Blowup Phenomena for a Modified Dullin-Gottwald-Holm Shallow Water System

Abstract

We discuss blowup phenomena for a modified two-component Dullin-Gottwald-Holm shallow water system. In this paper, some new blowup criteria of strong solutions involving the density and suitable integral form of the momentum are established.

1. Introduction

We consider the following two-component DGH type system: 𝑦_𝑡+2𝜔𝑢_𝑥+𝑢𝑦_𝑥+2𝑦𝑢_𝑥+𝛾𝑢_𝑥𝑥𝑥+𝑔𝜌𝜌_𝑥=0,𝑦=𝑢−𝛼²𝑢_𝑥𝑥,𝜌_𝑡+𝜌𝑢_𝑥=0, (1) where 𝑢=𝑢(𝑥,𝑡), (𝑥,𝑡)∈(ℝ,ℝ⁺) denotes the velocity field, 𝑔 is the downward constant acceleration of gravity in applications to shallow water waves, and 𝜌=(1−𝜕²_𝑥)(𝜌−𝜌₀), where 𝜌₀ is taken to be a constant. It is obvious that if 𝜌≡0, then (1) reduces to the well-known Dullin-Gottwald-Holm equation [1] (DGH equation for short). There are some contributions to DGH equation concerning the well-posedness, scattering problem, blowup phenomenon, and so forth; see, for example, [2–5] and references therein. We find that (1) is expressed in terms of an averaged filtered density component 𝜌 in analogy to the relation between momentum and velocity by setting 𝜌=(1−𝜕²_𝑥)(𝜌−𝜌₀) and the velocity component 𝑢. The idea is actually from the recent work [6]. Our modification breaks the structure of DGH2 system derived by following Ivanov’s approach [7] by the authors in [8]. The motivation of current research is stated as follows. From geometric point of view, (1) 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 𝑢 and the 𝐻¹-norm on filtered density. From a physical point of view, (1) admits wave breaking phenomena in finite time which attracts researchers’ interest. We also find that the 𝐻¹-norm of (𝑢,𝜌) is conserved with respect to time variable. This makes further different discussions on the singularities, unlike those for the DGH2 system or two-component Camassa-Holm system, possible. In the previous works [9–11] on the two-component Camassa-Holm equation and its modified version, blowup conditions were established in view of the negativity of initial velocity slope at some point; basically, the initial integral form of momentum is never involved. That is why we consider this kind of blowup condition in this paper. Precisely, we show the solutions blowup in finite time provided that the initial density and momentum satisfy certain sign conditions. To our knowledge, less results exist yet for the formation of singularities of (1) although the approaches we applied here are standard. The methods in previous works cannot be moved to this model parallelly. For convenience, let 𝑣=𝜌−𝜌₀ and Λ=(1−𝛼²𝜕²_𝑥)⁻¹; then the operator Λ can be expressed by its associated Green’s function 𝐺(𝑥)=(1/2𝛼)𝑒^{−|𝑥/𝛼|} with Λ𝑓(𝑥)=𝐺∗𝑓(𝑥)=_ℝ𝐺𝑥−𝑦𝑓𝑦𝑑𝑦. (2) Using this identity, system (1) takes an equivalent form of a quasilinear evolution equation of hyperbolic type as follows: 𝑢_𝑡+𝑢_𝑥𝛾𝑢−𝛼²=−𝜕_𝑥𝐺∗𝑢²+𝛼²2𝑢²_𝑥+𝛾2𝜔+𝛼²𝑔𝑢+2𝑣²−𝑔2𝑣²_𝑥,𝑣_𝑡+𝑢𝑣_𝑥=−𝐺∗𝑢_𝑥𝑣_𝑥𝑥+𝑢_𝑥𝑣. (3)

The current paper is based on some results on the Camassa-Holm equation [12–19] and its two-component generalizations [20–27]. We investigate further formation of singularities of solutions to (3) with the case of 𝑔=1 and 𝛼>0, just for simplicity mathematically. This paper is organized as follows. In Section 2, we recall some preliminary results on the well-posedness and blowup scenario. In Section 3, the detailed blowup conditions are presented.

2. Preliminaries

In this section, for completeness, we recall some elementary results and skip their proofs since they are not the main concern of this work. For convenience, in what follows, we let 𝜆=−𝛾/𝛼² and 2𝜅=2𝜔+𝛾/𝛼².

We can apply Kato’s theory [28] to establish the following local well-posedness theorem for (3).

