Blow-up Phenomena and Persistence Properties of Solutions to the Two-Component DGH Equation

Abstract

This paper is concerned with blow-up phenomena and persistence properties for an integrable two-component Dullin-Gottwald-Holm shallow water system. We give sufficient conditions on the initial data which guarantee blow-up phenomena of solutions in finite time for both periodic and nonperiodic cases, respectively. Furthermore, the persistence properties of solutions to the system are investigated.

1. Introduction

In 2001, Dullin et al. [1] derived a new equation by using the method of asymptotic analysis and a near-identity normal form transformation in water wave theory to describe the unidirectional propagation of surface waves in a shallow water regime, it reads as follows: 𝑚_𝑡+2𝜔𝑢_𝑥+𝑢𝑚_𝑥+2𝑚𝑢_𝑥=−𝛾𝑢_𝑥𝑥𝑥,𝑥∈ℝ,𝑡>0. (1) Since (1) was derived by Dullin et al., we call it DGH equation. Let 𝑚=𝑢−𝛼²𝑢_𝑥𝑥; then we can rewrite (1) as the following form: 𝑢_𝑡−𝛼²𝑢_𝑥𝑥𝑡+2𝜔𝑢_𝑥+3𝑢𝑢_𝑥+𝛾𝑢_𝑥𝑥𝑥=𝛼²2𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥, (2) where 𝛼² and 𝛾/2𝜔 are squares of length scales, the constant 𝜔=𝑔ℎ/2 is the critical shallow water speed for undisturbed water at rest at spatial infinity, ℎ is the mean fluid depth, and 𝑔 is the gravitational constant. This equation is connected with two separately integrable soliton equations for shallow water waves.

When 𝛼²=0, (2) becomes the KdV equation as follows: 𝑢_𝑡+2𝜔𝑢_𝑥+3𝑢𝑢_𝑥=−𝛾𝑢_𝑥𝑥𝑥. (3) Bourgain proved that solutions to the KdV equation are global as long as the initial data is square integrable [2], which is also shown in [3]. Another remarkable property is that it is integrable and the solitary waves are nonlinearly stable, and when 𝜔=0, there exists a smooth soliton solution [4].

Instead, taking 𝛾=0 in (2), it turns out to be the Camassa-Holm equation: 𝑢_𝑡−𝛼²𝑢_𝑥𝑥𝑡+2𝜔𝑢_𝑥+3𝑢𝑢_𝑥=𝛼²2𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥, (4) which was derived by Camassa and Holm in [5] by approximating directly the Hamiltonian for Euler’s equations in the shallow water regime in 1993. Many researches have been carried out on the Camassa-Holm equation in recent years. Local well-posedness has been proven by several authors; see [6–9]. Blow-up phenomena have been investigated in [10–15]. Besides, in [11], global solutions were also discussed. In [16], global existence of weak solutions was proved, but uniqueness was obtained only under a priori assumption. Furthermore, the authors in [17] studied the persistence properties of solution to the Camassa-Holm equation. In [18–20], the soliton solution was studied.

It is very interesting that (2) still preserves the bi-Hamiltonian structure and complete integrability. Indeed, (2) can be rewritten in two compatible Hamiltonian forms in terms of 𝑚=𝑢−𝛼²𝑢_𝑥𝑥 [1] as follows, 𝑚_𝑡=−𝐵₂𝛿𝐸𝛿𝑚=−𝐵₁𝛿𝐹,𝛿𝑚 (5) where 𝐵₁=𝜕_𝑥−𝛼²𝜕³_𝑥,𝐵₂=𝜕_𝑥(𝑚+𝜔)+(𝑚+𝜔)𝜕_𝑥+𝛾𝜕³_𝑥,𝐸(𝑢)=12𝑢²+𝛼²𝑢²_𝑥𝐹(𝑢)=1𝑑𝑥,2𝑢³+𝛼³𝑢𝑢²_𝑥+2𝜔𝑢²−𝛾𝑢²_𝑥𝑑𝑥. (6)𝐸(𝑢) and 𝐹(𝑢) are two conserved quantities.

In [21], Tian et al. studied the well-posedness of the Cauchy problem and the scattering problem of the DGH equation. The issue of passing to the limit as the dispersive parameter tends to zero for the solution of the DGH equation was investigated, and the convergence of the solutions to DGH equation as 𝛼²→0 was studied. Besides, data of the scattering problem for the equation could be explicitly expressed. The blow-up phenomenon of solutions to the DGH equation was the subject of [21–24].

In this paper, we are interested in the following Cauchy problem for an integrable two-component DGH equation. It reads as 𝑢_𝑡−𝛼²𝑢_𝑡𝑥𝑥+2𝜔𝑢_𝑥+3𝑢𝑢_𝑥+𝛾𝑢_𝑥𝑥𝑥=𝛼²2𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥−𝜎𝜌𝜌_𝑥,𝑥∈ℝ,𝑡>0,𝜌_𝑡+𝜌𝑢_𝑥𝑢()=0,𝑥∈ℝ,𝑡>0,0,𝑥=𝑢₀(𝑥),𝑥∈ℝ,𝜌(0,𝑥)=𝜌₀(𝑥),𝑥∈ℝ. (7) The variable 𝑢(𝑥,𝑡) describes the horizontal velocity of the fluid, and 𝜌(𝑥,𝑡) denotes the horizontal deviation of the surface from equilibrium, all measured in dimensionless units. This model can be derived by Constantin and Ivanov’s approach [25] from shallow water theory, which includes the two-component Camassa-Holm system [26–32] as its special case with 𝛼=1 and 𝛾=0 in (7). From a geometric point of view, (7) is the model for geodesic motion on the semidirect product Lie group of diffeomorphisms acting on densities, with respect to the 𝐻¹-norm of velocity and the 𝐿²-norm on the density. Mathematically, (7) admits not only breaking-wave solutions but also global in time solutions. In view of the context of hydrodynamics, we have the assumptions 𝑢→0, 𝜌→1 as 𝑥→∞, at any instant 𝑡. For convenience of later discussion, letting 𝜌=𝜌−1, 𝛼>0, and 𝜎=1, we have 𝜌→0 as 𝑥→∞. Then it follows that 𝑢_𝑡−𝛼²𝑢_𝑡𝑥𝑥+2𝜔𝑢_𝑥+3𝑢𝑢_𝑥+𝛾𝑢_𝑥𝑥𝑥−2𝛼²𝑢_𝑥𝑢_𝑥𝑥−𝛼²𝑢𝑢_𝑥𝑥𝑥+𝜌𝜌_𝑥+𝜌_𝑥𝜌=0,𝑥∈ℝ,𝑡>0,_𝑡+𝜌𝑢_𝑥+𝑢_𝑥𝑢()=0,𝑥∈ℝ,𝑡>0,0,𝑥=𝑢₀(𝑥)𝜌(),𝑥∈ℝ,0,𝑥=𝜌₀(𝑥),𝑥∈ℝ. (1) Define Λ=(1−𝛼²𝜕²_𝑥)⁻¹; then the operator Λ can be expressed by Λ𝑓=𝐺∗𝑓=_ℝ𝐺𝑥−𝑦𝑓𝑦𝑑𝑦, (9) where 𝐺(𝑥) is the Green function for Λ and ∗ denotes the convolution. It is easy to get 𝐺∗(1/2)=1/2. For the periodic case, 𝐺(𝑥)=cosh((𝑥/𝛼)−([𝑥]/𝛼)−(1/2𝛼))/2𝛼sinh(1/2𝛼) is the Green function for Λ in the unit circle. While, for the nonperiodic case, the Green function for Λ in ℝ is 𝐺(𝑥)=(1/2𝛼)𝑒^{−|𝑥/𝛼|}. Using this identity, we can rewrite (1) as the following nonlocal form: 𝑢_𝑡+𝑢_𝑥𝛾𝑢−𝛼²=−𝜕_𝑥𝐺∗𝑢²+𝛼²2𝑢²_𝑥+𝛾2𝜔+𝛼²1𝑢+2𝜌²,𝜌+𝜌𝑥∈ℝ,𝑡>0,_𝑡+𝜌𝑢_𝑥+𝑢_𝑥𝑢()=0,𝑥∈ℝ,𝑡>0,0,𝑥=𝑢₀(𝑥)𝜌(),𝑥∈ℝ,0,𝑥=𝜌₀(𝑥),𝑥∈ℝ. (10) We note that less deep results exist yet for this model except for the recent ones [33, 34]. Our motivation here is to explore some new blow-up conditions especially for the initial data were added on some different quantities. This paper is organized as follows. In Section 2, we recall some preliminary results on local well-posedness and blow-up scenario and present some useful conversation laws. In Section 3, we study the blow-up phenomenon of strong solutions in both periodic and nonperiodic cases. In Section 4, persistence properties of solutions to the system are studied.

