀‖_{𝐻𝑠×𝐻𝑠})>0 and a unique solution 𝑧=(𝑢,𝛾) to (3) (or (6)) such that 𝑧=⋅,𝑧₀∈𝐶[0,𝑇];𝐻^𝑠×𝐻^𝑠∩𝐶¹[0,𝑇];𝐻^𝑠−1×𝐻^𝑠−1. (7) Moreover, the solution depends continuously on the initial data; that is, the mapping 𝑢₀→𝑢(⋅,𝑢₀)∶𝐻^𝑠→𝐶([0,𝑇];𝐻^𝑠(ℝ))∩𝐶¹([0,𝑇];𝐻^𝑠−1(ℝ)) is continuous.

3. The Precise Blow-Up Scenario

In this section, we present the precise blow-up scenarios for solutions to (6).

Theorem 2. Let 𝑧₀=(𝑢,𝛾)∈𝐻^𝑠×𝐻^𝑠,𝑠>5/2, be given and assume that 𝑇 is the maximal existence time of the corresponding solution 𝑧=(𝑢,𝛾) to (6) with initial data 𝑧₀; if there exists 𝑀>0 such that 𝑢_𝑥(𝑡,⋅)_𝐿∞+𝛾(𝑡,⋅)_𝐿∞+𝛾_𝑥(𝑡,⋅)_𝐿∞≤𝑀,𝑡∈[0,𝑇), (8) then the 𝐻^𝑠×𝐻^𝑠 norm of 𝑧(𝑡,⋅) does not blow up on [0,𝑇).

The proof of the theorem is similar to the proof of Theorem 3 in [22]; we omit it here.

Consider the following differential equation: 𝑑𝑞(𝑥,𝑡)𝑞()[),𝑞()𝑑𝑡=𝑢𝑥,𝑡,𝑡,𝑡∈0,𝑇0,𝑡=𝑥,𝑥∈ℝ. (9) By applying the classical results on the theory of ordinary differential equations, we may derive the following properties of the solution 𝑞 of (9), which are crucial in the proof of global existence and blow-up of solutions.

Lemma 3 (see [23]). Let 𝑢₀∈𝐻^𝑠,𝑠≥5/2, and let 𝑇 be the maximal existence time of the corresponding solution 𝑢(𝑡,𝑥) to (9). Then (9) has a unique solution 𝑞∈𝐶¹([0,𝑇)×ℝ,ℝ). Moreover, the map 𝑞(𝑡,⋅) is an increasing diffeomorphism of ℝ with 𝑞_𝑥(𝑥,𝑡)=exp^𝑡₀𝑢_𝑥𝑞()𝑞𝑥,𝑠,𝑠𝑑𝑠>0,_𝑥(𝑥,0)=1,𝑥∈𝑅,0≤𝑡<𝑇. (10)

Lemma 4. Let 𝑧₀=(𝑢₀,𝛾₀)∈𝐻^𝑠×𝐻^𝑠 with 𝑠>5/2, and let 𝑇>0 be the maximal existence time of the corresponding solution 𝑧∈(𝑢,𝛾) to (5). Then, one has 𝜌𝑡,𝑞(𝑥,𝑡)𝑞_𝑥=𝜌₀𝑒^−𝜆𝑡,∀(𝑡,𝑥)∈[0,𝑇)×𝑅. (11) Moreover, if there exists 𝑀₁>0, such that 𝑢_𝑥(𝑡,𝑥)≥−𝑀₁ for all (𝑡,𝑥)∈[0,𝑇)×𝑅, then 𝜌(𝑡,⋅)_𝐿∞=𝜌𝑡,𝑞(𝑡,𝑥)_𝐿∞≤𝑒^𝑀1𝑇𝜌₀(⋅)_𝐿∞,∀𝑡∈[0,𝑇). (12)

Proof. Differentiating the left-hand side of (6) with respect to 𝑡, in view of (9) and the second equation of (5), we have 𝑑𝜌()𝑞𝑑𝑡𝑡,𝑞𝑥,𝑡_𝑥=𝜌_𝑡𝑞_𝑥+𝜌_𝑥𝑞_𝑡𝑞_𝑥+𝜌𝑞_𝑥𝑡=−𝜆𝜌𝑞_𝑥. (13) Solving the equation, we get (11).
By Lemma 3, in view of (11) and the assumption of the lemma, we obtain 𝜌(𝑡,⋅)_𝐿∞=𝜌𝑡,𝑞(𝑡,⋅)_∞=𝑒^{−𝜆𝑡−∫𝑡0𝑢𝑥(𝑠,⋅)𝑑𝑥}𝜌₀(⋅)_𝐿∞≤𝑒^𝑀1𝑇𝜌(⋅)_𝐿∞,∀𝑡∈[0,𝑇). (14)
The following result is proved only with regard to 𝑟=3, since we can obtain the same conclusion for the general case 𝑟>5/2 by using Theorem 1 and a simple density argument.

We now present a precise blow-up scenario for strong solutions to (5).

Theorem 5. Let 𝑦₀=(𝑢₀,𝛾₀)∈𝐻^𝑠×𝐻^𝑠, 𝑠>5/2, and let 𝑇 be the maximal existence of the corresponding solution 𝑧=(𝑢,𝛾) to (6). Then, the solution blows up in finite time if and only if liminf_{𝑡→𝑇,𝑥∈ℝ}𝑢_𝑥(𝑡,𝑥)=−∞ (15) or limsup_𝑡→𝑇𝛾_𝑥(𝑡,⋅)_𝐿∞=+∞. (16)