Theorem 1. Assume an initial data (𝑢₀,𝑣₀)∈𝐻^𝑠×𝐻^𝑠−1,𝑠≥5/2. Then there exists a maximal 𝑇=𝑇(‖𝑢₀,𝑣₀‖_{𝐻𝑠×𝐻𝑠−1})>0 and a unique solution (𝑢,𝑣)∈𝐶[0,𝑇);𝐻^𝑠×𝐻^𝑠−1∩𝐶¹[0,𝑇);𝐻^𝑠−1×𝐻^𝑠−2 (4) of system (3). Moreover, the solution (𝑢,𝑣) depends continuously on the initial value (𝑢₀,𝑣₀), and the maximal time of existence 𝑇>0 is independent of 𝑠.

The proof of Theorem 1 is similar to the one in [11]. Moreover, using the techniques in [11], one can get the criterion for finite time wave breaking to (3) as follows.

Theorem 2. Let (𝑢₀,𝑣₀)∈𝐻^𝑠×𝐻^𝑠−1 with 𝑠≥5/2, and let 𝑇>0 be the maximal time of existence of the solution (𝑢,𝑣) to (3) with initial data (𝑢₀,𝑣₀). Then the corresponding solution (𝑢,𝑣) blowsup in finite time if and only if lim_𝑡↑𝑇inf_𝑥∈ℝ𝑢_𝑥(𝑡,𝑥)=−∞. (5)

Lemma 3 (see [29]). Assume that a differentiable function 𝑦(𝑡) satisfies 𝑦^′(𝑡)≤−𝐶𝑦²(𝑡)+𝐾 (6) with constants 𝐶,𝐾>0. If the initial datum 𝑦(0)=𝑦₀<−𝐾/𝐶, then the solution to (9) goes to −∞ before 𝑡 tends to 1/(−𝐶𝑦₀+𝐾/𝑦₀).

Lemma 4 (see [19]). Suppose that Ψ(𝑡) is twice continuously differential satisfying Ψ^′′(𝑡)≥𝐶₀Ψ^′(𝑡)Ψ(𝑡),𝑡>0,𝐶₀Ψ(𝑡)>0,>0,Ψ^′(𝑡)>0. (7) Then Ψ blowsup in finite time. Moreover the blowup time can be estimated in terms of the initial datum as 2𝑇≤max𝐶₀Ψ(0),Ψ(0)Ψ^′(0). (8)

We also need to introduce the standard particle trajectory method for later use. Consider now the following two initial value problems: 𝑞_1,𝑡=𝑢𝑡,𝑞₁[),𝑞+𝜆,𝑡∈0,𝑇₁()𝑞0,𝑥=𝑥,𝑥∈ℝ,_2,𝑡=𝑢𝑡,𝑞₂[),𝑞,𝑡∈0,𝑇₂(0,𝑥)=𝑥,𝑥∈ℝ, (9)(10) where 𝑢∈𝐶¹([0,𝑇),𝐻^𝑠−1) is the first component of the solution (𝑢,𝑣) to system (3) with initial data (𝑢₀,𝑣₀)∈𝐻^𝑠×𝐻^𝑠−1(𝑠≥5/2), and 𝑇>0 is the maximal time of existence. By direct computation, we have 𝑞_{𝑖,𝑡𝑥}(𝑡,𝑥)=𝑢_𝑥𝑡,𝑞_𝑖(𝑡,𝑥)𝑞_𝑖,𝑥(𝑡,𝑥),𝑖=1,2. (11) Then, 𝑞_𝑖,𝑥(𝑡,𝑥)=exp^𝑡₀𝑢_𝑥𝜏,𝑞_𝑖(𝜏,𝑥)𝑑𝜏>0,𝑡>0,𝑥∈ℝ, (12) which means that 𝑞_𝑖(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 diffeomorphisms 𝑞_𝑖(𝑡,⋅); that is, ‖𝑣(𝑡,⋅)‖_𝐿^∞=𝑣𝑡,𝑞₁(𝑡,⋅)_𝐿∞=𝑣𝑡,𝑞₂(𝑡,⋅)_𝐿∞,𝑡∈[0,𝑇). (13) Similarly, inf_𝑥∈ℝ𝑣(𝑡,𝑥)=inf_𝑥∈ℝ𝑣𝑡,𝑞₁(𝑡,𝑥)=inf_𝑥∈ℝ𝑣𝑡,𝑞₂(),[),𝑡,𝑥𝑡∈0,𝑇sup_𝑥∈ℝ𝑣(𝑡,𝑥)=sup_𝑥∈ℝ𝑣𝑡,𝑞₁(𝑡,𝑥)=sup_𝑥∈ℝ𝑣𝑡,𝑞₂(,[).𝑡,𝑥)𝑡∈0,𝑇 (14)

3. Blowup Phenomenon

In this section, we show that blowup phenomenon is the only one way that singularity arises in smooth solutions. We start this section with the following useful lemma.