2. Preliminaries

We can use Kato’s theory [35] to establish the following local well-posedness theorem for (10); its proof is referred to in the discussions of [33, 34], so we omit it here.

Theorem 1. Assume an initial data (𝑢₀,𝜌₀)∈𝐻^𝑠×𝐻^𝑠−1,𝑠≥2. Then there exists a maximal 𝑇=𝑇(‖(𝑢₀,𝜌₀)‖_{𝐻𝑠×𝐻𝑠−1})>0 and a unique solution 𝑢,𝜌∈𝐶[0,𝑇);𝐻^𝑠×𝐻^𝑠−1∩𝐶¹[0,𝑇);𝐻^𝑠−1×𝐻^𝑠−2 (11) of system (10). Moreover, the solution (𝑢,𝜌) depends continuously on the initial value (𝑢₀,𝜌₀), and the maximal time of existence 𝑇>0 is independent of 𝑠.

Next, we will give some useful conserved quantities, which are important to discuss the blow-up criteria.

Proposition 2. Suppose (𝑢₀,𝜌₀)∈𝐻^𝑠×𝐻^𝑠−1,𝑠≥2; then the solution (𝑢,𝜌) of the system (10) guaranteed by Theorem 1 satisfies _ℝ𝑢²+𝛼²𝑢²_𝑥+𝜌²𝑑𝑥=_ℝ𝑢²₀+𝛼²𝑢²_0𝑥+𝜌²₀𝑑𝑥. (12)

Proof. Consider 𝑑𝑑𝑡_ℝ𝑢²+𝛼²𝑢²_𝑥+𝜌²=𝑑𝑥_ℝ2𝑢𝑢_𝑡+2𝛼²𝑢_𝑥𝑢_𝑥𝑡+2𝜌𝜌_𝑡=𝑑𝑥_ℝ2𝑢𝑢_𝑡−𝛼²𝑢_𝑥𝑥𝑡+2𝜌𝜌_𝑡𝑑𝑥. (13)
From the first equation of (10), we have 2𝑢𝑢_ℝ𝑡−𝛼²𝑢_𝑥𝑥𝑡=𝑑𝑥_ℝ2𝑢−2𝜔𝑢_𝑥−3𝑢𝑢_𝑥−𝛾𝑢_𝑥𝑥𝑥+2𝛼²𝑢_𝑥𝑢_𝑥𝑥+𝛼²𝑢𝑢_𝑥𝑥𝑥−𝜌𝜌_𝑥−𝜌_𝑥𝑑𝑥. (14) Similarly, from the second equation of (10), we can get _ℝ2𝜌𝜌_𝑡𝑑𝑥=_ℝ−2𝜌𝜌𝑢_𝑥+𝑢_𝑥𝑑𝑥. (15) So we can obtain 𝑑𝑑𝑡_ℝ𝑢²+𝛼²𝑢²_𝑥+𝜌²=𝑑𝑥_ℝ2𝑢−2𝜔𝑢_𝑥−3𝑢𝑢_𝑥−𝛾𝑢_𝑥𝑥𝑥+2𝛼²𝑢_𝑥𝑢_𝑥𝑥+𝛼²𝑢𝑢_𝑥𝑥𝑥−𝜌𝜌_𝑥−𝜌_𝑥−2𝜌𝜌𝑢_𝑥+𝑢_𝑥=𝑑𝑥_ℝ−2𝜔𝑢²_𝑥−2𝑢³_𝑥+𝛾𝑢²_𝑥𝑥+2𝛼²𝑢²𝑢_𝑥𝑥𝑥−2𝜌𝑢_𝑥−2𝜌𝜌𝑢_𝑥𝑑𝑥=0. (16) This completes the proof.

We also note that the quantities ∫_ℝ𝑢(𝑥,𝑡)𝑑𝑥 and ∫_ℝ𝜌(𝑥,𝑡)𝑑𝑥 are also conserved for 𝜌₀(𝑥)∈𝐿¹, 𝑢₀(𝑥)∈𝐿¹. These conservation laws will play important roles in considering the behavior of solutions. In what follows, we denote ∫_ℝ(𝑢²+𝛼²𝑢²_𝑥+𝜌²)𝑑𝑥 by ‖(𝑢,𝜌)‖²_{𝐻1𝛼×𝐿2}.

Moreover, using the techniques in [36], one can get the following criterion for finite time wave breaking to (10).

Theorem 3 (see [34]). Let (𝑢₀,𝜌₀)∈𝐻^𝑠×𝐻^𝑠−1 with 𝑠≥2, and let 𝑇>0 be the maximal time of existence of the solution (𝑢,𝜌) to the model (10) with initial data (𝑢₀,𝜌₀). Then the corresponding solution (𝑢,𝜌) blows up in finite time if and only if lim_𝑡↑𝑇inf_𝑥∈ℝ𝑢_𝑥(𝑡,𝑥)=−∞. (17)

We will then need to introduce the standard particle trajectory method for later use. Now consider the following initial value problem: 𝑞_𝑡[),𝑞()=𝑢𝑡,𝑞,𝑡∈0,𝑇0,𝑥=𝑥,𝑥∈ℝ, (18) where 𝑢∈𝐶¹([0,𝑇),𝐻^𝑠−1) is the first component of the solution (𝑢,𝜌) to system (10) with initial data (𝑢₀,𝜌₀)∈𝐻^𝑠×𝐻^𝑠−1,(𝑠≥2), and 𝑇>0 is the maximal time of existence. By direct calculation, we have 𝑞_𝑡𝑥(𝑡,𝑥)=𝑢_𝑥𝑡,𝑞(𝑡,𝑥)𝑞_𝑥(𝑡,𝑥). (19) Then, 𝑞_𝑥(𝑡,𝑥)=exp^𝑡₀𝑢_𝑥𝜏,𝑞(𝜏,𝑥)𝑑𝜏>0,𝑡>0,𝑥∈ℝ, (20) which means that 𝑞(𝑡,⋅)∶ℝ→ℝ is a diffeomorphism of the line for every 𝑡∈[0,𝑇). Consequently, the 𝐿^∞-norm of any function 𝑣(𝑡,⋅) is preserved under the family of the diffeomorphism 𝑞(𝑡,⋅); that is, ‖𝑣(𝑡,⋅)‖_𝐿^∞=𝑣(𝑡,𝑞(𝑡,⋅))_𝐿∞,𝑡∈[0,𝑇). (21) Similarly, inf_𝑥∈ℝ𝑣(𝑡,𝑥)=inf_𝑥∈ℝ𝑣𝑡,𝑞(𝑡,𝑥),𝑡∈[0,𝑇),sup_𝑥∈ℝ𝑣(𝑡,𝑥)=sup_𝑥∈ℝ𝑣𝑡,𝑞(𝑡,𝑥),𝑡∈[0,𝑇). (22)

3. Blow-Up Criteria

In this section, we establish sufficient conditions to guarantee the formation of singularities for the corresponding solutions to (10) in periodic and nonperiodic cases. These sufficient conditions are different from each other.

Hereinafter, we investigate the blow-up phenomenon with periodic setting; that is, 𝑥∈𝕊=ℝ/ℤ. First, we introduce the following lemmas for later use.