Proof. Multiplying the first equation in (5) by 𝑚=𝑢−𝑢_𝑥𝑥 and integrating by parts, we obtain 𝑑𝑑𝑡_ℝ𝑚²𝑑𝑥=2_ℝ𝑚𝑚_𝑡𝑑𝑥=2_ℝ𝑚−𝑢𝑚_𝑥−2𝑚𝑢_𝑥−𝜌𝛾_𝑥𝑑𝑥−2𝜆_ℝ𝑚²𝑑𝑥=−3_ℝ𝑚²𝑢_𝑥𝑑𝑥−2_ℝ𝑚𝜌𝛾_𝑥𝑑𝑥−2𝜆_ℝ𝑚²𝑑𝑥. (17) Repeating the same procedure to the second equation in (5), we get 𝑑𝑑𝑡_ℝ𝜌²𝑑𝑥=−_ℝ𝜌²𝑢_𝑥−2𝜆_ℝ𝜌²𝑑𝑥. (18) A combination of (17) and (18) yields 𝑑𝑑𝑡_ℝ𝑚²+𝜌²𝑑𝑥=−3_ℝ𝑚²𝑢_𝑥𝑑𝑥−2_ℝ𝑚𝜌𝛾_𝑥−𝑑𝑥_ℝ𝜌²𝑢_𝑥−2𝜆_ℝ𝑚²+𝜌²𝑑𝑥. (19) Differentiating the first equation in (5) with respect to 𝑥, multiplying by 𝑚_𝑥=𝑢_𝑥−𝑢_𝑥𝑥𝑥, and then integrating over ℝ, we obtain 𝑑𝑑𝑡_ℝ𝑚²_𝑥𝑑𝑥=−5_ℝ𝑚²_𝑥𝑢_𝑥𝑑𝑥+2_ℝ𝑚²𝑢_𝑥𝑑𝑥−2_ℝ𝑚_𝑥𝜌_𝑥𝛾_𝑥𝑑𝑥−2_ℝ𝑚_𝑥𝜌𝛾_𝑥𝑥𝑑𝑥−2𝜆_ℝ𝑚²_𝑥𝑑𝑥. (20) Similarly, 𝑑𝑑𝑡_ℝ𝜌²_𝑥𝑑𝑥=−3_ℝ𝜌²_𝑥𝑢_𝑥𝑑𝑥+_ℝ𝜌²𝑢_𝑥𝑥𝑥𝑑𝑥−2𝜆_ℝ𝜌²_𝑥𝑑𝑥. (21) A combination of (17)–(21) yields 𝑑𝑑𝑡_ℝ𝑚²+𝜌²+𝑚²_𝑥+𝜌²_𝑥𝑑𝑥=−_ℝ𝑚²𝑢_𝑥𝑑𝑥−5_ℝ𝑚²_𝑥𝑢_𝑥𝑑𝑥−2_ℝ𝑚𝜌𝛾_𝑥𝑑𝑥−2_ℝ𝑚_𝑥𝜌_𝑥𝛾_𝑥𝑑𝑥−2𝜆_ℝ𝑚²+𝜌²𝑑𝑥−2_ℝ𝑚_𝑥𝜌𝛾_𝑥𝑥−𝑑𝑥_ℝ𝜌²𝑢_𝑥𝑑𝑥−3_ℝ𝜌²_𝑥𝑢_𝑥+𝑑𝑥_ℝ𝜌²𝑢_𝑥𝑥𝑥𝑑𝑥−2𝜆_ℝ𝑚²_𝑥+𝜌²_𝑥𝑑𝑥=−_ℝ𝑚²𝑢_𝑥𝑑𝑥−5_ℝ𝑚²_𝑥𝑢_𝑥−𝑑𝑥_ℝ𝜌²𝑢_𝑥𝑑𝑥−3_ℝ𝜌²_𝑥𝑢_𝑥𝑑𝑥−2𝜆_ℝ𝑚²+𝜌²𝑑𝑥+_ℝ𝜌²𝑢_𝑥𝑥𝑥𝑑𝑥−2_ℝ𝑚𝜌𝛾_𝑥𝑑𝑥−2_ℝ𝑚_𝑥𝜌_𝑥𝛾_𝑥𝑑𝑥−2_ℝ𝑚_𝑥𝜌𝛾_𝑥𝑥𝑑𝑥−2𝜆_ℝ𝑚²_𝑥+𝜌²_𝑥𝑑𝑥. (22) Assume that there exist 𝑀₁>0 and 𝑀₂>0 such that 𝑢𝑥(𝑡,𝑥)≥𝑀₁ and ‖𝛾_𝑥(𝑡,⋅)‖_𝐿∞≤𝑀₂ for all (𝑡,𝑥)∈[0,𝑇)×𝑅; then it follows from Lemma 4 that 𝜌(𝑡,⋅)_𝐿∞≤𝑒^𝑀1𝑇𝜌₀(⋅)_𝐿∞. (23) Therefore, 𝑑𝑑𝑡_ℝ𝑚²+𝜌²+𝑚²_𝑥+𝜌²_𝑥≤𝑑𝑥5𝑀₁+2𝜆_ℝ𝑚²+𝜌²+𝑚²_𝑥+𝜌²_𝑥+𝑀𝑑𝑥₂+𝑒^𝑀1𝑇𝜌₀(⋅)_𝐿∞×_ℝ𝑚²+𝜌²+𝑚²_𝑥+𝜌²_𝑥+𝑢²_𝑥𝑥𝑥+𝛾²_𝑥𝑥≤𝑑𝑥5𝑀₁+2𝜆_ℝ𝑚²+𝜌²+𝑚²_𝑥+𝜌²_𝑥𝑀𝑑𝑥+2₂+𝑒^𝑀1𝑇𝜌₀(⋅)_𝐿∞×_ℝ𝑚²+𝜌²+𝑚²_𝑥+𝜌²_𝑥≤𝑑𝑥5𝑀₁+2𝜆+2𝑀₂+𝑒^𝑀1𝑇𝜌₀(⋅)_𝐿∞×_ℝ𝑚²+𝜌²+𝑚²_𝑥+𝜌²_𝑥𝑑𝑥. (24) The previous discussion shows that if there exist 𝑀₁>0 and 𝑀₂>0 such that 𝑢_𝑥(𝑡,𝑥)≥𝑀₁ and ‖𝛾_𝑥(𝑡,⋅)‖≤𝑀₂ for all (𝑡,𝑥)∈[0,𝑇)×ℝ, then there exist two positive constants 𝐾 and 𝑘 such that the following estimate holds: ‖𝑢(𝑡,⋅)‖²_𝐻^𝑠+‖𝑣(𝑡,⋅)‖²_𝐻^𝑠≤𝐾𝑒^𝑘𝑡,𝑡∈[0,𝑇). (25)
This inequality, Sobolev's embedding theorem, and Theorem 2 guarantee that the solution does not blow up in finite time.
On the other hand, we see that, if liminf_{𝑡→𝑇,𝑥∈ℝ}𝑢_𝑥(𝑡,𝑥)=−∞orlimsup_𝑡→𝑇𝛾_𝑥(𝑡,⋅)_𝐿∞=+∞, (26) then, by Sobolev's embedding theorem, the solution will blow up in finite time. This completes the proof of the theorem.