Lemma 5. Let 𝑋₀=(𝑢₀,𝑣₀)∈𝐻^𝑠×𝐻^𝑠−1, 𝑠≥2. 𝑇 is assumed to be the maximal existence time of the solution 𝑋=(𝑢,𝑣) to system (3) corresponding to the initial data 𝑋₀. Then for all 𝑡∈[0,𝑇), one has the following conservation law: 𝐸(𝑡)=_ℝ𝑢²+𝛼²𝑢²_𝑥+𝑣²+𝑣²_𝑥𝑑𝑥. (15)

Proof. We will prove that 𝐸(𝑡) is a conserved quantity with respect to time variable. Here we use the classical energy method. Multiplying the first equation in (3) by 𝑢(𝑥,𝑡) and integrating by parts, we obtain _ℝ𝑢𝑢_𝑡𝑑𝑥+_ℝ𝛼²𝑢_𝑥𝑢_𝑥𝑡𝑑𝑥=−_ℝ𝑢𝑣_𝑥𝑣−𝑣_𝑥𝑥𝑑𝑥. (16) Similarly, we have the following inequality for the second equation (3): _ℝ𝑣𝑣_𝑡+𝑣_𝑥𝑣_𝑥𝑡𝑑𝑥=_ℝ𝑢𝑣_𝑥𝑣−𝑣_𝑥𝑥𝑑𝑥. (17) This implies that _ℝ𝑢𝑢_𝑡+𝛼²𝑢_𝑥𝑢_𝑥𝑥+𝑣𝑣_𝑡+𝑣_𝑥𝑣_𝑥𝑥𝑑𝑥=0. (18) Thus, we have 𝑑𝑑𝑡_ℝ𝑢²+𝛼²𝑢²_𝑥+𝑣²+𝑣²_𝑥𝑑𝑥=2_ℝ𝑢𝑢_𝑡+𝛼²𝑢_𝑥𝑢_𝑥𝑥+𝑣𝑣_𝑡+𝑣_𝑥𝑣_𝑥𝑥𝑑𝑥=0. (19) This completes the proof.

Using this conservation law, we obtain ‖𝑢(⋅,𝑡)‖²_𝐿^∞_(ℝ)+‖𝑣(⋅,𝑡)‖²_𝐿^∞_(ℝ)≤1‖𝑢(2𝛼⋅,𝑡)‖²_{𝐻1𝛼(ℝ)}+12‖𝑣(⋅,𝑡)‖²_𝐻1(ℝ)≤𝐶₁𝐸(0), (20) where 𝐶₁1=max,12𝛼2. (21)

Theorem 6. Suppose that 𝑋₀=(𝑢₀,𝑣₀)∈𝐻^𝑠×𝐻^𝑠−1, 𝑠≥5/2, 𝜌₀(𝑥₀)=𝑦₀(𝑥₀)+𝜅=0, and the initial data satisfies the following conditions:(i)𝜌₀(𝑥)≥0𝑜𝑛(−∞,𝑥₀) and 𝜌₀(𝑥)≤0on (𝑥₀,∞),(ii)∫^𝑥0_−∞𝑒^𝜉/𝛼(𝑦₀(𝜉)+𝜅)𝑑𝜉>0 and ∫^𝑥0_−∞𝑒^−𝜉/𝛼(𝑦₀(𝜉)+𝜅)𝑑𝜉<0,for some point 𝑥₀∈ℝ. Then the solution to system (3) with the initial value 𝑋₀ blowsup in finite time.

