On Global Solutions for the Cauchy Problem of a Boussinesq-Type Equation

Abstract

We will give conditions which will guarantee the existence of global weak solutions of the Boussinesq-type equation with power-type nonlinearity 𝛾|𝑢|^𝑝 and supercritical initial energy. By defining new functionals and using potential well method, we readdressed the initial value problem of the Boussinesq-type equation for the supercritical initial energy case.

1. Introduction

This paper is devoted to the initial value problem of a Boussinesq-type equation: 𝑢_𝑡𝑡−𝑢_𝑥𝑥+𝑢_{𝑥𝑥𝑥𝑥}+𝑢_{𝑥𝑥𝑥𝑥𝑡𝑡}=𝑓(𝑢)_𝑥𝑥𝑢()(𝑥),𝑥∈𝑅,𝑡>0,𝑥,0=𝜑,𝑢_𝑡(𝑥,0)=𝜓(𝑥),𝑥∈𝑅, (1.1)(1.2) where 𝑓(𝑢)=𝛾|𝑢|^𝑝,𝛾>0. Equation (1.1) is of fundamental physical interest because it arises in the study of shallow water theory, nonlinear lattice theory, and some other physical phenomena. In the absence of the sixth-order term, (1.1) becomes the good Boussinesq equation and has been intensively studied from a mathematical viewpoint with various additional terms [1–9].

Concerning the initial value problem (1.1) and (1.2), it is important to cite the works of Xu et al. [10], Y. Z. Wang and Y. X. Wang, [11] and Wang and Xu [12].

Wang and Xu [12] investigated the Cauchy problem (1.1) and (1.2). They first proved the existence and nonexistence of global solutions of the problem for a general nonlinear function 𝑓(𝑢) provided that the antiderivative 𝐹 of 𝑓 satisfies 𝐹(𝑢)≥0 or 𝑓^′(𝑢) is bounded below. Then by the potential well method, they proved the global existence of solutions for a special case of the nonlinear term 𝑓(𝑢)=−𝛽|𝑢|^𝑝𝑢 with 0<𝐸(0)≤𝑑.

In [11], the initial value problem for a class of nonlinear wave equations of higher order: 𝑢_𝑡𝑡−𝑢_𝑥𝑥+𝑢_{𝑥𝑥𝑥𝑥}+𝑢_{𝑥𝑥𝑥𝑥𝑡𝑡}=𝜑(𝑢)_𝑥𝑥 (1.3) was considered. The global well-posedness of the initial value problem of (1.3) with 𝜑(𝑠)=𝛼|𝑠|^𝑝,𝛼≠0 was studied making use of the potential well method.

Very recently, Xu et al. [10] studied the multidimensional Boussinesq-type equation: 𝑢_𝑡𝑡−Δ𝑢+Δ²𝑢+Δ²𝑢_𝑡𝑡=Δ𝑓(𝑢), (1.4) with the nonlinear terms ±𝑎|𝑢|^𝑝,−𝑎|𝑢|^𝑝−1𝑢,±𝑎𝑢^2𝑘, and −𝑎𝑢^2𝑘+1. When 𝐸(0)<𝑑, the global existence and finite-time blowup of solutions were proved by the aid of the potential well method.

All of the above-mentioned three studies considered the cases 𝐸(0)<𝑑 or 0<𝐸(0)≤𝑑. The case 𝐸(0)>0 was also investigated in [12], but the conditions imposed on the nonlinear term are only valid for the odd-degree nonlinearities, that is, positivity condition is not enough for the nonlinear term 𝑓(𝑢)=𝛾|𝑢|^𝑝. In the present paper, we again investigate problem (1.1) and (1.2) and give the global existence of solutions for 𝑓(𝑢)=𝛾|𝑢|^𝑝 with supercritical initial energy. We emphasize that our main results would seem to be the first contribution to global well-posedness of the sixth-order Boussinesq equation with supercritical initial energy and this type of nonlinearity.