4. Blow-Up Results and Blow-Up Rate Estimate

In this section, we investigate the blow-up phenomena of strong solutions to (6). We now present the first blow-up result.

Lemma 6. Let 𝑧₀=(𝑢₀,𝛾₀)∈𝐻^𝑠×𝐻^𝑠,𝑠>5/2, and let 𝑇 be the maximal existence time of the solution 𝑧=(𝑢,𝛾) to (6) with the initial data 𝑧₀. Then, for all 𝑡∈[0,𝑇), one has ‖𝑢(𝑡,⋅)‖²_𝐻1+𝛾(𝑡,⋅)²_𝐻1=𝑒^−2𝜆𝑡𝑢₀²_𝐻1+𝛾₀²_𝐻1. (27) Moreover, ‖𝑢(𝑡,⋅)‖_𝐿^∞≤√22‖𝑢(𝑡,⋅)‖_𝐻¹≤√22𝑢₀²_𝐻1+𝛾₀²_𝐻1^1/2,𝛾(𝑡,⋅)_𝐿∞≤√22𝛾(𝑡,⋅)_𝐻1≤√22𝑢₀²_𝐻1+𝛾₀²_𝐻1^1/2. (28)

Proof. Denote 𝑓𝑢,𝛾=𝑢²+12𝑢²_𝑥+12𝛾²−12𝛾²_𝑥,𝑔=𝑔𝑢,𝛾=𝑢_𝑥𝛾_𝑥𝑥+𝑢_𝑥𝛾. (29) In view of the identity −𝜕²_𝑥𝑝∗𝑓=𝑓−𝑝∗𝑓, we can obtain, from (6), 𝑢_𝑡𝑥=−𝑢²_𝑥−𝑢𝑢_𝑥𝑥𝑣+𝑓−𝑝∗𝑓,_𝑡𝑥=−𝑢_𝑥𝛾_𝑥−𝑢𝛾_𝑥𝑥−𝜕_𝑥𝑝∗𝑔. (30) Therefore, an integration by parts yields 12𝑑‖𝑢‖𝑑𝑡^2𝐻1+𝛾²_𝐻1=_ℝ𝑢𝑢_𝑡+𝑢_𝑥𝑢_𝑡𝑥+𝛾𝛾_𝑡+𝛾_𝑥𝛾_𝑡𝑥=𝑑𝑥_ℝ𝑢−𝑢𝑢_𝑥−𝜕²_𝑥𝑝∗𝑓−𝜆𝑢+𝑢_𝑥−𝑢²_𝑥−𝑢𝑢_𝑥𝑥+𝑓−𝑝∗𝑓−𝜆𝑢_𝑥+𝛾−𝑢𝛾_𝑥−𝜆𝛾+𝛾_𝑥−𝑢𝛾_𝑥−𝑢𝛾_𝑥𝑥−𝜕_𝑥𝑝∗𝑔−𝜆𝛾_𝑥=𝑑𝑥_ℝ−12𝑢³_𝑥+𝑢_𝑥𝑢²+12𝑢²_𝑥+12𝛾²−12𝛾²_𝑥−𝑢𝛾𝛾_𝑥−𝛾𝑢_𝑥𝑥𝛾_𝑥+𝑢_𝑥𝛾−𝑢𝛾_𝑥𝛾²_𝑥−𝑢𝛾_𝑥𝛾_𝑥𝑥−𝜆𝑢²+𝑢²_𝑥+𝛾²+𝛾²_𝑥𝑑𝑥=−𝜆_ℝ𝑢²+𝑢²_𝑥+𝛾²+𝛾²_𝑥𝑑𝑥. (31) Thus, the statement of the conservation law follows. The remaining part of this lemma can be easily deduced from the conservation law. The proof of the lemma is complete.

Lemma 7 (see [37]). Let 𝑇>0 and 𝑣∈𝐶¹([0,𝑇);𝐻²). Then, for every 𝑡∈[0,𝑇), there exists at least one point 𝜉∈𝑅 with 𝑚(𝑡)∶=inf_𝑥∈𝑅𝑣_𝑥(𝑡,𝑥)=𝑣_𝑥(𝑡,𝜉(𝑡)). (32) The function 𝑚(𝑡) is absolutely continuous on (0,𝑇) with 𝑑𝑚𝑑𝑡=𝑣_𝑡𝑥(𝑡,𝜉(𝑡)),𝑎.𝑒.,𝑜𝑛(0,𝑇). (33)

Theorem 8. Let 𝑧₀=(𝑢₀,𝛾₀)∈𝐻^𝑠×𝐻^𝑠,𝑠>5/2, and let 𝑇 be the maximal existence time of the solution 𝑧=(𝑢,𝛾) to the (6) with the initial data 𝑧₀. If there exists some 𝑥₀∈ℝ such that 𝑢^′₀𝑥₀<−𝜆−𝜆²+‖𝑢₀‖²_𝐻1+‖𝛾₀‖²_𝐻1^1/2, (34) then the existence time 𝑇 is finite and the slope of 𝑢 tends to negative infinity as 𝑡 goes to 𝑇 while 𝑢 remains uniformly bounded on [0,𝑇].