Proof. Differentiating the first equation of (3) with respect to 𝑥, we obtain 𝑢_𝑥𝑡+𝑢²_𝑥+𝑢𝑢_𝑥𝑥+𝜆𝑢_𝑥𝑥+𝜕²_𝑥𝐺∗𝑢²+𝛼²2𝑢²_𝑥1+2𝜅𝑢+2𝑣²−12𝑣²_𝑥=0. (22) Applying the relation 𝜕²_𝑥(𝐺∗𝑓)=(1/𝛼²)(𝐺∗𝑓−𝑓) yields 𝑢_𝑥𝑡+𝑢²_𝑥+𝑢𝑢_𝑥𝑥+𝜆𝑢_𝑥𝑥+1𝛼²𝐺∗𝑢²+𝛼²2𝑢²_𝑥1+2𝜅𝑢+2𝑣²−12𝑣²_𝑥−1𝛼²𝑢²+𝛼²2𝑢²_𝑥1+2𝜅𝑢+2𝑣²−12𝑣²_𝑥=0. (23) From (23) we have 𝑑𝑢𝑑𝑡^𝑥𝑞₁𝑥₀=𝑢,𝑡,𝑡_𝑥𝑡+𝑢𝑢_𝑥𝑥+𝜆𝑢_𝑥𝑥𝑞₁𝑥₀,𝑡,𝑡=−𝑢²_𝑥𝑞₁𝑥₀−1,𝑡,𝑡𝛼²𝐺∗𝑢²+𝛼²2𝑢²_𝑥1+2𝜅𝑢+2𝑣²−12𝑣²_𝑥𝑞₁𝑥₀+1,𝑡,𝑡𝛼²𝑢²+𝛼²2𝑢²_𝑥1+2𝜅𝑢+2𝑣²−12𝑣²_𝑥𝑞₁𝑥₀1,𝑡,𝑡≤−2𝑢²_𝑥𝑞₁𝑥₀+1,𝑡,𝑡2𝛼²(𝑢+𝜅)²𝑞₁𝑥₀+1,𝑡,𝑡2𝛼²𝑣²−𝑣²_𝑥𝑞₁𝑥₀−1,𝑡,𝑡2𝛼²𝐺∗𝑣²−𝑣²_𝑥𝑞₁𝑥₀,𝑡,𝑡, (24) where we used the fact proved in [30] that 𝐺∗(𝑢+𝜅)²+𝛼²2𝑢²_𝑥≥12(𝑢+𝜅)². (25) In order to arrive at our result, we need the following three claims.
Claim  1. 𝑦(𝑞₁(𝑥₀,𝑡),𝑡))+𝜅=0 for all 𝑡 in its lifespan; 𝑞₁ is defined in (9).
It is worth noting the equivalent form of the first equation in (3) in what follows: 𝑦_𝑡+𝑢𝑦_𝑥+2𝑦𝑢_𝑥+𝜆𝑦_𝑥+2𝜅𝑢_𝑥+𝜌𝑣_𝑥=0. (26) From the previous equation, we can get 𝑑𝑦𝑞𝑑𝑡¹(𝑥,𝑡),𝑡+𝜅𝑞²_1,𝑥()=𝑦𝑥,𝑡_𝑡+𝑢𝑦_𝑥+2𝑦𝑢_𝑥+𝜆𝑦_𝑥+2𝜅𝑢_𝑥𝑞₁(𝑥,𝑡),𝑡𝑞²_1,𝑥()𝑞𝑥,𝑡=−𝜌₁(𝑥,𝑡),𝑡𝑣_𝑥𝑞₁(𝑥,𝑡),𝑡𝑞²_1,𝑥(𝑥,𝑡). (27) Since 𝑞₂(𝑥,⋅) defined by (10) is a diffeomorphism of the line for any 𝑡∈[0,𝑇), so there exists an 𝑥₃(𝑡)∈ℝ such that 𝑞₂𝑡,𝑥₃(𝑡)=𝑞₁𝑡,𝑥₀,𝑡∈[0,𝑡). (28) When 𝑡=0, we have 𝑥₃(0)=𝑞₂0,𝑥₃(0)=𝑞₁0,𝑥₀=𝑥₀. (29) Now we prove that 𝜌(𝑡,𝑞₁(𝑡,𝑥₀))=0. It is easy to get 𝑑𝜌𝑑𝑡𝑡,𝑞₂𝑡,𝑥₃(𝑡)=−𝜌𝑢_𝑥𝑡,𝑞₂𝑡,𝑥₃(𝑡). (30) Since 𝜌₀𝑥₀=0, (31) integrating the previous equation, we can obtain 𝜌𝑡,𝑞₂𝑡,𝑥₃(𝑡)=𝜌0,𝑞₂0,𝑥₃(0)𝑒^{−∫𝑡0𝑢𝑥(𝜏,𝑞2(𝜏,𝑥3(𝜏)))𝑑𝜏}=𝜌₀𝑥₀𝑒^{−∫𝑡0𝑢𝑥(𝜏,𝑞2(𝜏,𝑥3(𝜏)))𝑑𝜏}=0; (32) thus we have 𝜌𝑡,𝑞₁𝑡,𝑥₀=𝜌𝑡,𝑞₂𝑡,𝑥₃(𝑡)=0. (33) So we can get 𝑑𝑦𝑞𝑑𝑡¹𝑥₀,𝑡,𝑡+𝜅𝑞²_1,𝑥𝑥₀𝑞,𝑡=−𝜌₁𝑥₀,𝑡,𝑡𝑣_𝑥𝑞₁𝑥₀,𝑡,𝑡𝑞²_1,𝑥𝑥₀,𝑡=0; (34)
then we have 𝑦𝑞₁𝑥₀,𝑡,𝑡+𝜅=𝑦₀𝑥₀+𝜅=0. (35) Our claim is proved.