Lemma 4 (see [23]). For all 𝑓∈𝐻¹(𝕊), the following inequality holds: 𝐺∗𝑓²+𝛼²2𝑓²_𝑥(𝑥)≥𝐶₀𝑓²(𝑥), (23) with 𝐶₀=12+arctan(sinh(1/2𝛼))2sinh(1/2𝛼)+2arctan(sinh(1/2𝛼))sinh²().1/2𝛼 (24) Moreover, 𝐶₀ is the optimal constant obtained by the function 𝑓₀=𝑥1+arctansinh𝛼−[𝑥]𝛼−1𝑥2𝛼×sinh𝛼−[𝑥]𝛼−1×12𝛼1+arctansinh12𝛼sinh2𝛼⁻¹. (25)

Lemma 5 (see [23]). For all 𝑓∈𝐻¹(𝕊), the following inequality holds: max_𝑥∈𝕊𝑓²(𝑥)≤𝐶_1𝕊𝑓²+𝛼²𝑓²_𝑥𝑑𝑥, (26) where 𝐶₁=cosh(1/2𝛼)().2𝛼sinh1/2𝛼 (27) Moreover, 𝐶₁ is the minimum value, so in this sense, 𝐶₁ is the optimal constant which is obtained by the associated Green function 𝐺(𝑥)=cosh((𝑥/𝛼)−([𝑥]/𝛼)−(1/2𝛼))().2𝛼sinh1/2𝛼 (28)

Lemma 6 (see [37]). Assume that a differentiable function 𝑦(𝑡) satisfies 𝑦^′(𝑡)≤−𝐶𝑦²(𝑡)+𝐾 (29) with the constants 𝐶,𝐾>0. If the initial datum 𝑦(0)=𝑦₀<−𝐾/𝐶, then the solution to the previous equation goes to −∞ in finite time.

Lemma 7 (see [24]). Assume 𝑓(𝑥)∈𝐻^𝑠(𝕊), 𝑠>2. If ∫_𝕊𝑓(𝑥)𝑑𝑥=0, then 𝑓(𝑥)²_{𝐿∞(𝕊)}≤112_𝕊𝑓²_𝑥(𝑥)𝑑𝑥,_𝕊𝑓²(𝑥)1𝑑𝑥≤12_𝕊𝑓²_𝑥(𝑥)𝑑𝑥,_𝕊𝑓²(𝑥)𝑓²_𝑥(𝑥)1𝑑𝑥≤12_𝕊𝑓²_𝑥(𝑥)𝑑𝑥². (30)

Now we give some sufficient conditions on the initial data to guarantee finite time wave breaking for the periodic case.

Theorem 8. Assume that the initial data (𝑢₀,𝜌₀)∈𝐻^𝑠×𝐻^𝑠−1,𝑠≥2, satisfies _𝕊𝑢³_0𝑥𝑢𝑑𝑥<−⁰,𝜌₀²_{𝐻1𝛼×𝐿2}𝛼³3𝐴+6𝐶₁𝑢₀,𝜌₀²_{𝐻1𝛼×𝐿2}, (31) where 𝐶₁ is defined as in Lemma 5, and constant 𝐴 is determined later. Moreover, for convenience, we suppose that 2𝜔+𝛾/𝛼²=0. Then the corresponding strong solution to (10) blows up in finite time.

Proof. By assumption and Theorem 1, there exists a solution (𝑢,𝜌) to system (10) corresponding to the initial value (𝑢₀,𝜌₀)∈𝐻^𝑠×𝐻^𝑠−1,𝑠≥2. Suppose that the statement is not true; that is, the corresponding solution does not blow up in finite time, and then due to Theorem 3, there exists 𝑀₁≥0, such that inf_{()∈[)𝑡,𝑥0,𝑇×ℝ}𝑢_𝑥(𝑡,𝑥)≥−𝑀₁. (32) Let 𝑀(𝑡)=𝑢_𝑥()(𝑡)(),[),𝑡,𝑞𝑡,𝑥,𝛾=𝜌𝑡,𝑞𝑡,𝑥𝑡∈0,𝑇 (33) with 𝑞(𝑡,𝑥) determined in (18). Then we have the following inequality for each 𝑥∈ℝ and 𝑡∈[0,𝑇): 𝑀(𝑡)≥−𝑀₁. (34) Since 𝛾^′(𝑡)=−𝛾(𝑡)𝑀(𝑡), (35) we have, by solving it, 𝛾(𝑡)=𝛾(0)𝑒^{∫𝑡0−𝑀(𝜏)𝑑𝜏}. (36) So we can get 𝜌𝑡,𝑞(𝑡,𝑥)=𝛾(𝑡)=𝛾(0)𝑒^{∫𝑡0−𝑀(𝜏)𝑑𝜏}≤𝜌_0𝐿∞𝑒^𝑀1𝑡, (37) then we have 𝜌(𝑡,⋅)_𝐿∞≤𝜌_0𝐿∞𝑒^𝑀1𝑡. (38) Therefore, the following inequality holds: 𝜌+1²_𝐿∞≤𝜌_𝐿∞+1²≤𝜌_0𝐿∞𝑒^𝑀1𝑡+1²=̇𝐴. (39) With the notation of 𝜕²_𝑥(𝐺∗𝑓)=(1/𝛼²)(𝐺∗𝑓−𝑓), differentiating both sides of the first equation of (10) with respect to variable 𝑥, we obtain 𝑢_𝑡𝑥+𝑢²_𝑥+𝑢𝑢_𝑥𝑥+2𝜔𝑢_𝑥𝑥=1𝛼²𝑢²+𝛼²2𝑢²_𝑥+12𝜌²−1+𝜌𝛼²𝐺∗𝑢²+𝛼²2𝑢²_𝑥+12𝜌²+𝜌. (40) We note that 𝐺∗(1/2)=1/2; then (40) can be rewritten as follows: 𝑢_𝑡𝑥+𝑢²_𝑥+𝑢𝑢_𝑥𝑥+2𝜔𝑢_𝑥𝑥=1𝛼²𝑢²+𝛼²2𝑢²_𝑥+12𝜌+1²−1𝛼²𝐺∗𝑢²+𝛼²2𝑢²_𝑥+12𝜌+1². (41) Multiplying 𝑢²_𝑥 on both sides and integrating by parts with respect to 𝑥, one obtains 𝑑𝑑𝑡_𝕊𝑢³_𝑥1𝑑𝑥=−2_𝕊𝑢⁴_𝑥3𝑑𝑥+𝛼²_𝕊𝑢²_𝑥𝑢²+3𝑑𝑥𝛼²_𝕊𝑢²_𝑥12𝜌+1²−3𝑑𝑥𝛼²_𝕊𝑢²_𝑥𝐺∗𝑢²+𝛼²2𝑢²_𝑥+12𝜌+1²1𝑑𝑥≤−2_𝕊𝑢⁴_𝑥3𝑑𝑥+𝛼²_𝕊𝑢²_𝑥𝑢²+3𝑑𝑥𝛼²_𝕊𝑢²_𝑥12𝜌+1²𝑑𝑥, (42) where we have used the following identity: _𝕊𝑢²_𝑥𝑢𝑢_𝑥𝑥1𝑑𝑥=−3_𝕊𝑢⁴_𝑥𝑑𝑥. (43) Since _𝕊𝑢³_𝑥𝑑𝑥²≤1𝛼²_𝕊𝑢⁴_𝑥𝑑𝑥_𝕊𝛼²𝑢²_𝑥𝑑𝑥, (44) it follows that _𝕊𝑢⁴_𝑥𝛼𝑑𝑥≥²𝑢₀,𝜌₀²_{𝐻1𝛼×𝐿2}^𝕊𝑢³_𝑥𝑑𝑥². (45) We can also get 3𝛼²_𝕊𝑢²_𝑥𝑢²3𝑑𝑥≤𝛼⁴𝐶₁𝑢₀,𝜌₀⁴_{𝐻1𝛼×𝐿2},3𝛼²_𝕊𝑢²_𝑥12𝜌+1²=3𝑑𝑥2𝛼²_𝕊𝑢²_𝑥𝜌+1²≤3𝑑𝑥2𝛼⁴𝜌+1²_𝐿∞𝕊𝛼²𝑢²_𝑥≤3𝑑𝑥2𝛼⁴𝜌+1²_𝐿∞𝑢₀,𝜌₀²_{𝐻1𝛼×𝐿2}. (46) So we obtain 𝑑𝑑𝑡_𝕊𝑢³_𝑥𝛼𝑑𝑥≤−²2𝑢₀,𝜌₀²_{𝐻1𝛼×𝐿2}^𝕊𝑢³_𝑥𝑑𝑥²+3𝛼⁴𝐶₁𝑢₀,𝜌₀⁴_{𝐻1𝛼×𝐿2}+32𝛼⁴𝜌+1²_𝐿∞𝑢₀,𝜌₀²_{𝐻1𝛼×𝐿2}𝛼≤−²2𝑢₀,𝜌₀²_{𝐻1𝛼×𝐿2}^𝕊𝑢³_𝑥𝑑𝑥²+3𝛼⁴𝐶₁𝑢₀,𝜌₀⁴_{𝐻1𝛼×𝐿2}+32𝛼⁴𝐴𝑢₀,𝜌₀²_{𝐻1𝛼×𝐿2}. (47) Due to the theory of Riccati type differential equations in Lemma 6, if (31) holds, then we have _𝕊𝑢³_𝑥𝑑𝑥⟶−∞, (48) as 𝑡 tends to some 𝑇₀.
On the other hand, the majorization _𝕊𝑢³_𝑥𝑑𝑥≥inf𝑢_𝑥(𝑥,𝑡)_𝕊𝑢²_𝑥(𝛼)𝑑𝑥>𝑐inf𝑢_𝑥(𝑥,𝑡)𝑢₀,𝜌₀²_{𝐻1𝛼×𝐿2} (49) implies that lim_𝑡↑𝑇0inf𝑢_𝑥(𝑥,𝑡)=−∞. (50) It contradicts the assumption inf_{())𝑡,𝑥∈[0,𝑇×ℝ}𝑢_𝑥(𝑡,𝑥)≥−𝑀₁. (51) By Theorem 3, we know that the solution must blow up in finite time.