Proof. As mentioned earlier, here we only need to show that the previous theorem holds for 𝑠=3. Differentiating the first equation of (6) with respect to 𝑥, in view of 𝜕²_𝑥𝑝∗𝑓=𝑝∗𝑓−𝑓, we have 𝑢_𝑡𝑥+𝑢𝑢_𝑥𝑥1=−2𝑢²_𝑥+𝑢²+12𝛾²−12𝛾²_𝑥−𝑝∗𝑢²+12𝑢²_𝑥+12𝛾²−12𝛾²_𝑥−𝜆𝑢_𝑥. (35) Note that 𝑑𝑢_𝑥𝑡,𝑞(𝑡,𝑥)𝑑𝑡=𝑢_𝑥𝑡𝑡,𝑞(𝑡,𝑥)+𝑢_𝑥𝑥𝑡,𝑞(𝑡,𝑥)𝑞_𝑡(𝑡,𝑥)=𝑢_𝑥𝑡𝑡,𝑞(𝑡,𝑥)+𝑢𝑡,𝑞(𝑡𝑥)𝑢_𝑥𝑥𝑡,𝑞(𝑡,𝑥). (36) We know that 𝑝∗(𝑢²+(1/2)𝑢²_𝑥)≥(1/2)𝑢²and 𝑝∗𝛾²_𝑥𝐿∞≤𝑝_𝐿∞𝛾²_𝑥𝐿1=12𝛾²_𝑥𝐿1. (37) By (35) and (36) and the previous estimates, we deduce that 𝑑𝑢_𝑥𝑡,𝑞(𝑡,𝑥)1𝑑𝑡≤−2𝑢²_𝑥()+1𝑡,𝑞𝑡,𝑥2𝑢²()+1𝑡,𝑞𝑡,𝑥4𝛾²(+3𝑡,𝑞𝑡,𝑥)4𝑝∗𝛾²_𝑥𝑡,𝑞(𝑡,𝑥)−𝜆𝑢_𝑥1≤−2𝑢²_𝑥()+1𝑡,𝑞𝑡,𝑥2𝑢²()+1𝑡,𝑞𝑡,𝑥4𝛾²(+3𝑡,𝑞𝑡,𝑥)8𝛾_𝑥𝐿1−𝜆𝑢_𝑥1≤−2𝑢²_𝑥()+1𝑡,𝑞𝑡,𝑥4‖𝑢‖²_𝐻1+18𝛾²_𝐻1+38𝛾²_𝑥𝐿1−𝜆𝑢_𝑥1≤−2𝑢²_𝑥()+1𝑡,𝑞𝑡,𝑥2‖𝑢‖²_𝐻1+𝛾²_𝐻1−𝜆𝑢_𝑥1=−2𝑢²_𝑥𝑡,𝑞(𝑡,𝑥)−𝜆𝑢_𝑥+12𝑢₀²_𝐻1+𝛾₀²_𝐻1, (38) in view of Lemma 6. Take √𝐾=22𝑢₀²_𝐻1+𝛾₀²_𝐻1^1/2 (39) and define 𝑔(𝑡)=𝑢_𝑥(𝑡,𝑞(𝑡,𝑥₀)). It then follows from (38) that on [0,𝑇), 𝑔^′(𝑡)1≤−2𝑔²(𝑡)−𝜆𝑔+𝐾²1=−2𝑔(𝑡)+𝜆+𝜆²+2𝐾²×𝑔(𝑡)+𝜆−𝜆²+2𝐾². (40) Note that if 𝑔(0)≤−𝜆−𝜆²+2𝐾², then 𝑔(𝑡)≤−𝜆−𝜆²+2𝐾², forall𝑡∈[0,𝑇). Therefore, we can solve the previous inequality to obtain 𝑔(0)+𝜆+𝜆²+2𝐾²𝑔(0)+𝜆−𝜆²+2𝐾²𝑒^{𝜆2+2𝐾2𝑡}≤2−1𝜆²+2𝐾²𝑔(𝑡)+𝜆−𝜆²+2𝐾². (41) Due to 0<(𝑔(0)+𝜆+𝜆²+2𝐾²)/(𝑔(0)+𝜆−𝜆²+2𝐾²)<1, then there exists 𝑇, and 0<𝑇<(1/𝜆²+2𝐾²)ln((𝑔(0)+𝜆+𝜆²+2𝐾²)/(𝑔(0)+𝜆−𝜆²+2𝐾²)), such that lim_𝑡→𝑇𝑔(𝑡)=−∞. Applying Theorem 5, the solution 𝑧 does not exist globally in time.

Next, we give a blow-up result if 𝑢₀ and 𝛾₀ are odd.

Theorem 9. Let 𝑧₀=(𝑢₀,𝛾₀)∈𝐻^𝑠×𝐻^𝑠,𝑠>5/2, and let 𝑇 be the maximal existence time of the solution 𝑧=(𝑢,𝛾) to (6) with the initial data 𝑧₀. If 𝑢₀ and 𝛾₀ are odd, and furthermore 𝑢^′₀𝑥₀<−𝜆−𝜆²+12𝑢₀²_𝐻1+𝛾₀²_𝐻1^1/2, (42) then 𝑇 is finite and 𝑢_𝑥(𝑡,0)→−∞ as 𝑡 goes to 𝑇.