The plan of the paper is as follows. Section 2 contains some definitions and an abbreviated description of the global existence theory given in [10, 12]. Also it contains our first new functional and some theorems about sign preserving property of this functional. The global existence theory for supercritical initial energy is presented in Section 3, following the lines laid down by Kutev et al. [13]. For this purpose, we introduce the second new functional and give the invariance of this functional under the flow of (1.1) and (1.2). A few concluding remarks and an example are to be found in the last section.

The notation employed is standard and may be found explained in detail in [10].

2. Preliminaries

In this section, we give some definitions and some theorems from the papers mentioned in the first section. We also introduce a new functional, and we try to prove global well-posedness for supercritical energy by the aid of this functional.

Let 𝑠≥1. For the Cauchy problem (1.1) and (1.2), we define 𝐸(𝑡)=𝐸𝑢(𝑡),𝑢_𝑡(𝑡=1)2−𝜕²_𝑥^−1/2𝑢_𝑡²+𝑢_𝑥𝑡²+‖𝑢‖²_𝐻1+𝛾𝑝+1_𝑅|𝑢|^𝑝(0),𝐽(𝑢)=1𝑢𝑑𝑥=𝐸2‖𝑢‖²_𝐻1+𝛾𝑝+1_𝑅|𝑢|^𝑝𝐼(𝑢)=‖𝑢‖𝑢𝑑𝑥,_2𝐻1+𝛾_𝑅|𝑢|^𝑝𝑢𝑑𝑥,𝑑=inf_𝑢∈𝑁𝐽(𝑢),𝑁=𝑢∈𝐻¹∶𝐼(𝑢)=0,‖𝑢‖_𝐻1≠0, (2.1)(2.2)(2.3)(2.4) which are all well defined.

In the following, we show one more characterization of 𝑑 used before in the literature 𝑑=𝑝−12𝑝+1𝛾𝑆_𝑝^{𝑝+1−2/(𝑝−1)}, (2.5) where 𝑆_𝑝 is the imbedding constant from 𝐻¹(𝑅) into 𝐿^𝑝+1(𝑅) given by 𝑆_𝑝=sup_𝑢∈𝐻1‖𝑢‖_𝑝+1‖𝑢‖_𝐻1. (2.6) By the use of (2.1) and definition of 𝑆_𝑝, (2.5) can be easily obtained as follows: 12‖𝑢‖²_𝐻1−𝛾𝑆𝑝+1^𝑝𝑝+1‖𝑢‖_𝐻^(𝑝+1)/2₁≤𝐽(𝑢)≤𝐸(0). (2.7) The function ℎ(𝑘)=12𝛾𝑘−𝑆𝑝+1^𝑝𝑝+1𝑘^(𝑝+1)/2 (2.8) is strictly increasing in [0,𝑘₁) and strictly decreasing in (𝑘₁,∞), where 𝑘₁=𝛾^{−2/(𝑝+1)}𝑆_𝑝^{−2(𝑝+1)/(𝑝−1)}. By (2.4), we get 𝑑=max_{[)𝑘∈0,∞}ℎ(𝑘)=ℎ𝑘₁=𝑝−12𝑝+1𝛾𝑆_𝑝^{𝑝+1−2/(𝑝−1)}. (2.9) The following theorem is a generalization of Theorem 3.4 in [12] and Theorem 6.2 in [10].

Theorem 2.1. Assume that 𝜑∈𝐻¹,𝜓∈𝐻¹ and (−𝜕²_𝑥)^−1/2𝜑,(−𝜕²_𝑥)^−1/2𝜓∈𝐿²(𝑅).(i)If 𝐸(0)<0, then all weak solutions of (1.1) and (1.2) blow up in finite time;(ii)if 𝐸(0)=0, then all weak solutions of (1.1) and (1.2), except the trivial one blow up in finite time;(iii)let 0<𝐸(0)<𝑑. If 𝐼(𝜑)>0, then the weak solution of (1.1) and (1.2) is globally defined for every 𝑡∈[0,∞), if 𝐼(𝜑)<0, then the weak solution of (1.1) and (1.2) blows up in finite time.