Claim  2. For any fixed 𝑡, 𝑣²_𝑥(𝑥,𝑡)−𝑣²(𝑥,𝑡)≤𝑣²_𝑥(𝑞₁(𝑥₀,𝑡),𝑡)−𝑣²(𝑞₁(𝑥₀,𝑡),𝑡) for all 𝑥∈ℝ. For any fixed 𝑡, if 𝑥≤𝑞₁(𝑥₀,𝑡), then 𝑣²_𝑥(𝑥,𝑡)−𝑣²(𝑥,𝑡)=−^{𝑞1(𝑥0,𝑡)}_−∞𝑒^𝜉𝜌(𝜉,𝑡)𝑑𝜉−^𝑞1(𝑥0_𝑥^,𝑡)𝑒^𝜉𝜌()×𝜉,𝑡𝑑𝜉_{∞𝑞1𝑥0,𝑡}𝑒^−𝜉𝜌(𝜉,𝑡)𝑑𝜉+^𝑞1(𝑥0_𝑥^,𝑡)𝑒^−𝜉𝜌(𝜉,𝑡)𝑑𝜉=𝑣²_𝑥𝑞₁𝑥₀,𝑡,𝑡−𝑣²𝑞₁𝑥₀−,𝑡,𝑡_𝑥−∞𝑒^𝜉𝜌(𝜉,𝑡)𝑑𝜉^𝑞1(𝑥0_𝑥^,𝑡)𝑒^−𝜉𝜌()+𝜉,𝑡𝑑𝜉_{𝑞1(𝑥0𝑥,𝑡)}𝑒^𝜉𝜌(𝜉,𝑡)𝑑𝜉^∞_{𝑞1𝑥0,𝑡}𝑒^−𝜉𝜌(𝜉,𝑡)𝑑𝜉≤𝑣²_𝑥𝑞₁𝑥₀,𝑡,𝑡−𝑣²𝑞₁𝑥₀,𝑡,𝑡, (36) where the condition (i) is used. Similarly, for 𝑥≥𝑞₁(𝑥₀,𝑡), we also have 𝑣²_𝑥(𝑥,𝑡)−𝑣²(𝑥,𝑡)≤𝑣²_𝑥𝑞₁𝑥₀,𝑡,𝑡−𝑣²𝑞₁𝑥₀,𝑡,𝑡. (37) So Claim 2 is proved. Consequently, we can obtain 𝐺∗𝑣²−𝑣²_𝑥𝑞₁𝑥₀=1,𝑡,𝑡2𝛼_ℝ𝑒^{−|𝑞1(𝑥0,𝑡)−𝜉|/𝛼}𝑣²−𝑣²_𝑥)≥1(𝜉,𝑡𝑑𝜉2𝛼_ℝ𝑒^{−|𝑞1(𝑥0,𝑡)−𝜉|/𝛼}𝑣²−𝑣²_𝑥𝑞₁𝑥₀=𝑣,𝑡,𝑡𝑑𝜉₂−𝑣²_𝑥𝑞₁𝑥₀,𝑡,𝑡. (38) Thus, one can get 𝑑𝑢𝑑𝑡^𝑥𝑞₁𝑥₀1,𝑡,𝑡≤−2𝑢²_𝑥𝑞₁𝑥₀+1,𝑡,𝑡2𝛼²(𝑢+𝜅)²𝑞₁𝑥₀,𝑡,𝑡. (39)
Claim  3. (𝑢+𝜅)²(𝑞₁(𝑥₀,𝑡),𝑡)<𝛼²𝑢²_𝑥(𝑞₁(𝑥₀,𝑡),𝑡) for all 𝑡≥0. Furthermore, 𝑢_𝑥(𝑞₁(𝑥₀,𝑡),𝑡)<0 is strictly decreasing.
Suppose that there exists a 𝑡₀ such that (𝑢+𝜅)²(𝑞₁(𝑥₀,𝑡),𝑡)<𝛼²𝑢²_𝑥(𝑞₁(𝑥₀,𝑡),𝑡) on [0,𝑡₀) and (𝑢+𝜅)²(𝑞₁(𝑥₀,𝑡₀),𝑡₀)=𝛼²𝑢²_𝑥(𝑞₁(𝑥₀,𝑡₀),𝑡₀). From the expression of 𝑢(𝑥,𝑡) in terms of 𝑦(𝑥,𝑡), we can rewrite 𝑢(𝑥,𝑡)+𝜅 and 𝑢_𝑥(𝑥,𝑡) as follows: 𝑢()1𝑥,𝑡+𝜅=𝑒2𝛼^{−𝑥/𝛼𝑥}_−∞𝑒^𝜉/𝛼𝑦()+1𝜉,𝑡+𝜅𝑑𝜉𝑒2𝛼^𝑥/𝛼∞_𝑥𝑒^−𝜉/𝛼𝑦()𝑢𝜉,𝑡+𝜅𝑑𝜉,_𝑥()1𝑥,𝑡=−2𝛼²𝑒^{−𝑥/𝛼𝑥}_−∞𝑒^𝜉/𝛼𝑦()+1𝜉,𝑡+𝜅𝑑𝜉2𝛼²𝑒^𝑥/𝛼∞_𝑥𝑒^−𝜉/𝛼𝑦(𝜉,𝑡)+𝜅𝑑𝜉. (40) Letting 𝐼(𝑡)=𝑒^{−𝑞1(𝑥0,𝑡)/𝛼𝑞1(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼𝑦()(𝑡)𝜉,𝑡+𝜅𝑑𝜉,𝐼𝐼=𝑒^{𝑞1(𝑥0,𝑡)/𝛼∞}_{𝑞1𝑥0,𝑡}𝑒^−𝜉/𝛼𝑦(𝜉,𝑡)+𝜅𝑑𝜉, (41) then 𝑑𝐼(𝑡)1𝑑𝑡=−𝛼𝑢𝑞₁𝑥₀,𝑡,𝑡+𝜆𝑒^{−𝑞1(𝑥0,𝑡)/𝛼}×^{𝑞1(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼𝑦(𝜉,𝑡)+𝜅𝑑𝜉+𝑒^{−𝑞1(𝑥0,𝑡)/𝛼𝑞1(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼𝑦_𝑡(𝜉,𝑡)𝑑𝜉. (42) Integrating by parts, the first term of (42) yields −1𝛼𝑢𝑞₁𝑥₀,𝑡,𝑡+𝜆𝑒^{−𝑞1(𝑥0,𝑡)/𝛼}×^{𝑞1(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼𝑦()=𝜉,𝑡+𝜅𝑑𝜉𝛼𝑢𝑢_𝑥−𝑢²−𝜅𝑢𝑞₁𝑥₀−1,𝑡,𝑡𝛼𝑒^{−𝑞1(𝑥0,𝑡)/𝛼𝑞1(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼𝜆𝑦(𝜉,𝑡)+𝜅𝑑𝜉. (43) For the second term of (42), we have the following equation in the view of Claim 1: 𝑒^{−𝑞1(𝑥0,𝑡)/𝛼𝑞1(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼𝑦_𝑡(𝜉,𝑡)𝑑𝜉=−𝑒^{−𝑞1(𝑥0,𝑡)/𝛼𝑞1(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼𝑦+𝜅𝑢_𝑥+12𝑢²−𝛼²𝑢²_𝑥𝑥+𝜆𝑦+𝜅_𝑥+𝜅𝑢_𝑥+𝜌𝑣_𝑥=𝛼𝑑𝜉₂2𝑢²_𝑥−12𝑢²−𝜅𝑢𝑞₁𝑥₀,𝑡,𝑡−𝑒^{−𝑞1(𝑥0,𝑡)/𝛼𝑞1(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼𝜌𝑣_𝑥+1𝑑𝜉𝛼𝑒^{−𝑞1(𝑥0,𝑡)/𝛼𝑞1(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼1𝑦+𝜅𝑢+2𝑢²−𝛼²𝑢²_𝑥=𝛼+𝜆𝑦+𝜅+𝜅𝑢𝑑𝜉₂2𝑢²_𝑥−12𝑢²1−𝜅𝑢+2𝑣²_𝑥−𝑣²𝑞₁𝑥₀+1,𝑡,𝑡𝑒2𝛼^{−𝑞1(𝑥0,𝑡)/𝛼𝑞1(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼𝑣²−𝑣²_𝑥+1𝑑𝜉𝛼𝑒^{−𝑞1(𝑥0,𝑡)/𝛼𝑞1(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼32𝑢²−𝛼²2𝑢²_𝑥−𝛼²𝑢𝑢_𝑥𝑥=𝛼+𝜆𝑦+𝜅+2𝜅𝑢𝑑𝜉₂2𝑢²_𝑥−𝛼𝑢𝑢_𝑥1−𝜅𝑢+2𝑣²_𝑥−𝑣²𝑞₁𝑥₀+1,𝑡,𝑡𝑒2𝛼^{−𝑞1(𝑥0,𝑡)/𝛼𝑞1(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼𝑣²−𝑣²_𝑥+1𝑑𝜉𝛼𝑒^{−𝑞1(𝑥0,𝑡)/𝛼𝑞1(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼𝑢²+𝛼²2𝑢²_𝑥+𝜆𝑦+𝜅+2𝜅𝑢𝑑𝜉. (44) Here we have used −𝛼𝑒^{−𝑞1(𝑥0,𝑡)/𝛼𝑞1(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼𝑢𝑢_𝑥𝑥(𝜉,𝑡)𝑑𝜉=−𝛼𝑢𝑢_𝑥𝑞₁𝑥₀+1,𝑡,𝑡2𝑢²𝑞₁𝑥₀,𝑡,𝑡+𝛼𝑒^{−𝑞1(𝑥0,𝑡)/𝛼𝑞1(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼𝑢²_𝑥−12𝛼²𝑢²(𝜉,𝑡)𝑑𝜉. (45) Combining the previous equations together, and with the help of (38), (42) reads as 