Abstract

We are concerned with the singularly perturbed Boussinesq-type equation including the singularly perturbed sixth-order Boussinesq equation, which describes the bidirectional propagation of small amplitude and long capillary-gravity waves on the surface of shallow water for bond number (surface tension parameter) less than but very close to 1/3. The nonexistence of global solution to the initial boundary value problem for the singularly perturbed Boussinesq-type equation is discussed and two examples are given.

1. Introduction

In the numerical study of the ill-posed Boussinesq equation, Darapi and Hua [1] proposed the singularly perturbed Boussinesq equation as a dispersive regularization of the ill-posed classical Boussinesq equation (1), where is a small parameter. The authors use both filtering and regularization techniques to control growth of the errors and to provide better approximate solutions of this equation. Dash and Daripa [2] presented a formal derivation of (2) from two-dimensional potential flow equations for water waves through an asymptotic series expansion for small amplitude and long wave length. The physical relevance of (2) in the context of water waves was also addressed in [2]; it was shown that (2) actually describes the bidirectional propagation of small amplitude and long capillary-gravity waves on the surface of shallow water for bond number (surface tension parameter) less than but very close to . On the basis of far-field analysis and heuristic arguments, Daripa and Dash [3] proved that the traveling wave solutions of (2) are weakly nonlocal solitary waves characterized by small amplitude fast oscillations in the far-field and obtained weakly nonlocal solitary wave solutions of (2). Feng [4] investigated the generalized Boussinesq equation including the singularly perturbed Boussinesq equation where , , and ( ) are all real constants. It is easily seen that the choices , , , , and lead (3) to the singularly perturbed Boussinesq equation (2). By the means of two proper ansatzs, the author obtained explicit traveling solitary wave solutions of the generalized Boussinesq equation (3). To the best of our knowledge, however, there have not been any discussions on global solutions of the initial boundary value problem for (2) in the literature; recently, Song et al. [5] discussed the initial boundary value problem for the singularly perturbed Boussinesq-type equation with the initial boundary value conditions or with where, and in the sequel , is a given nonlinear function, and are real numbers, and are given initial value functions, and . By virtue of the Galerkin method and prior estimates, under the assumption “ is bounded below and satisfies some smooth condition,” the existence and uniqueness of the global generalized solution and the global classical solution of the initial boundary value problem (4), (5) and (4), (6) are proved, respectively. But if is not bounded below, does the above-mentioned problem have any global solution? In this paper, we employ the energy method and the Jensen inequality to prove that the global solutions of the initial boundary value problem (4), (5) and (4), (6) cease to exist in a finite time, respectively. At last, we show that the global solution of the initial boundary value problem (2), (6) blows up in a finite time.

The paper is organized as follows. In Section 2, the main results are stated. The nonexistence of global solution of problem (4), (5) and (4), (6) is discussed in Section 3. In Section 4, we study the initial boundary problem (2), (6) and give two examples satisfying the theorems (Theorems 16).

2. Main Theorems

Throughout this paper, we use the abbreviations . In the following we state the main results of this paper, where the existence of Theorems 14 has been proved in [5].

Theorem 1 (see [5]). Assume that , , , ( ), , and is bounded below; namely, there exists a constant such that , for any . Then, for any , the initial boundary value problem (4), (5) admits a unique global generalized solution with

Theorem 2 (see [5]). Assume that the assumptions of Theorem 1 hold, , , and . Then, the initial boundary value problem (4), (5) admits a unique global classical solution .

Theorem 3 (see [5]). Assume that , , ( ), , ( ), and is bounded below. Then, for any , the initial boundary value problem (4), (6) admits a unique global generalized solution with

Theorem 4 (see [5]). Assume that the assumptions of Theorem 3 hold, , , , and ( ). Then, the initial boundary value problem (4), (6) admits a unique global classical solution .

Theorem 5. Assume that (1) , , where , , , and are constants, and (2) , , , and where . Then the solution of initial boundary value problem (4), (5) blows up in a finite time ; namely, where is defined in the proof.

Theorem 6. Assume that (1) , , and one of the following conditions holds: (i) is a convex and even function, , where and are real numbers, (ii) is a convex function, , where is a real number and is an even number, and (2) , . Then the solution of the initial boundary value problem (4), (6) blows up in a finite time ; namely, where is defined in the proof.