Remark 2.2. For the critical and subcritical initial energy case, the functional 𝐼(𝜑) determines the behavior of solutions of (1.1) and (1.2). But, the numerical results in [13] showed that in the supercritical case the sign of the functional 𝐼(𝜑) cannot guarantee the global well-posedness of the problem. Due to the choice of initial data, for 𝐼(𝜑)>0, there are some solutions that blow up in finite time, and for 𝐼(𝜑)<0, there are some solutions that exist globally. A new functional 𝐼_𝛿 was considered in [6]. In the rest of this section, we prove that even if we take a more general functional than 𝐼_𝛿, it will be insufficient in determining the behavior of problem (1.1) and (1.2) in supercritical case. For a satisfactory result, we have to take into account the initial velocity in a new functional. This will be done in the next section.

Now, we define the first new functional: 𝐼_𝜎(𝑢)=(1−𝜎)‖𝑢‖²_𝐻1+𝛾_𝑅|𝑢|^𝑝𝑢𝑑𝑥=𝐼(𝑢)−𝜎‖𝑢‖²_𝐻1, (2.10) for 𝜎>−(𝑝−1)/2. The depth 𝐷_𝜎 and 𝑁_𝜎 are as follows: 𝐷_𝜎=inf_{𝑢∈𝑁𝜎}𝐽(𝑢),𝑁_𝜎=𝑢∈𝐻¹∶𝐼_𝜎(𝑢)=0,‖𝑢‖_𝐻1≠0. (2.11) Obviously, taking 𝜎=0 corresponds to the functional 𝐼(𝑢). Moreover in the case of 𝜎<−(𝑝−1)/2 we have 𝐷_𝜎<0. From Theorem 2.1, we know that in this case all weak solutions of (1.1) and (1.2) blow up in a finite time.

For 𝜎>−(𝑝−1)/2, we have the following lemmas.

Lemma 2.3. Assume that 𝑢∈𝐻¹(𝑅). If 𝐼_𝜎(𝑢)<0, then ‖𝑢‖_𝐻1>𝑟(𝜎), and if 𝐼_𝜎(𝑢)=0, then ‖𝑢‖_𝐻1≥𝑟(𝜎) or ‖𝑢‖_𝐻1=0, where 𝑟(𝜎)=((1−𝜎)/𝛾𝑆_𝑝^𝑝+1)^1/(𝑝−1).

Proof. First, from 𝐼_𝜎(𝑢)<0, we have ‖𝑢‖_𝐻1≠0. Hence, by (1−𝜎)‖𝑢‖²_𝐻1<−𝛾_𝑅|𝑢|^𝑝𝑢𝑑𝑥≤𝛾𝑆_𝑝^𝑝+1‖𝑢‖_𝐻^𝑝+1₁, (2.12) we have ‖𝑢‖_𝐻1>𝑟(𝜎).
If ‖𝑢‖_𝐻1=0, then 𝐼_𝜎(𝑢)=0, if 𝐼_𝜎(𝑢)=0 and ‖𝑢‖_𝐻1≠0, then from (1−𝜎)‖𝑢‖²_𝐻1=−𝛾_𝑅|𝑢|^𝑝𝑢𝑑𝑥≤𝛾S_𝑝^𝑝+1‖𝑢‖_𝐻^𝑝+1₁, (2.13) we have ‖𝑢‖_𝐻1≥𝑟(𝜎).

Lemma 2.4. If ‖𝑢‖_𝐻1<𝑟(𝜎), then 𝐼_𝜎(𝑢)≥0.

Proof. From ‖𝑢‖_𝐻1<𝑟(𝜎), we obtain −𝛾_𝑅|𝑢|^𝑝𝑢𝑑𝑥≤𝛾𝑆_𝑝^𝑝+1‖𝑢‖_𝐻^𝑝+1₁<(1−𝜎)‖𝑢‖²_𝐻1 (2.14) from which follows 𝐼_𝜎(𝑢)≥0.

The properties of 𝐼(𝑢) have been studied in detail. Particularly, analogous results to above lemmas are obtained in [10] (Lemma 3.3), for 𝜎=0.