Next, we find if the initial velocity has zero mean and initial energy is sufficiently large, wave breaking can occur.

Theorem 9. Suppose 𝑧₀=(𝑢₀,𝜌₀)∈𝐻^𝑠×𝐻^𝑠−1,𝑠>5/2, and 𝛼 satisfies 0<sinh(1/2𝛼)/𝛼<3. If the initial energy satisfies ∫_𝕊(𝑢²₀+𝛼²𝑢²_0𝑥+𝜌²₀)𝑑𝑥>𝐶₀ for some positive constant 𝐶₀, and then ∫_𝕊𝑢₀(𝑥)𝑑𝑥=0 holds for nontrival 𝑧₀. Besides, assume that 2𝜔+𝛾/𝛼²=0 just for convenience. Then the solution to the system (10) with initial value 𝑧₀ blows up in finite time.

Proof. From the first equation of (10), we know that 𝑑𝑑𝑡_𝕊𝑢³_𝑥1𝑑𝑥=−2_𝕊𝑢⁴_𝑥3𝑑𝑥+𝛼²_𝕊𝑢²_𝑥𝑢²3𝑑𝑥+𝛼²_𝕊𝑢²_𝑥12𝜌²−3+𝜌𝑑𝑥𝛼²_𝕊𝑢²_𝑥𝐺∗𝑢²+𝛼²2𝑢²_𝑥+12𝜌²+𝜌𝑑𝑥. (52) On the other hand, we know that ∫_𝕊𝑢(𝑥,𝑡)𝑑𝑥=0 in view of the hypothesis, and the following inequality holds: 1()(𝑥)≤()2𝛼sinh1/2𝛼≤𝐺cosh1/2𝛼().2𝛼sinh1/2𝛼 (53) Using Lemma 7 and (53), we obtain 𝑑𝑑𝑡_𝕊𝑢³_𝑥1𝑑𝑥=−2_𝕊𝑢⁴_𝑥3𝑑𝑥+𝛼²_𝕊𝑢²_𝑥𝑢²+3𝑑𝑥𝛼²_𝕊𝑢²_𝑥12𝜌+1²−3𝑑𝑥𝛼²_𝕊𝑢²_𝑥𝐺∗𝑢²+𝛼²2𝑢²_𝑥+12𝜌+1²1𝑑𝑥≤−2_𝕊𝑢⁴_𝑥1𝑑𝑥+4𝛼²_𝕊𝑢²_𝑥𝑑𝑥²+3𝛼²_𝕊𝑢²_𝑥12𝜌+1²−3𝑑𝑥2_𝕊𝑢²_𝑥𝐺∗𝑢²_𝑥1𝑑𝑥≤−2_𝕊𝑢⁴_𝑥3𝑑𝑥+𝛼²_𝕊𝑢²_𝑥12𝜌+1²+1𝑑𝑥4𝛼²−34𝛼sinh(1/2𝛼)_𝕊𝑢²_𝑥𝑑𝑥². (54) Similar to Theorem 8, assume that the solution (𝑢,𝜌) of system (10) does not blow up; that is, there is 𝑀₁≥0, such that inf_{()∈[)𝑡,𝑥0,𝑇×ℝ}𝑢_𝑥(𝑡,𝑥)≥−𝑀₁, (55) then 𝜌(𝑡,⋅)_𝐿∞≤𝜌_0𝐿∞exp𝑀₁𝑡=̇𝜅. (56) So we can get 𝑑𝑑𝑡_𝕊𝑢³_𝑥1𝑑𝑥≤−2_𝕊𝑢⁴_𝑥3𝑑𝑥+𝛼²_𝕊𝑢²_𝑥12𝜌+1²+1𝑑𝑥4𝛼²−34𝛼sinh(1/2𝛼)_𝕊𝑢²_𝑥𝑑𝑥²1≤−2_𝕊𝑢⁴_𝑥3()𝑑𝑥+𝜅+1²2𝛼²_𝕊𝑢²_𝑥+1𝑑𝑥4𝛼²−34𝛼sinh(1/2𝛼)_𝕊𝑢²_𝑥𝑑𝑥². (57) If 0<sinh(1/2𝛼)/𝛼<3, then (1/4𝛼²)−(3/4𝛼sinh(1/2𝛼))<0. Since ∫_𝕊(𝑢²₀+𝛼²𝑢²_0𝑥+𝜌²₀)𝑑𝑥>𝐶₀ for some constant 𝐶₀, due to Lemma 7, there is some 𝛿^⋆>0 such that 3(𝜅+1)²2𝛼²_𝕊𝑢²_𝑥𝑑𝑥≤𝛿^⋆_𝕊𝑢²_𝑥𝑑𝑥²,3()−14𝛼sinh1/2𝛼4𝛼²−𝛿^⋆>0. (58) Moreover, Lemma 7 also implies that _𝕊𝑢²_𝑥𝑑𝑥≥1212𝛼²‖𝑢(𝑥+1)‖²_{𝐻1𝛼(𝕊)}. (59) Hence, 𝑑𝑑𝑡_𝕊𝑢³_𝑥1𝑑𝑥≤−2_𝕊𝑢⁴_𝑥𝑑𝑥−14412𝛼²+1²×3()−14𝛼sinh1/2𝛼4𝛼²−𝛿^⋆‖𝑢(𝑥)‖⁴_{𝐻1𝛼(𝕊)}. (60) Note that ‖𝑢(𝑥)‖_𝐻¹_𝛼(𝕊) is bounded. In view of Hölder’s inequality, there exists _𝕊𝑢⁴_𝑥𝑑𝑥≥_𝕊𝑢³_𝑥𝑑𝑥^4/3. (61) For simplicity of notations, we denote by 𝜑(𝑡) and 𝜇 the following quantities: _𝕊𝑢³_𝑥𝑑𝑥,14412𝛼²+1²3()−14𝛼sinh1/2𝛼4𝛼²−𝛿^⋆, (62) respectively. Therefore we have 𝑑𝜑(𝑡)1𝑑𝑡≤−2𝜑^4/3(𝑡)−𝜇‖𝑢(𝑥)‖⁴_{𝐻1𝛼(𝕊)}. (63) First, we can easily get 𝜑(𝑡)≤𝜑(0)−𝜇‖𝑢‖⁴_{𝐻1𝛼(𝕊)}𝑡, and it is not difficult to find that there exists a time 𝑡₀ such that 𝜑(𝑡₀)<0. Then for all 𝑡>𝑡₀, we have 𝑑𝜑(𝑡)1𝑑𝑡≤−2𝜑^4/3(𝑡),with𝜑𝑡₀<0. (64) Solving this inequality yields 𝜑(𝑡)≤𝜑^−1/3𝑡₀+16𝑡−𝑡₀⁻³, (65) which goes to −∞ as 𝑡 tends to −6𝜑^−1/3(𝑡₀)+𝑡₀; that is, there exists a time 𝑇≤−6𝜑^−1/3(𝑡₀)+𝑡₀ such that lim_{𝑡↑𝑇𝕊}𝑢³_𝑥𝑑𝑥=−∞. (66) Since _𝕊𝑢³_𝑥𝑑𝑥≥inf𝑢_𝑥(𝑥,𝑡)_𝕊𝑢²_𝑥𝑑𝑥≥𝑐(𝛼)inf𝑢_𝑥(𝑥,𝑡)‖𝑢‖²_{𝐻1𝛼(𝕊)}, (67) it shows that lim_𝑡↑𝑇inf_𝑥∈ℝ𝑢_𝑥(𝑥,𝑡)=−∞. (68) Then it contradicts the assumption inf_{()𝑡,𝑥∈[0,𝑇)×ℝ}𝑢_𝑥(𝑥,𝑡)>−𝑀₁. (69) By Theorem 3, we know that the solution must blow up in finite time. This finishes the proof.

