Cauchy Problem of the Singularly Perturbed Sixth Order Boussinesq Type Equation

In this paper, the existence and uniqueness of the global generalized solution and the global classical solution for the Cauchy problem of the singularly perturbed sixth order Boussinesq type equation are proved.


Introduction
In this paper, we consider the following Cauchy problem 4 6 ( ) 0 ) where ( , ) u x t is the unknown function, subscript x and t indicate partial derivatives, ( ) s σ is the given function, 0 α > and 0 β > are real numbers, 0 ( ) u x and 1 ( ) u x are given functions defined on R .There are also several equations which are closely related to Equation (1.1).In the numerical study of the ill-posed Boussinesq equation 2 ( ) In [1], Darapi and Hua proposed the singularly perturbed Boussinesq equa- as a dispersive regularization of the ill-posed classical Boussinesq Equation (1.3), where 0 δ > is small parameter.The authors use both filtering and regulariza- tion techniques to control growth of errors and to provide better approximate solutions of this equation.Dash and Daripi presented a formal derivation of (1.4) from two-dimensional potential flow equations for water waves through an asymptotic series expansion for small amplitude and long wave length in [2] [3].The physical relevance of equation (1.4) describes the bi-directional 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.
In [4], Feng investigated the generalized Boussinesq equation including the singularly perturbed Boussinesq equation In order to prove that the Cauchy problem (1.1), (1.2) has a unique global solution, we shall consider the following auxiliary problem ( ) 0 ) First of all, we shall prove that the periodic boundary value problem of Equa-

Periodic Boundary Value Problem of (1.6), (1.7)
To obtain the global solution for the Cauchy problem (1.6), (1.7), we first discuss the following periodic boundary value problem on T Q 4 6 ( ) ( ) ( 2 ) ) where ( ) ) Multiplying both sides of (2.4) and (2.6) by ( ) s y x , summing up for and integrating on Ω , we have .
Lemma 2.1.(Adams [7]) There exist constants 0 ε > and ( ) 0 C ε > such that for any integers j and 0 m j m , ≤ ≤ , the following inequality holds ) s N = , , , .Moreover, we have the following estimate where and in the sequel are constants which depend on T , but not on N and Ω .
Using (2.9) and the Leray-Schauder fixed point theorem [8] to the both sides of (2.17), integrating the product over [0 ] t , , Cauchy inequality, Lemma 2.3, (2.9) and (2.18), we have Proof: First we give the definition of the generalized solution, which ( ) v x t , satisfies the identity and the periodic boundary value conditions (2.2), (2.3) in the classical sense.
By Lemma 2.4, we have It follows from Sobolev embedding theorem, when 3 k = , we know We can select a subsequence of { ( )} N v x t , and a function ( ) v x t , and N → ∞ , the subsequence uniformly converges to the limiting function ( ) Proof: Differentiating (2.7) with respect to t , we have Multiplying both sides of (2.25) by ( 1) ( ) , summing up for 1 2 s N = , , , , integrating by parts and using the Holder inequality, combining (2.15), we conclude
Proof: Let us take a real sequence { }( 1) For every s , let us construct periodic functions 0 ( ) s v x and 1 ( ) we consider the following periodic boundary value problem ( )  ( ) ) By the same method as in the estimates (2.15), (2.16), we have where and in the sequel ( )( 8 9 ) i C T i = , , are constants independent of s N and s D .By the Sobolev imbedding theorem when 4 k = , we get By virtue of (3.9) and Ascoli-Arzela theorem, we can select from { ( )} weakly converge to limiting functions From the corollary of the resonance theorem [10], it follows that the estimates (3.8), (3.9) still hold for ( ) are all real constants.By the means of two proper ansatzs, the author obtained explicit traveling solitary wave solutions.In [5], Song et al. studied the existence and uniqueness of the global generalized solution and the global classical for the initial boundary value problem of Equation (1.1).In [6], Song et al. also studied the nonexistence of the global solutions for the initial boundary value problem of Equation (1.1).The aim of the present article is to prove that, by virtue of the Galerkin method and prior estimates, the problem (1.1), (1.2) has a unique global generalized solution and a unique global classical solution.
Using the method of Theorem 2.1, when 7 k ≥ , the periodic boundary value problem (2.1) -(2.3) has a global classical solution ( ) v x t , .It is easy to prove the uniqueness of solution for the problem (2.1) -(2.3). d which is the generalized solution of the prob- lem (3.1)-(3.3).Using Ascoli-Arzela theorem, we can select from { ( )} It follows from (3.8) that we can select from ( ) , the Cauchy problem (1.6), (1.7) has a unique global generalized solution.Theorem 3.2.Assume that the assumptions of Theorem 3.1 hold, If Using the method of Theorem 3.1, when 7 k ≥ , the Cauchy problem (1.6), (1.7) has a global classical solution.It is easy to prove the uniqueness of solution H. Li et al.