Theorem 2.5. Let 𝐷_𝜎 be defined as above. Then for 𝜎>−(𝑝−1)/2, one has 𝐷_𝜎=𝑏(𝜎)𝑟²(𝜎), (2.15) where 𝑏(𝜎)=(1/2)−((1−𝜎)/(𝑝+1)). If one writes 𝐷_𝜎 in terms of 𝑑, one obtains the following statement: 𝐷_𝜎=𝑏(𝜎)(1−𝜎)^2/(𝑝−1)2𝑝+1𝑝−1𝑑. (2.16)

Proof. If 𝑢∈𝑁_𝜎, we have by Lemma 2.3 that ‖𝑢‖_𝐻1≥𝑟(𝜎). In the proof of Lemma 2.3, the inequality (2.12) is an equality if and only if 𝑢 is a minimizer of the imbedding 𝐻¹ into 𝐿^𝑝+1. Since ‖𝑢‖_𝑝+1=𝑆_𝑝‖𝑢‖_𝐻1 is attained only for 𝑢=(cosh((𝑝−1)/2)𝑥)^{−2/(𝑝−1)} [14], and it has a constant sign, we have inf_{𝑢∈𝑁𝜎}‖𝑢‖_𝐻1=𝑟(𝜎). (2.17) Hence from inf_{𝑢∈𝑁𝜎}𝐽(𝑢)=inf_{𝑢∈𝑁𝜎}12‖𝑢‖²_𝐻1+𝛾𝑝+1_𝑅|𝑢|^𝑝𝑢𝑑𝑥=inf_{𝑢∈𝑁𝜎}12−(1−𝜎)‖𝑢‖𝑝+1^2𝐻1+1𝐼𝑝+1^𝜎(𝑢=1)2−(1−𝜎)𝑝+1inf_{𝑢∈𝑁𝜎}‖𝑢‖²_𝐻1, (2.18) and by definition of 𝐷_𝜎, we obtain 𝐷_𝜎=𝑏(𝜎)𝑟²(𝜎).

As properties of 𝐷_𝜎, the following corollary can be obtained by a simple computation.

Corollary 2.6. 𝐷_𝜎 is strictly increasing on 𝜎∈(−(𝑝−1)/2,0)∪(1,∞) and strictly decreasing on (0,1). Moreover lim_𝜎→1𝐷_𝜎=0, and 𝐷_𝜎0=0, where 𝜎₀=−(𝑝−1)/2.

The following theorems show the invariance of 𝐼_𝜎 under the flow of (1.1) and (1.2) in the framework of weak solutions for 0<𝐸(0)<𝑑 and 𝐸(0)=𝑑, respectively.

Theorem 2.7. Assume that 𝜑,𝜓∈𝐻¹(𝑅),(−𝜕²_𝑥)^−1/2𝜓∈𝐿²(𝑅). Let 0<𝐸(0)<𝑑. Then the sign of 𝐼_𝜎 is invariant under the flow of (1.1) and (1.2) for 𝜎∈(𝜎₁,𝜎₂], where (𝜎₁,𝜎₂] is the maximal interval such that 𝐷_𝜎=𝐸(0).

Proof. In consequence of assumption 𝐸(0)>0, we have ‖𝑢‖_𝐻1>0. If for 𝜎∈(𝜎₁,𝜎₂], the sign of 𝐼_𝜎 is changeable, then we must have a 𝜎∈(𝜎₁,𝜎₂] such that 𝐼_𝜎=0. First, we prove the theorem for 𝜎∈(𝜎₁,𝜎₂). Notice from (2.16) that 𝜎₁<0 and 𝜎₂∈(0,1). Thus, we have 𝐷_𝜎2≥𝐸(0)≥𝐽(𝑢)≥𝐷_𝜎. Since 𝐷_𝜎 is strictly decreasing on (0,1), from Corollary 2.6 we have 𝐷_𝜎>𝐷_𝜎2, which contradicts with the previous inequality. So, the theorem is proved for 𝜎∈(𝜎₁,𝜎₂). For 𝜎=𝜎₂, assume that 𝐼_𝜎2=0 and there exists some 𝜎∈(0,𝜎₂) such that 𝐼_𝜎=𝐼_𝜎2+(𝜎₂−𝜎)‖𝑢‖²_𝐻1. Hence 𝐼_𝜎=(𝜎₂−𝜎)‖𝑢‖²_𝐻1>0 and for some 𝑡^′>0, we get 𝐼_𝜎(𝑢(𝑡^′))=0. Then, by the similar argument used for 𝜎∈(𝜎₁,𝜎₂), we obtain a contradiction. Thus, the proof is finished.