From now on, we give some blow-up criteria for the nonperiodic case.

Theorem 10. Assume that the initial data (𝑢₀,𝜌₀)∈𝐻^𝑠×𝐻^𝑠−1,𝑠≥2, satisfy 𝑢^′₀𝑥₀<−2𝐾₀, (70) where 𝐾₀ is a constant determined later. And assume that 2𝜔+𝛾/𝛼²=0 just for convenience. Then 𝑇 is finite and the slope of 𝑢 tends to negative infinity as 𝑡 goes to 𝑇 while 𝑢 is uniformly bounded on [0,𝑇).

Proof. Let (𝑢,𝜌) be the solution to system (10) with the initial data (𝑢₀,𝜌₀) and let 𝑇 be the maximal existence time of solution. Defining 𝑚(𝑡)=inf_𝑥∈ℝ𝑢_𝑥(𝑡,𝑥)=𝑢_𝑥((𝑡,𝛾(𝑡)((𝑡,𝑡,𝜉))=𝜌𝑡,𝜉)) (71) one can get that 𝑚 is almost everywhere differentiable on [0,𝑇) with 𝑑𝑚(𝑡)𝑑𝑡=𝑢_𝑡𝑥(𝑡,𝜉(𝑡)). (72) Note that 𝑢_𝑥𝑥(𝑡,𝜉(𝑡))=0, and with (40), we have 𝑚_𝑡=−𝑚²+1𝛼²𝑢²+𝛼²2𝑢²_𝑥+12𝜌²−1+𝜌𝛼²𝐺∗𝑢²+𝛼²2𝑢²_𝑥+12𝜌²+𝜌, (73) and with the relation 1/2=𝐺∗(1/2), we can get 𝑚_𝑡=−𝑚²+1𝛼²𝑢²+𝛼²2𝑢²_𝑥+12𝜌+1²−1𝛼²𝐺∗𝑢²+𝛼²2𝑢²_𝑥+12𝜌+1²1≤−2𝑚²+1𝛼²𝑢²+12𝜌+1². (74) Since 𝑢 and 𝜌 are both bounded, then there exists a 𝐾₀>0, so that (1/𝛼²)(𝑢²+(1/2)(𝜌+1)²)<𝐾₀, and thus we have 𝑚_𝑡1≤−2𝑚²+𝐾₀. (75) Note that if 𝑚(0)<−2𝐾₀, then 𝑚(𝑡)<−2𝐾₀ for all 𝑡∈[0,𝑇). So there exists a 𝑇 such that lim_𝑡↑𝑇𝑚(𝑡)=−∞. (76) This completes the proof of the theorem.

The authors in [25] claimed that the corresponding solution to the two-component Camassa-Holm equation exists globally in time provided that some initial data satisfy smallness conditions. However, we find surprisingly the smallness of initial energy can also lead to breaking wave solutions. This is stated as follows.

Theorem 11. Let (𝑢₀,𝜌₀)∈𝐻^𝑠(ℝ)×𝐻^𝑠−1(ℝ) with 𝑠>5/2 and let 𝑇>0 be the maximal time of existence of the solution (𝑢,𝜌) to (10) with initial data (𝑢₀,𝜌₀). Assume there is some 𝑥₀∈ℝ such that 𝜌₀(𝑥₀)=−1, and there holds 0<𝛾<1−2𝜔𝛼² for suitable 𝛼 and 𝛾. Moreover, 𝑢₀,𝜌₀²_{𝐻1𝛼×𝐿2}<𝛼⁵1−2𝜔𝛼²−𝛾2𝛼⁴+2𝜔𝛼².+𝛾 (77) Then the corresponding solution to (10) blows up in finite time in the following sense, and there exists a 𝑇₀ with 0<𝑇₀≤2+4𝛼⁷+4𝛼⁷𝑢_0,𝑥𝑥₀𝛼⁵−2𝜔𝛼⁷−𝛾𝛼⁵−2𝛼⁴+2𝜔𝛼²+𝛾𝑢₀,𝜌₀²_{𝐻1𝛼×𝐿2}, (78) such that lim_𝑡↑𝑇0infinf_𝑥∈ℝ𝑢_𝑥(𝑥,𝑡)=−∞. (79)

