Nonexistence of Global Solutions to the Initial Boundary Value Problem for the Singularly Perturbed Sixth-Order Boussinesq-Type Equation

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.


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  > 0 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 1/3.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 () =  +  0   ,  (2+2) = ( 2+2 )/( 2+2 ),  and   ( = 1, 2, . . ., ) are all real constants.It is easily seen that the choices  0 = 1,  = 2,  = 2,  1 = 1, and  2 =  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 Journal of Applied Mathematics with the initial boundary value conditions or with where, and in the sequel    =   /  , () is a given nonlinear function,  > 0 and  > 0 are real numbers,  0 () and  1 () are given initial value functions, and Ω = (0, 1).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 1-6).

Main Theorems
Throughout this paper, we use the abbreviations ‖ ⋅ ‖ = ‖ ⋅ ‖  2 (Ω) .In the following we state the main results of this paper, where the existence of Theorems 1-4 has been proved in [5].

Initial Boundary Value Problem
(2), ( 6) and Some Examples 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  ̸ = 0 and  > 1 are all real numbers,  0 () =  1 () =  0 cos , and  0 > 0 is a constant.
(1) If  > 0 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  > 0 and  is an even number, then () (=   ) is a convex and even function, and we can take  1 suitable large such that −  2 ∫  (66)