Proof. As mentioned earlier, here we only need to show that the previous theorem holds for 𝑠=3. Note that (6) is the invariant under the transformation(𝑢,𝑥)→(−𝑢,−𝑥) and (𝛾,𝑥)→(−𝛾,−𝑥). Thus, we deduce that if 𝑢₀(𝑥) and 𝛾₀(𝑥) are odd, then 𝑢(𝑡,𝑥) and 𝛾(𝑡,𝑥) are odd for any 𝑡∈[0,𝑇). By continuity with respect to 𝑥 of 𝑧 and 𝑧_𝑥𝑥, we have 𝑢(𝑡,0)=𝑢_𝑥𝑥(𝑡,0)=𝛾(𝑡,0)=𝛾_𝑥𝑥(𝑡,0)=0,∀𝑡∈[0,𝑇). (43) Hence, in view of (35) and Lemma 6, we obtain 𝑢_𝑡𝑥()1𝑡,0=−2𝑢²_𝑥()−1𝑡,02𝛾²_𝑥−𝑝∗𝑢²+12𝑢_𝑥+12𝛾²−12𝛾²_𝑥(𝑡,0)−𝜆𝑢_𝑥1≤−2𝑢²_𝑥()+1𝑡,02𝑝∗𝛾²_𝑥(𝑡,0)−𝜆𝑢_𝑥1≤−2𝑢²_𝑥(𝑡,0)−𝜆𝑢_𝑥+14𝑢₀²_𝐻1+𝛾₀²_𝐻1. (44) Take 1𝐾=2𝑢₀²_𝐻1+𝛾₀²_𝐻1^1/2 (45) and define 𝑔(𝑡)=𝑢_𝑥(𝑡,𝑞(𝑡,𝑥₀)). It then follows from (38) that on [0,𝑇), 𝑔^′(𝑡)1≤−2𝑔²(𝑡)−𝜆𝑔+𝐾²1=−2𝑔(𝑡)+𝜆+𝜆²+2𝐾²×𝑔(𝑡)+𝜆−𝜆²+2𝐾². (46) Note that if 𝑔(0)≤−𝜆−𝜆²+2𝐾², then 𝑔(𝑡)≤−𝜆−𝜆²+2𝐾², for all 𝑡∈[0,𝑇). Therefore, we can solve the previous inequality to obtain 𝑔(0)+𝜆+𝜆²+2𝐾²𝑔(0)+𝜆−𝜆²+2𝐾²𝑒^{𝜆2+2𝐾2𝑡}≤2−1𝜆²+2𝐾²𝑔(𝑡)+𝜆−𝜆²+2𝐾². (47) Due to 0<(𝑔(0)+𝜆+𝜆²+2𝐾²)/(𝑔(0)+𝜆−𝜆²+2𝐾²)<1, then there exists 𝑇, and 0<𝑇<(1/𝜆²+2𝐾²)ln((𝑔(0)+𝜆+𝜆²+2𝐾²)/(𝑔(0)+𝜆−𝜆²+2𝐾²)), such that lim_𝑡→𝑇𝑔(𝑡)=−∞. Applying Theorem 5, the solution 𝑧 does not exist globally in time.

Next, we give more insight into the blow-up rate for the wave-breaking solutions to (6).

Theorem 10. Let 𝑧₀=(𝑢₀,𝛾₀)∈𝐻^𝑠×𝐻^𝑠,𝑠≥5/2,𝑧=(𝑢,𝛾) be the corresponding solution to (6) with initial data 𝑧₀ and satisfy ‖𝛾_𝑥(𝑡,𝑥)‖_𝐿^∞≤𝑀,for all (𝑡,𝑥)∈[0,𝑇)×𝑅, and 𝑇 be the maximal existence time of the solution. Then let one has lim_𝑡→𝑇inf_𝑥∈ℝ𝑢_𝑥(𝑡,𝑥)(𝑇−𝑡)=−2. (48)

Proof. By Lemma 6, we get the uniform bound of 𝑢. Set 𝑚(𝑡)=inf_𝑥∈ℝ𝑢_𝑥(𝑡,𝑥). By the proof of Theorem 8 (or Theorem 9), we find a constant 𝐾>0 such that 𝑔^′(𝑡)+12𝑔(𝑡)+𝜆𝑔(𝑡)≤𝐾, (49) where 𝐾 depends only on ‖𝑢₀‖_𝐻1 and ‖𝛾₀‖_𝐻1. It follows that 1−𝐾−2𝜆²≤𝑔^′(𝑡)+12𝑔(𝑡)+𝜆²1≤𝐾+2𝜆²,a.e.,on(0,𝑇). (50) Choose 𝜖∈(0,1/2). Since liminf_𝑟→𝑇(𝑦(𝑡)+𝜆)=−∞ by Theorem 5, there is some 𝑡₀∈(0,𝑇) with 𝑔(𝑡₀)+𝜆<0 and (𝑔(𝑡₀)+𝜆)²>(𝐾+(1/2)𝜆²)/𝜖. Let us first prove that 𝑔(𝑡)+𝜆²>1𝜖1𝐾+2𝜆²,𝑡∈𝑡₀,𝑇. (51) Since 𝑔 is locally Lipschitz, there is some 𝛿>0 such that 𝑔(𝑡)+𝜆²>1𝜖1𝐾+2𝜆²,𝑡∈𝑡₀,𝑡₀+𝛿. (52)
Note that 𝑔 is locally Lipschitz and therefore absolutely continuous. Integrating the previous relation on (𝑡₀,𝑡₀+𝛿) yields that 𝑔𝑡₀+𝛿+𝜆≤𝑔𝑡₀+𝜆<0. (53) It follows from the previous inequality that 𝑔(𝑡₀+𝛿)+𝜆²≥𝑔𝑡₀+𝜆²>1𝜖1𝐾+2𝜆². (54) By (50)-(51), we infer that 12𝑔−𝜖≤−^′(𝑡)(𝑚+𝜆)²≤12+𝜖,a.e.,on(0,𝑇). (55) For 𝑡∈(𝑡₀,𝑇), integrating (55) on  (𝑡,𝑇) to get 12()1−𝜖𝑇−𝑡≤−𝑔(𝑡)≤1+𝜆2+𝜖(𝑇−𝑡),𝑡∈𝑡₀,𝑇. (56) Since 𝑔(𝑡)+𝜆<0 on [𝑡₀,𝑇), it follows that 1()𝑔(𝑡)()≤11/2+𝜖≤−+𝜆𝑇−𝑡()𝑡1/2+𝜖,𝑡∈₀,𝑇. (57) By the arbitrariness of 𝜖∈(0,1/2), the statement of the theorem follows.