Theorem 2.8. If on top of all the assumptions of Theorem 2.7, we suppose that 𝐸(0)=𝑑. Then the sign of 𝐼₀ (recall that when 𝐸(0)=𝑑, we have 𝜎₁=𝜎₂=0) is invariant with respect to (1.1) and (1.2) for every 𝑡∈[0,∞).

Proof. The theorem states that if 𝐼₀(𝜑)≥0, then 𝐼₀(𝑢(𝑡))≥0, contrary if 𝐼₀(𝜑)<0, then 𝐼₀(𝑢(𝑡))<0. We only give the proof of the first statement, the second is similar. To check that 𝐼₀ does not change sign, we proceed as follows. Let 𝑢(𝑡) be any weak solution of problem (1.1),(1.2) with 𝐸(0)=𝑑. If the first statement is false, then there must exist a 𝑡^′>0 such that 𝐼₀(𝑢(𝑡^′))<0. It follows from Lemma 2.3 that ‖𝑢(𝑡^′)‖_𝐻1>0. From the energy identity, we get (0)=1𝑑=𝐸2−𝜕²_𝑥^−1/2𝑢_𝑡𝑡^′2+𝑢_𝑥𝑡𝑡^′2+𝐽𝑢𝑡^′≥𝐽𝑢𝑡^′≥inf_𝑢∈𝑁0𝐽(𝑢)=𝑑. (2.19) If 𝑢(𝑡^′)∈𝑁₀, then 𝑢(𝑡^′) must be a minimizer of 𝐽(𝑢) for 𝑢∈𝑁₀, and we have 𝐼₀(𝑢(𝑡^′))=0. This, however, is impossible, since it violates 𝐼₀(𝑢(𝑡^′))<0. Thus the lemma is proved.

Now, we give a lemma for 𝜎>1, which states similar results to Lemmas 2.3 and 2.4, and can be proved similarly.

Lemma 2.9. Assume that 𝑢∈𝐻¹(𝑅). For 𝜎>1, if 𝐼_𝜎(𝑢)>0, then ‖𝑢‖_𝐻1>𝑠(𝜎), and if 𝐼_𝜎(𝑢)=0, then ‖𝑢‖_𝐻1≥𝑠(𝜎) or ‖𝑢‖_𝐻1=0, where 𝑠(𝜎)=((𝜎−1)/𝛾𝑆_𝑝^𝑝+1)^1/(𝑝−1). Moreover, if ‖𝑢‖_𝐻1<𝑠(𝜎), then 𝐼_𝜎(𝑢)≤0 and 𝐼_𝜎(𝑢)=0 if and only if ‖𝑢‖_𝐻1=0.

Theorem 2.10. Assume that 𝜑,𝜓∈𝐻¹(𝑅),(−𝜕²_𝑥)^−1/2𝜓∈𝐿²(𝑅). If 𝐸(0)>0, then 𝐼_𝜎(𝑢(𝑡))≤0 for every 𝑡>0 and 𝜎≥𝜎_𝑚, where 𝜎_𝑚 is the maximal positive root of 𝐷_𝜎=𝐸(0).