Proof. As in Theorem 8, we define 𝑀(𝑡)=inf_𝑥∈ℝ𝑢_𝑥(𝑡,𝑥)=𝑢_𝑥(𝑡,𝜉(𝑡)),𝑡∈[0,𝑇), (80) so that 𝑢_𝑥𝑥(𝑡,𝜉(𝑡))=0,a.e.𝑡∈[0,𝑇). (81) We recall that 𝑞(𝑡,⋅) defined by (18) is a diffeomorphism of the line for any 𝑡∈[0,𝑇), so there exists an 𝑥(𝑡)∈ℝ such that 𝑞(𝑡,𝑥(𝑡))=𝜉(𝑡),𝑡∈[0,𝑇). (82) Differentiating the first equation of (10), we can get the expression of 𝑀_𝑡 as follows: 𝑀_𝑡𝑡,𝑞(𝑡,𝑥(𝑡))=−𝑀²+1𝛼²𝑢²+𝛼²2𝑢²_𝑥+12𝜌+1²−1𝛼²𝐺∗𝑢²+𝛼²2𝑢²_𝑥+12𝜌+1²−𝛾2𝜔+𝛼²𝜕²_𝑥1𝐺∗𝑢=−2𝑀²+1𝛼²𝑢²+12𝛼²𝜌+1²−1𝛼²𝐺∗𝑢²+𝛼²2𝑢²_𝑥−12𝛼²𝐺∗𝜌+1²−𝛾2𝜔+𝛼²𝜕²_𝑥𝐺∗𝑢. (83) Besides, along the trajectory of 𝑞(𝑡,𝑥(𝑡)), we have 𝑑𝜌𝑡,𝑞+1𝑞_𝑥𝑑𝑡=𝑞_𝑥𝜌_𝑡+𝜌_𝑥𝑢+𝜌𝑢_𝑥+𝑢_𝑥=0. (84) Then choose 𝜉(0)=𝑥₀ and 𝜌(𝜉(0))=𝜌₀(𝑥₀)=−1, which implies that 𝜌(𝑡,𝜉(𝑡))≡−1. (85) Therefore we can obtain 𝑀_𝑡1=−2𝑀²+1𝛼²𝑢²−1𝛼²𝐺∗𝑢²+𝛼²2𝑢²_𝑥−12𝛼²𝐺∗𝜌+1²−𝛾2𝜔+𝛼²𝜕²_𝑥1𝐺∗𝑢≤−2𝑀²+1𝛼²𝑢²−12𝛼²𝑢²−12𝛼²𝐺∗𝜌+1²−𝛾2𝜔+𝛼²𝜕²_𝑥1𝐺∗𝑢=−2𝑀²+12𝛼²𝑢²−12𝛼²𝐺∗𝜌+1²−𝛾2𝜔+𝛼²𝜕²_𝑥𝐺∗𝑢, (86) where we use the fact 𝐺∗𝑢²+𝛼²2𝑢²_𝑥≥12𝑢². (87) We define 1𝑓=2𝛼²𝑢²−12𝛼²𝐺∗𝜌+1²−𝛾2𝜔+𝛼²𝜕²_𝑥𝐺∗𝑢. (88) For any 𝑥∈ℝ, then 𝑢²(𝑥)=^𝑥0_−∞𝑢𝑢_𝑥𝑑𝑥+^𝑥0_∞𝑢𝑢_𝑥≤1𝑑𝑥2𝛼^𝑥0_−∞𝑢²+𝛼²𝑢²_𝑥𝑑𝑥+^𝑥0_∞𝑢²+𝛼²𝑢²_𝑥=1𝑑𝑥‖𝑢‖2𝛼^2𝐻1𝛼≤12𝛼𝑢,𝜌²_{𝐻1𝛼×𝐿2}. (89) We also have 𝐺∗𝜌≤‖𝐺‖_𝐿²𝜌_𝐿2=12√𝛼𝜌_𝐿2≤14+14𝛼𝑢,𝜌²_{𝐻1𝛼×𝐿2},𝜕²_𝑥𝐺∗𝑢=𝐺_𝑥∗𝑢_𝑥≤𝐺_𝑥𝐿2𝑢_𝑥𝐿2=12√𝛼³𝑢_𝑥𝐿2≤14+14𝛼⁵𝑢,𝜌²_{𝐻1𝛼×𝐿2}. (90) Thus, we can get 1𝑓=2𝛼²𝑢²−12𝛼²𝐺∗𝜌²−1𝛼²−1𝐺∗𝜌2𝛼²−𝛾2𝜔+𝛼²𝜕²_𝑥≤1𝐺∗𝑢2𝛼²𝑢²+1𝛼²−1𝐺∗𝜌2𝛼²+𝛾2𝜔+𝛼²𝜕²_𝑥≤1𝐺∗𝑢4𝛼³𝑢,𝜌²_{𝐻1𝛼×𝐿2}+14𝛼³𝑢,𝜌²_{𝐻1𝛼×𝐿2}+14𝛼⁵𝛾2𝜔+𝛼²𝑢,𝜌²_{𝐻1𝛼×𝐿2}−14𝛼²+14𝛾2𝜔+𝛼²=̇−𝑎<0. (91) Therefore, when ‖(𝑢₀,𝜌₀)‖²_𝐻1×𝐿2<(𝛼⁵−2𝜔𝛼⁷−𝛾𝛼⁵)/(2𝛼⁴+2𝜔𝛼²+𝛾) holds, we always have 𝑓<0. In this case, we can obtain 𝑀^′(𝑡)1≤−2𝑀²−𝑎<0,𝑡∈[𝑡,𝑇), (92) so 𝑀(𝑡) is decreasing strictly in such interval. Assume that the solution (𝑢,𝜌) of (10) does not blow up; that is, it exists globally in time; that is, 𝑇=∞. Then integrating (92) over [0,𝑡₁] with 𝑡₁=(1+|𝑢_0,𝑥(𝑥₀)|)/𝑎 yields 𝑀𝑡₁=𝑀(0)−^𝑡1₀𝑀^′(𝑡)𝑑𝑡≤𝑢_0,𝑥𝑥₀−𝑎𝑡₁≤−1. (93) We can also find that 𝑀^′(𝑡)≤−(1/2)𝑀², which leads to −𝑑1𝑑𝑡𝑀(𝑡)=1𝑀²(𝑡)𝑑𝑀(𝑡)1𝑑𝑡≤−2,𝑡∈𝑡₁,𝑇. (94) Integrating both sides, we have −1𝑀(𝑡)1−1≤−𝑀(𝑡)+1𝑀𝑡₁1≤−2𝑡−𝑡₁, (95) where we use the result 𝑀(𝑡₁)≤−1 obtained previously. Solve the inequality 𝑀(𝑡)≤2𝑡−𝑡₁−2⟶−∞,as𝑡⟶𝑡₁+2. (96) That implies that 𝑇≤𝑡₁+2<∞, which leads to a contradiction as 𝑡→∞. This completes the proof.

4. Persistence Properties

Attention now is turned to determining the persistence properties of solutions to the two-component DGH equation. We show that certain decay properties of the initial data persist in corresponding solution in later time as long as it exists. Precisely, we prove that the corresponding solution (𝑢,𝜌) and its first order spatial derivatives preserve the exponential decay as their initial values do. This is a very important investigation since we can obtain the detailed asymptotic behavior of solution from the initial values. The main idea comes from the recent works of Zhou et al. [17, 38]. Now, we state our result as follows.

Theorem 12. Assume that for some 𝑇>0 and 𝑠>5/2, (𝑢,𝜌)∈𝐶([0,𝑇];𝐻^𝑠×𝐻^𝑠−1) is a strong solution of the initial value problem to the system (10) and 2𝜔+𝛾/𝛼²=0 just for convenience. If 𝑢₀=𝑢(𝑥,0), 𝜌₀=𝜌(𝑥,0) satisfy 𝑢₀(𝑥),𝜕_𝑥𝑢₀(𝑥),𝜌₀(𝑥),𝜕_𝑥𝜌₀(𝑥)∼𝑂𝑒^−𝜃𝑥as𝑥⟶∞, (97) for some 𝜃∈(0,1). Then |𝑢(𝑥,𝑡)|,𝑢_𝑥(𝑥,𝑡),𝜌(𝑥,𝑡),𝜌_𝑥(𝑥,𝑡)∼𝑂𝑒^−𝜃𝑥as𝑥⟶∞ (98) uniformly in the time interval [0,𝑇].

Notation. Consider 𝑢(𝑥,𝑡)∼𝑂𝑒^−𝜃𝑥,as𝑥⟶∞,iflim_𝑥→∞|𝑢(𝑥)|𝑒^−𝜃𝑥𝑢()𝑒=𝐿,𝑥,𝑡∼o^−𝜃𝑥,as𝑥⟶∞,iflim_𝑥→∞|𝑢(𝑥)|𝑒^−𝜃𝑥=0. (99)