𝑑𝐼(𝑡)=𝛼𝑑𝑡2²𝑢²_𝑥−𝑢²−2𝜅𝑢𝑞₁𝑥₀+1,𝑡,𝑡𝛼𝑒^{−𝑞1(𝑥0,𝑡)/𝛼𝑞1(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼𝑢²+𝛼²2𝑢²_𝑥+1+2𝜅𝑢𝑑𝜉2𝑣²_𝑥−𝑣²𝑞₁𝑥₀+1,𝑡,𝑡𝑒2𝛼^{−𝑞1(𝑥0,𝑡)/𝛼}×^{𝑞1(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼𝑣²−𝑣²_𝑥=𝛼𝑑𝜉₂2𝑢²_𝑥−(𝑢+𝜅)²𝑞₁𝑥₀+1,𝑡,𝑡𝛼𝑒^{−𝑞1(𝑥0,𝑡)/𝛼}×^{𝑞1(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼𝛼²2𝑢²_𝑥+(𝑢+𝜅)²+1𝑑𝜉2𝑣²_𝑥−𝑣²𝑞₁𝑥₀+1,𝑡,𝑡𝑒2𝛼^{−𝑞1(𝑥0,𝑡)/𝛼}×^{𝑞1(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼𝑣²−𝑣²_𝑥≥1𝑑𝜉2𝛼²𝑢²_𝑥−(𝑢+𝜅)²𝑞₁𝑥₀,𝑡,𝑡>0,on0,𝑡₀, (46) where Claim 2 and the inequality [30] ^𝑥_−∞𝑒^𝜉/𝛼(𝑢+𝜅)²+𝛼²2𝑢²_𝜉()𝛼𝜉,𝑡𝑑𝜉≥2𝑒^𝑥/𝛼(𝑢+𝜅)² (47) have been used. From the continuity property, we have 𝑒^{−𝑞1(𝑥0,𝑡0)/𝛼𝑞1(𝑥0,𝑡0)}_−∞𝑒^𝜉/𝛼𝑦𝜉,𝑡₀+𝜅𝑑𝜉>𝑒^{−𝑥0/𝛼𝑥0}_−∞𝑒^𝜉/𝛼𝑦₀(𝜉)+𝜅𝑑𝜉>0. (48) Similarly, 𝑑𝐼𝐼(𝑡)≤1𝑑𝑡2(𝑢+𝜅)²−𝛼²𝑢²_𝑥𝑞₁𝑥₀,𝑡,𝑡<0,on0,𝑡₀. (49) Thus, by continuity property, 𝑒^{𝑞1(𝑥0,𝑡0)/𝛼𝑞1(𝑥0,𝑡0)}_−∞𝑒^−𝜉/𝛼𝑦𝜉,𝑡₀+𝜅𝑑𝜉<𝑒^{𝑥0/𝛼𝑥0}_−∞𝑒^−𝜉/𝛼𝑦₀(𝜉)+𝜅𝑑𝜉<0. (50) Summarizing (48) and (50), we obtain 𝛼²𝑢²_𝑥𝑞₁𝑥₀,𝑡₀,𝑡₀−(𝑢+𝜅)²𝑞₁𝑥₀,𝑡₀,𝑡₀1=−𝛼^{2𝑞1(𝑥0,𝑡0)}_−∞𝑒^𝜉/𝛼𝑦𝜉,𝑡₀×+𝜅𝑑𝜉_{∞𝑞1𝑥0,𝑡0}𝑒^−𝜉/𝛼𝑦𝜉,𝑡₀1+𝜅𝑑𝜉>−𝛼^2𝑥0_−∞𝑒^𝜉/𝛼𝑦₀(𝜉)+𝜅𝑑𝜉^∞_𝑥0𝑒^−𝜉/𝛼𝑦₀(𝜉)+𝜅𝑑𝜉=𝛼²𝑢²_0𝑥𝑥₀−𝑢₀+𝜅²𝑥₀>0. (51) That is a contradiction. On the other hand, from the expression of 𝑢_𝑥(𝑥,𝑡) in terms of 𝑦(𝑥,𝑡), we can easily get that 𝑢_𝑥(𝑞₁(𝑥₀,𝑡),𝑡)<0. So we complete the proof of Claim 3.
Furthermore, due to (46) and (49), we can obtain 𝑑𝛼𝑑𝑡²𝑢²_𝑥−(𝑢+𝜅)²𝑞₁𝑥₀1,𝑡,𝑡=−𝛼²𝑑𝑑𝑡^{𝑞1(𝑥0,𝑡)}_−∞𝑒^𝜉/𝛼𝑦()×𝜉,𝑡+𝜅𝑑𝜉_{∞𝑞1𝑥0,𝑡}𝑒^−𝜉/𝛼𝑦()1𝜉,𝑡+𝜅𝑑𝜉≥−2𝛼²𝛼²𝑢²_𝑥−(𝑢+𝜅)²𝑞₁𝑥₀,𝑡,𝑡𝑒^{𝑞1(𝑥0,𝑡)/𝛼}×^𝑥_−∞𝑒^−𝜉/𝛼𝑦()+1𝜉,𝑡+𝜅𝑑𝜉2𝛼²𝛼²𝑢²_𝑥−(𝑢+𝜅)²𝑞₁𝑥₀,𝑡,𝑡𝑒^{−𝑞1(𝑥0,𝑡)/𝛼}×^∞_𝑥𝑒^𝜉/𝛼𝑦(𝜉,𝑡)+𝜅𝑑𝜉=−𝑢_𝑥𝑞₁𝑥₀,𝑡,𝑡𝛼²𝑢²_𝑥−(𝑢+𝜅)²𝑞₁𝑥₀,𝑡,𝑡. (52) Integrating (39) and then substituting it into the previous inequality, we have 𝑑𝛼𝑑𝑡²𝑢²_𝑥−(𝑢+𝜅)²𝑞₁𝑥₀≥1,𝑡,𝑡2𝛼²𝛼²𝑢²_𝑥−(𝑢+𝜅)²𝑞₁𝑥₀×,𝑡,𝑡_𝑡0𝛼²𝑢²_𝑥−(𝑢+𝜅)²𝑞₁𝑥₀,𝜏,𝜏𝑑𝜏−2𝛼²𝑢_0𝑥𝑥₀. (53) Let Ψ(𝑡)=∫^𝑡₀(𝛼²𝑢²_𝑥−(𝑢+𝜅)²)(𝑞₁(𝑥₀,𝜏),𝜏)𝑑𝜏−2𝛼²𝑢_0𝑥(𝑥₀); then we can complete the proof with the help of Lemma 4.