5. Asymptotic Profile

In this section, we focus on the persistence property of the solution to (6) in 𝐿^∞-space. Precisely, we give an asymptotic description on how the solutions behave under the initial values possess algebraic decay at infinity. Recently, the asymptotic behavior for the celebrated Camassa-Holm equation was investigated in [38]. We notice that in [39], the authors showed that the solution of the Camassa-Holm equation and its first-order spatial derivative retain exponential decay at infinity as their initial values behave. After all, the exponential decay of initial value is a faster way; this motivates us to establish the decay rate of solution if its initial value decays algebraically. We show that the strong solution of (6) corresponding to initial data with a slower algebraically decaying way will keep this behavior in the 𝑥-variable at infinity in its lifespan. In order to achieve our result, we first recall the following lemma.

Lemma 11 (see [40]). For a function Φ_𝑁(𝑋) defined next, there exists a constant 𝐶_𝜃 which only depends on 𝜃∈(0,1], such that for any positive integer 𝑁≥2Φ_𝑁(𝑥)_𝑅𝑒^{−|𝑥−𝑦|}1Φ_𝑁𝑦𝑑𝑦≤𝐶_𝜃, (58) where Φ_𝑁(𝑥)=𝑥1,𝑥≤1,_𝜃(),𝑁,𝑥∈1,𝑁_𝜃,𝑋≥𝑁. (59)

Theorem 12. Assume that 𝑋₀(𝑥)=(𝑢₀(𝑥),𝛾₀(𝑥))^𝑇∈𝐻^𝑠×𝐻^𝑠 with 𝑠>5/2 satisfies that for some 𝜃∈(0,1]𝑋₀(𝑥),𝑋_0𝑥(𝑥)∼𝑂𝑥^−𝜃𝑎𝑠𝑥↑∞. (60) Then, the corresponding strong solution 𝑋(𝑥)=(𝑢(𝑥),𝛾(𝑥))^𝑇∈𝐶([0,𝑇);𝐻^𝑠×𝐻^𝑠) to (6) satisfies that |𝑋(𝑥)|,𝑋_𝑥(𝑥)∼𝑂𝑥^−𝜃𝑎𝑠𝑥↑∞, (61) uniformly in the time interval [0,𝑇).

Notation. One has 𝑓(𝑥)∼𝑂𝑔(𝑥)as𝑥↑∞iflim_𝑥→∞𝑓(𝑥)𝑔(𝑥)=𝐿, (62)
where 𝐿 is a nonnegative constant. In order to shorten the presentation in the sequel, we introduce 𝐹𝑢,𝛾=𝑢²+12𝑢²_𝑥+12𝛾²−12𝛾²_𝑥,𝐻𝑢,𝛾=𝑢_𝑥𝛾_𝑥𝑥+𝑢_𝑥𝛾. (63)

