`Abstract and Applied AnalysisVolumeÂ 2012Â (2012), Article IDÂ 535031, 10 pageshttp://dx.doi.org/10.1155/2012/535031`
Research Article

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

Department of Mathematics, Dicle University, 21280 Diyarbakir, Turkey

Received 4 May 2012; Accepted 11 June 2012

Abstract

We will give conditions which will guarantee the existence of global weak solutions of the Boussinesq-type equation with power-type nonlinearity 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: where . 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 [1â€“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 provided that the antiderivative of satisfies or is bounded below. Then by the potential well method, they proved the global existence of solutions for a special case of the nonlinear term with .

In [11], the initial value problem for a class of nonlinear wave equations of higher order: was considered. The global well-posedness of the initial value problem of (1.3) with was studied making use of the potential well method.

Very recently, Xu et al. [10] studied the multidimensional Boussinesq-type equation: with the nonlinear terms , and . When , 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 or . The case 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 . In the present paper, we again investigate problem (1.1) and (1.2) and give the global existence of solutions for 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 . For the Cauchy problem (1.1) and (1.2), we define which are all well defined.

In the following, we show one more characterization of used before in the literature where is the imbedding constant from into given by By the use of (2.1) and definition of , (2.5) can be easily obtained as follows: The function is strictly increasing in and strictly decreasing in , where . By (2.4), we get The following theorem is a generalization of Theorem 3.4 in [12] and Theorem 6.2 in [10].

Theorem 2.1. Assume that and .(i)If , then all weak solutions of (1.1) and (1.2) blow up in finite time;(ii)if , then all weak solutions of (1.1) and (1.2), except the trivial one blow up in finite time;(iii)let . If , then the weak solution of (1.1) and (1.2) is globally defined for every , if , 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 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 cannot guarantee the global well-posedness of the problem. Due to the choice of initial data, for , there are some solutions that blow up in finite time, and for , there are some solutions that exist globally. A new functional was considered in [6]. In the rest of this section, we prove that even if we take a more general functional than , 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: for . The depth and are as follows: Obviously, taking corresponds to the functional . Moreover in the case of we have . 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 , we have the following lemmas.

Lemma 2.3. Assume that . If , then , and if , then or , where .

Proof. First, from , we have . Hence, by we have .
If , then , if and , then from we have .

Lemma 2.4. If , then .

Proof. From , we obtain from which follows .

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

Theorem 2.5. Let be defined as above. Then for , one has where . If one writes in terms of , one obtains the following statement:

Proof. If , we have by Lemma 2.3 that . In the proof of Lemma 2.3, the inequality (2.12) is an equality if and only if is a minimizer of the imbedding into . Since is attained only for [14], and it has a constant sign, we have Hence from and by definition of , we obtain .

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

Corollary 2.6. is strictly increasing on and strictly decreasing on . Moreover , and , where .

The following theorems show the invariance of under the flow of (1.1) and (1.2) in the framework of weak solutions for and , respectively.

Theorem 2.7. Assume that ,. Let . Then the sign of is invariant under the flow of (1.1) and (1.2) for , where is the maximal interval such that .

Proof. In consequence of assumption , we have . If for , the sign of is changeable, then we must have a such that . First, we prove the theorem for . Notice from (2.16) that and . Thus, we have . Since is strictly decreasing on , from Corollary 2.6 we have , which contradicts with the previous inequality. So, the theorem is proved for . For , assume that and there exists some such that . Hence and for some , we get . Then, by the similar argument used for , 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 . Then the sign of (recall that when , we have ) is invariant with respect to (1.1) and (1.2) for every .

Proof. The theorem states that if , then , contrary if , then . We only give the proof of the first statement, the second is similar. To check that does not change sign, we proceed as follows. Let be any weak solution of problem (1.1),(1.2) with . If the first statement is false, then there must exist a such that . It follows from Lemma 2.3 that . From the energy identity, we get If , then must be a minimizer of for , and we have . This, however, is impossible, since it violates . Thus the lemma is proved.

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

Lemma 2.9. Assume that . For , if , then , and if , then or , where . Moreover, if , then and if and only if .

Theorem 2.10. Assume that . If , then for every and , where is the maximal positive root of .

Proof. We give the proof of the theorem for and separately. First we prove the theorem for . Proceeding by contradiction, assume that there exists some such that . By Lemma 2.3, we have , and there exists a value such that . Then by (2.1), . From definition of for , we have . A contradiction occurs, which proves the theorem for . For implies that the theorem is true for every .

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

Corollary 2.11. Suppose . Let and . Then for every .

Remark 2.12. We tried to characterize the behavior of solutions for in terms of initial displacement. We constituted a new functional and proved the sign invariance of for and . But the case is still an open question, because from Theorem 2.10, we concluded that in this case 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 In order to simplify notation, we rewrite as 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 and . For some defined as above, one have Moreover, is positive provided that is positive, for every .

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 Direct computations yield For contradiction, assume that there exists some such that . Since , we conclude that is strictly decreasing on . Moreover, (3.3) implies and therefore in , from which follows that is strictly decreasing on . By the energy identity and , we have Theorem 2.10, Corollary 2.11, and yield The use of the above inequality in (3.6) shows that This can be rephrased in terms of and as From the monotonicity of and , we get which contradicts (3.3). Thus, the theorem is proved.

Theorem 3.2. Assume that . Suppose that and (3.3) holds for some . Then, the weak solution of (1.1) and (1.2) is globally defined for every .

Proof. The proof of this theorem follows from adding some arguments to the local existence result of Corollary 2.10 of [10]. If , then (Lemma 2.8 of [10]). implies from the sign preserving property of that , thereby for every . From the energy identity we have Therefore and are bounded for every . 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 or is bounded below, that is, . As the authors have mentioned in an example, this condition is valid only for odd-degree nonlinearities, namely for is a nonnegative integer. For , this condition cannot guarantee the global existence. For example, if we take , then the antiderivative and derivative of are and , respectively. For , we have and , 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 , which is more general than the ones introduced before in some papers for the fourth-order Boussinesq equation. However, in the case of we are not able to prove the sign invariance of , because we see that for is always nonpositive. Eventually, a satisfactory result comes from which includes not only the initial displacement , but also the initial velocity .

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