Remark 7. We note that if the condition (i) is replaced by the following one:(i^′)𝜌₀(𝑥)≤0on(−∞,𝑥₀) and 𝜌₀(𝑥)≥0on(𝑥₀,∞),then Claim 2 also holds; that is, the theorem always holds with anyone of (i) and (i^').

As a corollary of Theorem 6, we have the following.

Theorem 8. Suppose that 𝑋₀=(𝑢₀,𝑣₀)∈𝐻^𝑠×𝐻^𝑠−1,𝑠≥5/2, and the initial data satisfies the following conditions:(i)𝜌₀(𝑥)≥0  on (−∞,𝑥₀) and 𝜌₀(𝑥)≤0  on (𝑥₀,∞)(or 𝜌₀(𝑥)≤0  on (−∞,𝑥₀) and 𝜌₀(𝑥)≥0  on (𝑥₀,∞)),(ii)𝑢^′₀(𝑥₀)≤−(𝐶₁𝐸₀+|𝜅|)/𝛼,for some point 𝑥₀∈ℝ. Then the solution to system (3) with the initial value 𝑋₀ blowsup in finite time.

Proof. As shown in Theorem 6, condition (i) guarantees that 𝑣²_𝑥(𝑥,𝑡)+𝑣²(𝑥,𝑡)≤(𝑣²_𝑥−𝑣²)(𝑞₁(𝑥₀,𝑡),𝑡) for all 𝑥∈ℝ. Then, 𝑑𝑢𝑑𝑡^𝑥𝑞₁𝑥₀1,𝑡,𝑡≤−2𝑢²_𝑥𝑞₁𝑥₀+1,𝑡,𝑡2𝛼²(𝑢+𝜅)²𝑞₁𝑥₀1,𝑡,𝑡≤−2𝑢²_𝑥𝑞₁𝑥₀+1,𝑡,𝑡2𝛼²𝐶₁𝐸₀+|𝜅|²1∶=−2𝑢²_𝑥𝑞₁𝑥₀,𝑡,𝑡+𝐾², (54) where 𝐾>0 is a constant. By setting 𝜑(𝑡)=𝑢_𝑥(𝑞₁(𝑥₀,𝑡),𝑡), we obtain 𝑑𝜑1𝑑𝑡=−2𝜑²+𝐾². (55) Applying Lemma 3, we have lim_𝑡↑𝑇𝜑(𝑡)1=−∞with𝑇=−(1/2)𝜑₀−𝐾²/𝜑₀, (56) when 𝜑₀<−√2𝐾=−𝐶₁𝐸₀+|𝜅|𝛼. (57) This completes the proof.

Acknowledgments

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