Proof. First, we figure out the estimates on ‖𝑢(𝑥,𝑡)‖_𝐿^∞, ‖𝑢_𝑥(𝑥,𝑡)‖_𝐿∞, ‖𝜌(𝑥,𝑡)‖_𝐿∞, and ‖𝜌_𝑥(𝑥,𝑡)‖_𝐿∞. Then we use the weight function to obtain the desired result.
Step 1. Estimate for ‖𝑢(𝑥,𝑡)‖_𝐿^∞,‖𝜌(𝑥,𝑡)‖_𝐿∞.
Multiplying the first equation of (10) by 𝑢^2𝑛−1 with 𝑛∈ℤ and then integrating both sides with respect to 𝑥-variable, we can get _ℝ𝑢^2𝑛−1𝑢_𝑡+𝑑𝑥_ℝ𝑢^2𝑛−1𝑢𝑢_𝑥𝑑𝑥+_ℝ𝑢^2𝑛−12𝜔𝑢_𝑥+𝑑𝑥_ℝ𝑢^2𝑛−1𝜕_𝑥𝑝∗𝑢²+𝛼²2𝑢²_𝑥+12𝜌²+𝜌𝑑𝑥=0. (100) The first term of the previous identity is _ℝ𝑢^2𝑛−1𝑢_𝑡1𝑑𝑥=𝑑2𝑛‖𝑢(𝑡𝑑𝑡)‖_𝐿^2𝑛_2𝑛=‖𝑢(𝑡)‖_𝐿^2𝑛−1_2𝑛𝑑‖𝑢(𝑡𝑑𝑡)‖_𝐿^2𝑛, (101) and the estimate of the second term is _ℝ𝑢^2𝑛−1𝑢𝑢_𝑥𝑑𝑥≤𝑢_𝑥(𝑡)_𝐿∞‖𝑢(𝑡)‖_𝐿^2𝑛_2𝑛. (102)
In view of Hölder’s inequality, we can obtain the following estimate for the third and the last terms in (100): _ℝ2𝜔𝑢^2𝑛−1𝑢_𝑥𝑑𝑥≤2𝜔‖𝑢(𝑡)‖_𝐿^2𝑛−1_2𝑛𝑢_𝑥(𝑡)_𝐿2𝑛,_ℝ𝑢^2𝑛−1𝜕_𝑥𝑝∗𝑢²+𝛼²2𝑢²_𝑥+12𝜌²≤‖𝑢(𝑡+𝜌𝑑𝑥)‖_𝐿^2𝑛−1_2𝑛𝜕_𝑥𝑝∗𝑢²+𝛼²2𝑢²_𝑥+12𝜌²+𝜌_𝐿2𝑛. (103) Putting all the previous inequalities into (100) yields 𝑑‖𝑢(𝑡𝑑𝑡)‖_𝐿^2𝑛≤𝑢_𝑥(𝑡)_𝐿∞‖𝑢(𝑡)‖_𝐿^2𝑛+2𝜔𝑢_𝑥(𝑡)_𝐿2𝑛+𝜕_𝑥𝑝∗𝑢²+𝛼²2𝑢²_𝑥+12𝜌²+𝜌_𝐿2𝑛. (104) Thus, according to the Sobolev embedding theorem, there exists a constant 𝑀=sup_{[]𝑡∈0,𝑇}‖𝑢(𝑥,𝑡)‖_𝐻^𝑠, (105) such that applying Gronwall’s inequality gives us ‖𝑢(𝑡)‖_𝐿^2𝑛≤‖𝑢(0)‖_𝐿^2𝑛+^𝑡₀2𝜔𝑢_𝑥(𝜏)_𝐿2𝑛+^𝑡₀𝜕_𝑥𝑝∗𝑢²+𝛼²2𝑢²_𝑥+12𝜌²+𝜌_𝐿2𝑛𝑑𝜏𝑒^𝑀𝑡. (106) For any 𝑓∈𝐿¹(ℝ)∩𝐿^∞(ℝ), we know that lim_𝑞→∞𝑓_𝐿𝑞=𝑓_𝐿∞. (107) Thus, taking the limits in (106), we get ‖𝑢(𝑡)‖_𝐿^∞≤‖𝑢(0)‖_𝐿^∞+^𝑡₀2𝜔𝑢_𝑥(𝜏)_𝐿∞+𝑑𝜏_𝑡0𝜕_𝑥𝑝∗𝑢²+𝛼²2𝑢²_𝑥+12𝜌²+𝜌_𝐿∞𝑑𝜏𝑒^𝑀𝑡. (108) Similarly, for the second equation of (10), we have _ℝ𝜌^2𝑛−1𝜌_𝑡𝑑𝑥+_ℝ𝜌^2𝑛−1𝜌_𝑥+𝑢𝑑𝑥_ℝ𝜌^2𝑛−1𝜌𝑢_𝑥𝑑𝑥+_ℝ𝜌^2𝑛−1𝑢_𝑥𝑑𝑥=0, (109) and using the same method, one can get 𝑑𝜌(𝑡𝑑𝑡)_𝐿2𝑛≤𝑢_𝑥𝐿∞𝜌(𝑡)_𝐿2𝑛+‖𝑢‖_𝐿^∞𝜌_{𝑥𝐿2𝑛}+𝑢_{𝑥𝐿2𝑛}. (110) Using the Gronwall’s inequality, one gets 𝜌(𝑡)_𝐿2𝑛≤𝜌(0)_𝐿2𝑛+^𝑡₀‖𝑢‖_𝐿^∞𝜌_{𝑥𝐿2𝑛}+𝑢_{𝑥𝐿2𝑛}𝑑𝜏𝑒^𝑀𝑡. (111) Taking the limits, one gets 𝜌(𝑡)_𝐿∞≤𝜌(0)_𝐿∞+^𝑡₀‖𝑢‖_𝐿^∞𝜌_𝑥𝐿∞+𝑢_𝑥𝐿∞𝑑𝜏𝑒^𝑀𝑡. (112)
Step 2. Estimate for ‖𝑢_𝑥(𝑥,𝑡)‖_𝐿∞, ‖𝜌_𝑥(𝑥,𝑡)‖_𝐿∞.
We will establish an estimate on ‖𝑢_𝑥(𝑥,𝑡)‖_𝐿∞ using the same method. Differentiating the first equation (10) with respect to 𝑥-variable produces the following equation: 𝑢_𝑥𝑡+𝑢𝑢_𝑥𝑥+𝑢²_𝑥+2𝜔𝑢_𝑥𝑥+𝜕²_𝑥𝑝∗𝑢²+𝛼²2𝑢²_𝑥+12𝜌²+𝜌=0. (113) Then, multiplying (113) by 𝑢_𝑥^2𝑛−1 with 𝑛∈ℤ⁺, integrating the result in the 𝑥-variable, and considering the second term in the previous identity with integration by parts, one can get _ℝ𝑢𝑢_𝑥𝑥𝑢_𝑥^2𝑛−1𝑑𝑥=_ℝ𝑢𝑢_𝑥^2𝑛2𝑛_𝑥1𝑑𝑥=−2𝑛_ℝ𝑢_𝑥𝑢_𝑥^2𝑛𝑑𝑥, (114) so we have _ℝ𝑢_𝑥^2𝑛−1𝑢_𝑥𝑡1𝑑𝑥−2𝑛_ℝ𝑢_𝑥𝑢_𝑥^2𝑛𝑑𝑥+_ℝ𝑢_𝑥^2𝑛−1𝑢²_𝑥+𝑑𝑥_ℝ𝑢_𝑥^2𝑛−12𝜔𝑢_𝑥𝑥+𝑑𝑥_ℝ𝑢_𝑥^2𝑛−1𝜕²_𝑥𝑝∗𝑢²+𝛼²2𝑢²_𝑥+12𝜌²+𝜌𝑑𝑥=0. (115) Similarly, we can get the following inequality: 𝑑𝑢𝑑𝑡^𝑥(𝑡)_𝐿2𝑛≤2𝑢_𝑥(𝑡)_𝐿∞𝑢_𝑥(𝑡)_𝐿2𝑛+2𝜔𝑢_𝑥𝑥(𝑡)_𝐿2𝑛+𝜕²_𝑥𝑝∗𝑢²+𝛼²2𝑢²_𝑥+12𝜌²+𝜌_𝐿2𝑛. (116) Using Gronwall’s inequality, we can similarly get 𝑢_𝑥(𝑡)_𝐿2𝑛≤𝑒^2𝑀𝑡𝑢_𝑥(0)_𝐿2𝑛+^𝑡₀2𝜔𝑢_𝑥𝑥(𝜏)_𝐿2𝑛+𝑑𝜏_𝑡0𝜕²_𝑥𝑝∗𝑢²+𝛼²2𝑢²_𝑥+12𝜌²+𝜌_𝐿2𝑛𝑑𝜏. (117) Taking the limits, one gets 𝑢_𝑥(𝑡)_𝐿∞≤𝑒^2𝑀𝑡𝑢_𝑥(0)_𝐿∞+^𝑡₀2𝜔𝑢_𝑥𝑥(𝜏)_𝐿∞+𝑑𝜏_𝑡0𝜕²_𝑥𝑝∗𝑢²+𝛼²2𝑢²_𝑥+12𝜌²+𝜌_𝐿∞𝑑𝜏. (118) Differentiating