Proof. We give the proof of the theorem for 𝜎=𝜎_𝑚 and 𝜎>𝜎_𝑚 separately. First we prove the theorem for 𝜎=𝜎_𝑚. Proceeding by contradiction, assume that there exists some 𝑡^′>0 such that 𝐼_𝜎𝑚(𝑢(𝑡^′))>0. By Lemma 2.3, we have ‖𝑢‖_𝐻1>0, and there exists a value 𝜎,𝜎>𝜎_𝑚 such that 𝐼_𝜎(𝑢(𝑡^′))=0. Then by (2.1), 𝐷_𝜎𝑚=𝐸(0)≥𝐽(𝑢(𝑡^′))≥inf_{𝑢∈𝑁𝜎}𝐽(𝑢)=𝐷_𝜎. From definition of 𝐷_𝜎 for 𝜎>𝜎_𝑚>1, we have 𝐷_𝜎>𝐷_𝜎𝑚. A contradiction occurs, which proves the theorem for 𝜎=𝜎_𝑚. For 𝜎≥𝜎_𝑚,𝐼_𝜎𝑚(𝑢(𝑡))≥𝐼_𝜎(𝑢(𝑡)) implies that the theorem is true for every 𝜎≥𝜎_𝑚.

The following corollary is a direct consequence of Theorems 2.1 and 2.10.

Corollary 2.11. Suppose 𝜑,𝜓∈𝐻¹,(−𝜕²_𝑥)^−1/2𝜓∈𝐿²(𝑅). Let 0<𝐸(0)<𝑑 and 𝐼₀(𝜑)>0. Then 0<𝐼₀(𝑢(𝑡))<𝜎_𝑚‖𝑢‖²_𝐻1, (2.20) for every 𝑡>0.

Remark 2.12. We tried to characterize the behavior of solutions for 𝐸(0)>0 in terms of initial displacement. We constituted a new functional 𝐼_𝜎(𝑢) and proved the sign invariance of 𝐼_𝜎(𝑢) for 0<𝐸(0)<𝑑 and 𝐸(0)=𝑑. But the case 𝐸(0)>0 is still an open question, because from Theorem 2.10, we concluded that in this case 𝐼_𝜎(𝑢) is always nonpositive. Due to numerical results of [13], we know that such a functional to prove global existence must include the initial velocity too. We will introduce this new functional in the next section.

3. Global Existence for Supercritical Initial Energy

In this section, we state the main result of the paper. The functional we introduce here, which was used before in [13] in a similar form, permits us to establish global existence of solutions for (1.1) and (1.2) in the supercritical initial energy case 𝐻(𝜐,𝜔)=‖𝜐‖²_𝐻1+𝛾_𝑅|𝜐|^𝑝𝜐𝑑𝑥−−𝜕²_𝑥^−1/2𝜔²−𝜔_𝑥²=𝐼₀(𝑣)−−𝜕²_𝑥^−1/2𝜔²−𝜔_𝑥². (3.1) In order to simplify notation, we rewrite 𝐻(𝑢(⋅,𝑡),𝑢_𝑡(⋅,𝑡)) as 𝐻(𝑢,𝑡)=𝐻𝑢(⋅,𝑡),𝑢_𝑡(⋅,𝑡). (3.2) Once we have proved the invariance of the above functional with respect to (1.1) and (1.2), then global existence can be proved by the aid of invariance of this functional.

Theorem 3.1. Assume that 𝜑,𝜓∈𝐻¹(𝑅),(−𝜕²_𝑥)^−1/2𝜑,(−𝜕²_𝑥)^−1/2𝜓∈𝐿²(𝑅) and 𝐸(0)>0. For some 𝜎>𝜎_𝑚,𝜎_𝑚 defined as above, one have −𝜕²_𝑥^−1/2𝜓,−𝜕²_𝑥^−1/2𝜑+𝜓_𝑥,𝜑_𝑥+12−𝜕²_𝑥^−1/2𝜑²+12𝜑_𝑥²+𝑝+1𝜎𝜎𝐸(0)𝑝−1+𝑝+3≤0. (3.3) Moreover, 𝐻(𝑢,𝑡) is positive provided that 𝐻(𝑢,0) is positive, for every 𝑡∈[0,∞).