3. Nonexistence of Global Solutions of Problem (4), (5) and (4), (6)

We first quote the following lemmas.

Lemma 7 (see Li [6]). Assume that , , , , and . Then as .

Lemma 8 (Jensen inequality [7]). Assume that is a convex function, , and is a continuous function, , . Then
Integrating both sides of (4) over and using (5) and the assumption of Theorem 1, we obtain , . Let ; then and satisfies where and .

Proof of Theorem 5. Multiplying both sides of (13) by , integrating by parts, and using condition (2) of Theorem 5, we have where Let By virtue of condition (1) of Theorem 5 and noting that we obtain It follows from (20) that where and . Combining (20) with (22) leads to Making use of the Hölder inequality, we get Substituting (24) into (23) and using the Poincaré inequality and the inequality ( ), we conclude that when , Choose such that and thus (21) and (22) imply that and , as . Multiplying both sides of (25) by , we have where Equation (27) implies that Since there exists a such that By (31), integrating both sides of (29) over , we obtain Namely, In order to use Lemma 7, we consider the following initial value problem of the Bernoulli equation: We can obtain the solution of the initial value problem (34) as follows: where By (36), we know that and It follows from (22) that Choose sufficiently large such that . Combining (37) with (9), we obtain Hence By using the continuity of , there exists a finite time , such that . Therefore, as . By virtue of Lemma 7, we deduce that , . Hence as . It follows from (41) that Therefore, Theorem 5 is proved.

Proof of Theorem 6. Let Multiplying both sides of (4) by , integrating by parts over , and making use of the Jensen inequality and condition (1) of Theorem 6, we have and, from (6) and condition (2) of Theorem 6, we get Thus, we claim that
In fact, if it is not true, then there exists a such that , and . Then is monotonically increasing on ; that is, , . By using (45) and condition (2) of Theorem 6, we obtain and hence is monotonically increasing on , which contradicts the assumption . So claim (47) is valid.
Multiplying both sides of (45) by and integrating the product over lead to Since and , . It follows from (49) that and (51) implies that the interval of the existence of is finite; namely, and as ; that is, as . By the Hölder inequality, we have , as . Theorem 6 is proved.

4. Initial Boundary Value Problem (2), (6) and Some Examples

By virtue of the Galerkin method [8] we can prove that initial boundary value problem (2), (6) admits a unique local generalized solution and a unique local classical solution. Moreover, by using Theorem 6, we obtain the following theorem.

Theorem 9. Assume that is the generalized solution of initial boundary value problem (2), (6) and the following condition holds: Then where

Proof. A simple verification shows that all conditions of Theorem 6 are satisfied and thus Theorem 9 is proved immediately.

Example 1. We consider the following equation: with the initial boundary value conditions or with where and are all real numbers, , and is a constant.
(1) If and , a simple calculation shows that , ( ), and ; that is, is bounded below. And and satisfy the conditions of Theorems 2 and 4, respectively; then by Theorems 2 and 4 we know that the initial boundary value problem (56), (57) and (56), (58) admits a unique global classical solution, respectively.
(2) If and , we have , ; taking and , then , ; obviously, , , , and We can take suitable large such that where . Thus, all assumptions of Theorem 5 are satisfied; then by Theorem 5 we conclude that the solution of initial boundary value problem (56), (57) must blow up in a finite time ; namely,

Example 2. We consider the following equation: with the initial boundary value conditions or with where is a real number and is a positive integer, , and is a constant.
(1) If and is an odd number, a simple verification shows that all conditions of Theorems 2 and 4 are satisfied; then by Theorems 2 and 4 we know that the initial boundary value problem (62), (63) and (62), (64) admits a unique global classical solution, respectively.
(2) If and is an even number, then ( ) is a convex and even function, and we can take suitable large such that Thus, by Theorem 6, we deduce that the solution of initial boundary value problem (62), (64) must blow up in a finite time ; namely,

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the Natural Science Foundation of Henan Province of China (No. 122300410166 and No. 102300410275), the Natural Science Foundation of China (No. 11271336 and No. 11326167), and the Foundation of Henan Educational Committee (No. 13A110119).