the second equation of (10), one has 𝜌_𝑡𝑥+𝜌_𝑥𝑥𝑢+2𝜌_𝑥𝑢_𝑥+𝜌𝑢_𝑥𝑥+𝑢_𝑥𝑥=0, (119) and using the same method, we can obtain _ℝ𝜌_𝑥^2𝑛−1𝜌_𝑡𝑥𝑑𝑥+_ℝ𝜌_𝑥^2𝑛−1𝜌_𝑥𝑥𝑢𝑑𝑥+2_ℝ𝜌_𝑥^2𝑛−1𝜌_𝑥𝑢_𝑥+𝑑𝑥_ℝ𝜌_𝑥^2𝑛−1𝜌𝑢_𝑥𝑥𝑑𝑥+_ℝ𝜌_𝑥^2𝑛−1𝑢_𝑥𝑥𝑑𝑥=0. (120) In a similar way, we should deal with the second term as follows: _ℝ𝜌_𝑥^2𝑛−1𝜌_𝑥𝑥𝑢𝑑𝑥=_ℝ𝜌_𝑥^2𝑛2𝑛_𝑥=1𝑢𝑑𝑥2𝑛_ℝ𝜌_𝑥^2𝑛𝑢_𝑥𝑑𝑥, (121) thus we can get the estimates of ‖𝜌_𝑥(𝑥,𝑡)‖_𝐿∞ as follows: 𝜌_𝑥(𝑡)_𝐿∞≤𝑒^3𝑀𝑡𝜌_𝑥(0)_𝐿∞+^𝑡₀𝑢_{𝑥𝑥𝐿∞}𝜌_𝐿∞+𝑢_{𝑥𝑥𝐿∞}𝑑𝜏. (122)
In what follows, we use the weight function to get the desired result. We shall now introduce the weight function 𝜑_𝑁(𝑥) with 𝑁∈ℤ⁺, which is independent of 𝑡 as follows: 𝜑_𝑁(𝑥)=𝑒1,𝑥≤0,_𝜃𝑥(),𝑒,𝑥∈0,𝑁_𝜃𝑁,𝑥≥𝑁. (123) Note that 0≤𝜑^′_𝑁(𝑥)≤𝜑_𝑁(𝑥). Multiplying the first equation of (10) and (106) by 𝜑_𝑁, we have 𝜑_𝑁𝑢_𝑡+𝜑_𝑁𝑢𝑢_𝑥+2𝜔𝜑_𝑁𝑢_𝑥+𝜑_𝑁𝜕_𝑥𝑝∗𝑢²+𝛼²2𝑢²_𝑥+12𝜌²𝜑+𝜌=0,_𝑁𝑢_𝑥𝑡+𝜑_𝑁𝑢𝑢_𝑥𝑥+𝜑_𝑁𝑢²_𝑥+2𝜔𝜑_𝑁𝑢_𝑥𝑥+𝜑_𝑁𝜕²_𝑥𝑝∗𝑢²+𝛼²2𝑢²_𝑥+12𝜌²+𝜌=0. (124) In order to get the estimate of 𝑢_𝑥𝜑_𝑁, we need to eliminate the second derivatives as follows: _ℝ𝜑_𝑁𝑢𝑢_𝑥𝑥𝑢_𝑥𝜑_𝑁^2𝑛−1=𝑑𝑥_ℝ𝑢𝑢_𝑥𝜑_𝑁^2𝑛−1𝑢_𝑥𝜑_𝑁𝑥−𝑢_𝑥𝜑^′_𝑁=𝑑𝑥_ℝ𝑢(𝑢_𝑥𝜑_𝑁)^2𝑛2𝑛_𝑥𝑑𝑥−_ℝ𝑢𝑢_𝑥𝜑^′_𝑁𝑢_𝑥𝜑_𝑁^2𝑛−1‖𝑢(𝑡𝑑𝑥≤2)‖_𝐿^∞+𝑢_𝑥(𝑡)_𝐿∞𝑢_𝑥𝜑_𝑁𝐿^2𝑛_2𝑛, (125) where we use the fact 0≤𝜑^′_𝑁(𝑥)≤𝜑_𝑁(𝑥). Hence, as in the weightless case, we get the following inequality in view of the estimates of ‖𝑢(𝑥,𝑡)‖_𝐿^∞ and ‖𝑢_𝑥(𝑥,𝑡)‖_𝐿∞: 𝑢(𝑡)𝜑_𝑁𝐿∞+𝑢_𝑥(𝑡)𝜑_𝑁𝐿∞≤𝑒^2𝑀𝑡𝑢(0)𝜑_𝑁𝐿∞+𝑢_𝑥(0)𝜑_𝑁𝐿∞+𝑒^{2𝑀𝑡𝑡}₀𝜑_𝑁𝜕_𝑥𝑝∗𝑢²+𝛼²2𝑢²_𝑥+12𝜌²+𝜌_𝐿∞𝑑𝜏+𝑒^{2𝑀𝑡𝑡}₀𝜑_𝑁𝜕²_𝑥𝑝∗𝑢²+𝛼²2𝑢²_𝑥+12𝜌²+𝜌_𝐿∞𝑑𝜏. (126) Similarly, we get estimates for 𝜌𝜑_𝑁 and 𝜌_𝑥𝜑_𝑁 as follows: 𝜌𝜑_𝑁𝐿∞+𝜌_𝑥𝜑_𝑁𝐿∞≤𝑒^2𝑀𝑡𝜌(0)𝜑_𝑁𝐿∞+𝜌_𝑥(0)𝜑_𝑁𝐿∞. (127) On the other hand, a simple calculation shows that there exists 𝐶>0, depending only on 𝜃∈(0,1) such that for any 𝑁∈ℤ⁺, 𝜑_𝑁(𝑥)_ℝ𝑒^{−|(𝑥−𝑦)/𝛼|}1𝜑_𝑁𝑦𝑑𝑦≤𝐶. (128) Therefore for any appropriate function 𝑔, one can get 𝜑_𝑁𝜕_𝑥𝐺(𝑥)∗𝑔²(𝑥=1)𝜑2𝛼^𝑁(𝑥)_ℝ𝑒^{−|(𝑥−𝑦)/𝛼|}𝑔²𝑦=1𝑑𝑦𝜑2𝛼^𝑁(𝑥)_ℝ𝑒^{−|(𝑥−𝑦)/𝛼|}1𝜑_𝑁𝑦𝜑_𝑁𝑦𝑔𝑦𝑔𝑦≤1𝑑𝑦𝜑2𝛼^𝑁(𝑥)_ℝ𝑒^{−|(𝑥−𝑦)/𝛼|}1𝜑_𝑁𝑦𝑑𝑦𝑔𝜑_𝑁𝐿∞𝑔_𝐿∞≤𝐶^′𝑔𝜑_𝑁𝐿∞𝑔_𝐿∞, (129) and using the same method, 𝜑_𝑁𝜕²_𝑥𝐺(𝑥)∗𝑔²(𝑥)≤𝐶^′𝑔𝜑_𝑁𝐿∞𝑔_𝐿∞. (130) Besides, 𝜑_𝑁𝜕²_𝑥=𝜑𝑝∗𝜌_𝑁𝑝_𝑥∗𝜌_𝑥≤𝜑_𝑁𝑝_𝑥𝐿2𝜌_𝑥𝐿2≤𝐶^′𝜑_𝑁𝜌_𝑥𝐿∞. (131) In a similar way, we have 𝜑_𝑁𝜕_𝑥𝑝∗𝜌≤𝐶^′𝜑_𝑁𝜌_𝐿∞. (132) Then we can get the following estimate: 𝑢(𝑡)𝜑_𝑁𝐿∞+𝑢_𝑥(𝑡)𝜑_𝑁𝐿∞+𝜌(𝑡)𝜑_𝑁𝐿∞+𝜌_𝑥(𝑡)𝜑_𝑁𝐿∞≤𝜅₀𝑢(0)𝜑_𝑁𝐿∞+𝑢_𝑥(0)𝜑_𝑁𝐿∞+𝜌(0)𝜑_𝑁𝐿∞+𝜌_𝑥(0)𝜑_𝑁𝐿∞+𝜅₀^𝑡₀𝑢(𝑡)𝜑_𝑁𝐿∞+𝑢_𝑥(𝑡)𝜑_𝑁𝐿∞+𝜌(𝑡)𝜑_𝑁𝐿∞+𝜌_𝑥(𝑡)𝜑_𝑁𝐿∞𝑑𝑡. (133) Using Gronwall’s inequality of integral form, one can get 𝑢(𝑡)𝜑_𝑁𝐿∞+𝑢_𝑥(𝑡)𝜑_𝑁𝐿∞+𝜌(𝑡)𝜑_𝑁𝐿∞+𝜌_𝑥(𝑡)𝜑_𝑁𝐿∞≤𝜅₀𝑢(0)𝜑_𝑁𝐿∞+𝑢_𝑥(0)𝜑_𝑁𝐿∞+𝜌(0)𝜑_𝑁𝐿∞+𝜌_𝑥(0)𝜑_𝑁𝐿∞. (134) Taking the limit as 𝑁 goes to infinity, we can obtain 𝑢(𝑥,𝑡)𝑒^𝜃𝑥+𝑢_𝑥(𝑥,𝑡)𝑒^𝜃𝑥+𝜌(𝑥,𝑡)𝑒^𝜃𝑥+𝜌_𝑥(𝑥,𝑡)𝑒^𝜃𝑥≤𝜅₀𝑢(0)𝑒^𝜃𝑥+𝑢_𝑥(0)𝑒^𝜃𝑥+𝜌(0)𝑒^𝜃𝑥+𝜌_𝑥(0)𝑒^𝜃𝑥. (135) This completes the proof.

Acknowledgments

This work was partially supported by Zhejiang Provincial Natural Science Foundation of China under LQ12A01009 and LY12A01014 and NSFC under Grant no. 11226172.