Proof. Looking for the global solution is equivalent to showing that there is no blow up. So, we modify a blow up technique for the proof [15]. To this end, we define 𝜃(𝑡)=−𝜕²_𝑥^−1/2𝑢²+𝑢_𝑥². (3.4) Direct computations yield 𝜃^′(𝑡)=2−𝜕²_𝑥^−1/2𝑢_𝑡,−𝜕²_𝑥^−1/2𝑢+2𝑢_𝑥𝑡,𝑢_𝑥,𝜃^′′(𝑡)=2−𝜕²_𝑥^−1/2𝑢_𝑡²+2−𝜕²_𝑥^−1/2𝑢_𝑡𝑡,−𝜕²_𝑥^−1/2𝑢+2𝑢_𝑥𝑡𝑡,𝑢_𝑥+2𝑢_𝑥𝑡(𝑡)²=2−𝜕²_𝑥^−1/2𝑢_𝑡²+2𝑢_𝑥𝑡(𝑡)²+2−𝜕²_𝑥⁻¹𝑢_𝑡𝑡,𝑢−2𝑢_{𝑥𝑥𝑡𝑡},𝑢=2−𝜕²_𝑥^−1/2𝑢_𝑡²+2𝑢_𝑥𝑡(𝑡)²−2𝐼₀(𝑢)=−2𝐻(𝑢,𝑡). (3.5) For contradiction, assume that there exists some 𝑡^′>0 such that 𝐻(𝑢,𝑡^′)=0. Since 𝜃^′′(𝑡)<0, we conclude that 𝜃^′(𝑡) is strictly decreasing on [0,𝑡^′). Moreover, (3.3) implies 𝜃^′(0)<0 and therefore 𝜃^′(𝑡)<0 in [0,𝑡^′], from which follows that 𝜃(𝑡) is strictly decreasing on [0,𝑡^′]. By the energy identity and 𝐻(𝑢,𝑡^′)=0, we have 𝐸(0)=12−𝜕²_𝑥^−1/2𝑢_𝑡𝑡^′2+𝑢_𝑥𝑡𝑡^′2+𝑝−12𝑢𝑡𝑝+1^′2𝐻1+1𝐼𝑢𝑡𝑝+1^′=12+1𝑝+1−𝜕²_𝑥^−1/2𝑢_𝑡𝑡^′2+𝑢_𝑥𝑡𝑡^′2+𝑝−12𝑢𝑡𝑝+1^′2𝐻1. (3.6) Theorem 2.10, Corollary 2.11, and 𝐻(𝑢,𝑡^′)=0 yield ‖𝑢‖²_𝐻1≥𝜎_𝑚⁻¹𝐼₀𝑢𝑡^′≥𝜎⁻¹−𝜕²_𝑥^−1/2𝑢_𝑡𝑡^′2+𝑢_𝑥𝑡𝑡^′2. (3.7) The use of the above inequality in (3.6) shows that 𝐸(0)≥12+1+𝑝+1𝑝−12𝑝+1𝜎−𝜕²_𝑥^−1/2𝑢_𝑡𝑡^′2+𝑢_𝑥𝑡𝑡^′2. (3.8) This can be rephrased in terms of 𝜃(𝑡) and 𝜃^′(𝑡) as 𝐸(0)≥𝑝+3𝜎+𝑝−12𝑝+1𝜎−𝜕²_𝑥^−1/2𝑢_𝑡𝑡^′+𝑢𝑡^′2+𝑢_𝑥𝑡𝑡^′+𝑢𝑡^′2−2−𝜕²_𝑥^−1/2𝑢_𝑡𝑡^′,−𝜕²_𝑥^−1/2𝑢𝑡^′−2𝑢_𝑥𝑡𝑡^′,𝑢_𝑥𝑡^′−−𝜕²_𝑥⁻12𝑢𝑡^′2−𝑢_𝑥𝑡^′2. (3.9) From the monotonicity of 𝜃(𝑡) and 𝜃^′(𝑡), we get 𝐸(0)>𝑝+3𝜎+𝑝−1𝜎−𝑝+1−𝜕²_𝑥^−1/2𝜓,−𝜕²_𝑥^−1/2𝜑−𝜓_𝑥,𝜑_𝑥−12−𝜕²_𝑥^−1/2𝜑²−12𝜑_𝑥², (3.10) which contradicts (3.3). Thus, the theorem is proved.