Proof. The first step is devoted to giving estimates on ‖𝑢(𝑥,𝑡)‖_∞ and ‖𝛾(𝑥,𝑡)‖_∞, where ‖⋅‖_𝑝 is the standard 𝐿^𝑝(ℝ) norm.
Multiplying the first equation of (6) by 𝑢^2𝑛−1 with 𝑛∈𝑍⁺ and integrating both sides with respect to 𝑥 variable, we obtain _ℝ𝑢^2𝑛−1𝑢_𝑡𝑑𝑥+_ℝ𝑢^2𝑛−1𝑢𝑢_𝑥+𝑑𝑥_ℝ𝑢^2𝑛−1𝜕_𝑥(𝐺∗𝐹)𝑑𝑥+𝜆_ℝ𝑢^2𝑛𝑑𝑥=0. (64) The first term in (64) is _ℝ𝑢^2𝑛−1𝑢_𝑡1𝑑𝑥=𝑑2𝑛‖𝑢‖𝑑𝑡^2𝑛2𝑛=‖𝑢‖^2𝑛−1_2𝑛𝑑‖𝑢‖𝑑𝑡^2𝑛; (65) for the second term of (64), we have _ℝ𝑢^2𝑛−1𝑢𝑢_𝑥𝑑𝑥≤𝑢_𝑥∞‖𝑢‖^2𝑛_2𝑛. (66) It follows from the Höder inequality that _ℝ𝑢^2𝑛−1𝜕_𝑥(𝐺∗𝐹(𝑢))𝑑𝑥≤‖𝑢‖^2𝑛−1_2𝑛𝜕_𝑥(𝐺∗𝐹)_2𝑚. (67) Therefore, 𝑑‖𝑢‖𝑑𝑡^2𝑛≤𝑢_𝑥∞+𝜆‖𝑢‖_2𝑛+𝜕_𝑥(𝐺∗𝐹)_2𝑛. (68) Similarly, for the estimate of ‖𝛾(𝑥)‖_∞, we have by multiplying 𝛾^2𝑛−1 and integration _ℝ𝛾^2𝑛−1𝛾_𝑡𝑑𝑥+_ℝ𝛾^2𝑛−1𝑢𝛾_𝑥+𝑑𝑥_ℝ𝛾^2𝑛−1(𝐺∗𝐻)𝑑𝑥+𝜆_ℝ𝛾^2𝑛−1𝑑𝑑𝑥=0,𝛾𝑑𝑡^2𝑛≤𝛾_𝑥∞+𝜆‖𝑢‖_2𝑛+‖𝐺∗𝐻‖_2𝑛. (69) By the Sobolev embedding theorem, there exists a constant 𝑀>0 such that 𝑑𝛾𝑑𝑡^2𝑛+‖𝑢‖_2𝑛≤𝑀𝛾_2𝑛+‖𝑢‖_2𝑛+𝜕_𝑥(𝐺∗𝐻)_2𝑛+‖𝐺∗𝐻‖_2𝑛. (70) In view of Gronwall's inequality, we have the estimate 𝛾_2𝑛+‖𝑢‖_2𝑛≤𝑒^𝑀𝑡𝛾_02𝑛+𝑢_02𝑛+^𝑡₀𝜕_𝑥(𝐺∗𝐻)_2𝑛+‖𝐺∗𝐻‖_2𝑛𝑑𝜏. (71) We can take limits as 𝑁 goes to infinity to obtain 𝛾_∞+‖𝑢‖_∞≤𝑒^𝑀𝑡𝛾_0∞+𝑢_0∞+^𝑡₀𝜕_𝑥(𝐺∗𝐻)_∞+‖𝐺∗𝐻‖_∞𝑑𝜏. (72)
The second step is to establish estimates for ‖𝑢_𝑥‖_∞ and ‖𝛾_𝑥‖_∞ by using the same method as previously mentioned. Differentiating the first equation of (6) with respect to 𝑥 produces the following equation: 𝑢_𝑥𝑡+𝑢²_𝑥+𝑢𝑢_𝑥𝑥+𝜕²_𝑥(𝐺∗𝐹)+𝜆𝑢_𝑥=0. (73) Multiplying (73) by 𝑢_𝑥^2𝑛−1, and then integrating by parts, one obtains _ℝ𝑢_𝑥^2𝑛−1𝑢_𝑥𝑡𝑑𝑥+_ℝ𝑢_𝑥^2𝑛+11𝑑𝑥−2𝑛_ℝ𝑢_𝑥^2𝑛𝑢_𝑥+_ℝ𝑢_𝑥^2𝑛−1𝜕²_𝑥(𝐺∗𝐹)𝑑𝑥+𝜆_ℝ𝑢_𝑥^2𝑛=0. (74) Similarly, one can obtain 𝑢_𝑥∞≤𝑒^𝑀𝑡𝑢_0𝑥∞+^𝑡₀𝜕²_𝑥(𝐺∗𝐹)_∞𝑑𝜏. (75) For the second equation of (6), we may get _ℝ𝛾_𝑥^2𝑛−1𝛾_𝑥𝑡𝑑𝑥+_ℝ𝑢_𝑥𝛾_𝑥^2𝑛+𝑑𝑥_ℝ𝑢𝛾_𝑥𝑥𝛾_𝑥^2𝑛−1𝑑𝑥+_ℝ𝛾_𝑥^2𝑛−1𝜕_𝑥(𝐺∗𝐻)𝑑𝑥+𝜆∫_ℝ𝛾_𝑥^2𝑛𝑑=0,𝛾𝑑𝑡^𝑥2𝑛≤𝑢_𝑥∞+𝜆𝛾_𝑥2𝑛+𝛾_𝑥𝑥∞‖𝑢‖_2𝑛+𝜕_𝑥(𝐺∗𝐻)_2𝑛, (76) and by Gronwall's inequality 𝛾_𝑥∞≤𝑒^𝑀𝑡𝛾_0𝑥∞+^𝑡₀‖𝑢‖_∞+𝜕_𝑥(𝐺∗𝐻)_∞𝑑𝜏. (77) In order to arrive at our result, we introduce a weighted continuous function which is independent on 𝑡 as follows: Φ_𝑁(𝑥)=𝑥1,𝑥≤1,_𝜃(),𝑁,𝑥∈1,𝑁_𝜃,𝑋≥𝑁, (78) where 𝜃∈(0,1],𝑁∈𝑍⁺,𝑁>2. It is trivial that 0≤Φ^′_𝑁(𝑥)≤Φ_𝑁(𝑥),a.e.,𝑥∈𝑅, (79) where the derivative is with respect to the variable 𝑥. From the first equation of (6) and (73), we have Φ_𝑁𝑢_𝑡+Φ_𝑁𝑢𝑢_𝑥+Φ_𝑁𝜕_𝑥(𝐺∗𝐹)+𝜆Φ_𝑁Φ𝑢=0,_𝑁𝑢_𝑥𝑡+Φ_𝑁𝑢²_𝑥+Φ_𝑁𝑢𝑢_𝑥𝑥+Φ_𝑁𝜕²_𝑥(𝐺∗𝐹)+𝜆Φ_𝑁𝑢_𝑥=0. (80)(81) Next, in order