Theorem 3.2. Assume that 𝜑,𝜓∈𝐻¹(𝑅),(−𝜕²_𝑥)^−1/2𝜑,(−𝜕²_𝑥)^−1/2𝜓∈𝐿²(𝑅). Suppose that 𝐸(0)>0,𝐻(𝑢,0)>0 and (3.3) holds for some 𝜎>𝜎_𝑚. Then, the weak solution of (1.1) and (1.2) is globally defined for every 𝑡∈[0,∞).

Proof. The proof of this theorem follows from adding some arguments to the local existence result of Corollary 2.10 of [10]. If (−𝜕²_𝑥)^−1/2𝜓∈𝐿², then (−𝜕²_𝑥)^−1/2𝑢_𝑡∈𝐿² (Lemma 2.8 of [10]). 𝐻(𝑢,0)>0 implies from the sign preserving property of 𝐻(𝑢,𝑡) that 𝐻(𝑢,𝑡)>0, thereby 𝐼₀(𝑢)>0 for every 𝑡>0. From the energy identity we have 𝐸(0)=12−𝜕²_𝑥^−1/2𝑢_𝑡²+𝑢_𝑥𝑡²+𝑝−12‖𝑢‖𝑝+1^2𝐻1+1𝐼(𝑢)≥1𝑝+12−𝜕²_𝑥^−1/2𝑢_𝑡²+12𝑢_𝑥𝑡²+𝑝−12‖𝑢‖𝑝+1^2𝐻1. (3.11) Therefore ‖𝑢‖_𝐻1 and ‖𝑢_𝑡‖_𝐻1 are bounded for every 𝑡>0. The previously mentioned local existence theory completes the proof.

4. Final Remarks

Remark 4.1. In a section of the paper of Y. Z. Wang and Y. X. Wang [11], problem (1.1) and (1.2) was studied with the supercritical initial energy. A global existence result was obtained under the assumption 𝐹(𝑢)≥0 or 𝑓^′(𝑢) is bounded below, that is, 𝑓^′(𝑢)≥𝐴₀. As the authors have mentioned in an example, this condition is valid only for odd-degree nonlinearities, namely for 𝑓(𝑢)=𝛽𝑢^2𝑝+1,𝑢∈𝑅,𝛽>0,𝑝 is a nonnegative integer. For 𝑓(𝑢)=𝛽|𝑢|^𝑝, this condition cannot guarantee the global existence. For example, if we take 𝑓(𝑢)=(3/2)𝑢², then the antiderivative and derivative of 𝑓 are 𝐹(𝑢)=(1/2)𝑢³ and 𝑓^′(𝑢)=3𝑢, respectively. For 𝑓(𝑢)=𝛽|𝑢|^𝑝, we have 𝐹(𝑢)=(𝛽/(𝑝+1))|𝑢|^𝑝𝑢 and 𝑓^′(𝑢)=𝑎𝑝|𝑢|^𝑝−2𝑢, which do not always satisfy the positivity condition. To remedy this, we generate a new functional for potential well method, which also contains the initial velocity different from the previous ones, and use the invariance of this functional with respect to problem (1.1) and (1.2).

Remark 4.2. In Section 2, we introduce the first new functional 𝐼_𝜎(𝑢), which is more general than the ones introduced before in some papers for the fourth-order Boussinesq equation. However, in the case of 𝐸(0)>0 we are not able to prove the sign invariance of 𝐼_𝜎(𝑢), because we see that for 𝐸(0)>0,𝐼_𝜎(𝑢) is always nonpositive. Eventually, a satisfactory result comes from 𝐻(𝑢,𝑡) which includes not only the initial displacement 𝜑, but also the initial velocity 𝜓.