to obtain the estimates on ‖𝑢Φ_𝑁‖_∞ and ‖𝑢_𝑥Φ_𝑁‖_∞, we apply a similar technique that was used before for ‖𝑢(𝑥,𝑡)‖_∞ and ‖𝑢_𝑥(𝑥,𝑡)‖_∞ step by step to (80) and (81). For (81), we need to eliminate the term with the second-order derivative in order to attain the estimate for 𝑢_𝑥Φ_𝑁. Using integration by parts, we obtain _ℝ𝑢_𝑥Φ_𝑁^2𝑛−1Φ_𝑁𝑢𝑢_𝑥𝑥=𝑑𝑥_𝑅𝑢_𝑥Φ_𝑁^2𝑛−1𝑢Φ_𝑁𝑢_𝑥𝑥−Φ^′_𝑁𝑢_𝑥=𝑑𝑥_ℝ𝑢(𝑢_𝑥Φ_𝑁)^2𝑛2𝑛_𝑥𝑑𝑥−_ℝ𝑢𝑢_𝑥Φ_𝑁^2𝑛−1Φ^′_𝑁𝑢_𝑥𝑢𝑑𝑥≤2_𝑥∞+‖𝑢‖_∞𝑢_𝑥Φ_𝑁^2𝑛_2𝑛, (82) where (79) is used directly. Therefore, with these preparations, it holds that 𝑢Φ_𝑁∞+𝑢_𝑥Φ_𝑁∞≤𝑒^2𝑀𝑡𝑢₀Φ_𝑁∞+𝑢_0𝑥Φ_𝑁∞+𝑒^{2𝑀𝑡𝑡}₀Φ_𝑁𝜕_𝑥(𝐺∗𝐻)_∞+Φ_𝑁𝜕²_𝑥(𝐺∗𝐹)_∞𝑑𝜏. (83) For the second equation of (6), we have Φ_𝑁𝛾_𝑡+Φ_𝑁𝑢𝛾_𝑥+Φ_𝑁Φ𝐺∗𝐻+𝜆𝛾=0,_𝑁𝛾_𝑥𝑡+Φ_𝑁𝑢_𝑥𝛾_𝑥+Φ_𝑁𝑢𝛾_𝑥𝑥+Φ_𝑁𝜕_𝑥𝐺∗𝐻+𝜆𝛾_𝑥=0. (84) Therefore, 𝑑𝑑𝑡𝛾Φ_𝑁2𝑛≤‖𝑢‖_∞Φ_𝑁𝛾_2𝑛+1𝑢2𝑛^𝑥∞𝛾Φ_𝑁2𝑛+Φ_𝑁𝐺∗𝐻_2𝑛,𝑑𝛾𝑑𝑡^𝑥Φ_𝑁2𝑛≤3‖𝑢‖_∞+𝑢_𝑥∞𝛾_𝑥Φ_𝑁2𝑛+Φ_𝑁𝜕_𝑥(𝐺∗𝐻)_2𝑛. (85) Then by (85), we obtain 𝛾Φ_𝑁∞+𝛾_𝑥Φ_𝑁∞≤𝑒^3𝑀𝑡𝛾₀Φ_𝑁∞+𝛾_0𝑥Φ_𝑁∞+𝑒^{3𝑀𝑡𝑡}₀Φ_𝑁𝜕_𝑥(𝐺∗𝐻)_∞+Φ_𝑁𝜕²_𝑥(𝐺∗𝐹)_∞𝑑𝜏. (86) On the other hand, for a suitable function 𝑓, one obtains, due to Lemma 11, Φ_𝑁𝜕_𝑥𝐺∗𝑓²≤Φ_𝑁(𝑥)_𝑅𝑒^{−|𝑥−𝑦|}1Φ_𝑁𝑦Φ_𝑁𝑦𝑓𝑦𝑓𝑦≤𝑑𝑦𝑓Φ_𝑁∞𝑓_∞Φ_𝑁(𝑥)_𝑅𝑒^{−|𝑥−𝑦|}1Φ_𝑁𝑦𝑑𝑦≤𝐶_𝜃𝑓Φ_𝑁∞𝑓_∞. (87) Similarly, Φ_𝑁𝜕²_𝑥𝐺∗𝑓²≤𝐶_𝜃𝑓Φ_𝑁∞𝑓_∞. (88) Note that there are two additional quantities in (86) to be dealt with. Let us estimate ‖Φ_𝑁(𝐺∗𝐻)‖_∞ first. One has Φ_𝑁(=1𝐺∗𝐻)2Φ_𝑁(𝑥)_𝑅𝑒^{−|𝑥−𝑦|}𝑢_𝑥𝛾_𝑥𝑥+𝑢_𝑥𝛾𝑑𝑦≤𝐶_𝜃𝑢_𝑥Φ_𝑁∞𝛾_∞+𝛾_𝑥∞. (89) It is similar to the remaining ‖Φ_𝑁𝜕_𝑥(𝐺∗𝐻)‖_∞. Then, combining (87)–(89) with (83) and (86), it follows that there exists a constant 𝐶=𝐶(𝑀,𝑇)>0 such that Γ(𝑡)≤𝐶Γ(0)+𝐶^𝑡₀𝜁(𝜏)Γ(𝜏)≤𝐶Γ(0)+𝑑𝜏_𝑡0Γ(𝜏)𝑑𝜏, (90) where Γ(𝑡)=𝑢Φ_𝑁∞+𝑢_𝑥Φ_𝑁∞+𝛾Φ_𝑁∞+𝛾_𝑥Φ_𝑁‖_∞,𝜁(𝑡)=‖𝑢‖_∞+𝑢_𝑥∞+𝛾_∞+𝛾_𝑥∞ (91) are introduced just for simplicity. Next for any 𝑁∈𝑍⁺,𝑡∈[0,𝑇], and 𝑥>0, we have, by Gronwall's inequality, Γ(𝑡)≤𝐶₀Γ(0)≤𝐶₀𝑢₀(𝑥)𝑥^𝜃_∞+𝑢_0𝑥(𝑥)𝑥^𝜃_∞+𝛾₀(𝑥)𝑥^𝜃_∞+𝛾_0𝑥(𝑥)𝑥^𝜃_∞. (92) Finally, passing limit as 𝑁 goes to infinity in the previous inequality, we obtain 𝑢(𝑥,𝑡)𝑥^𝜃_∞+𝑢_𝑥(𝑥,𝑡)𝑥^𝜃_∞+𝛾(𝑥,𝑡)𝑥^𝜃_∞+𝛾_𝑥(𝑥,𝑡)𝑥^𝜃_∞≤𝑢₀(𝑥)𝑥^𝜃_∞+𝑢_0𝑥(𝑥)𝑥^𝜃_∞+𝛾₀(𝑥)𝑥^𝜃_∞+𝛾₀(𝑥)𝑥^𝜃_∞. (93) We complete the proof.

Acknowledgments

The authors would like to say thanks to Professor Zhaoyang Yin and Professor Yong Zhou for sending several of their papers to the authors. This work was partially supported by NSF of China (11371384), NSF of Chongqing (cstc2013jcyjA0940), and NSF of Fuling (FLKJ, 2013ABA2036).