Journal of Applied Mathematics and Physics
Vol.05 No.09(2017), Article ID:79079,10 pages
10.4236/jamp.2017.59137
Cauchy Problem of the Singularly Perturbed Sixth Order Boussinesq Type Equation
Hong Li, Changming Song, Li Chen
College of Science, Zhongyuan University of Technology, Zhengzhou, China
Received: July 20, 2017; Accepted: September 12, 2017; Published: September 15, 2017
ABSTRACT
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.
Keywords:
Boussinesq Equation, Cauchy Problem, Global Generalized Solution, Global Classical Solution
1. Introduction
In this paper, we consider the following Cauchy problem
(1.1)
(1.2)
where is the unknown function, subscript and indicate partial derivatives, is the given function, and are real numbers, and are given functions defined on .
There are also several equations which are closely related to Equation (1.1). In the numerical study of the ill-posed Boussinesq equation
(1.3)
In [1], Darapi and Hua proposed the singularly perturbed Boussinesq equation
(1.4)
as a dispersive regularization of the ill-posed classical Boussinesq Equation (1.3), where is small parameter. The authors use both filtering and regularization 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
(1.5)
where and 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.
In order to prove that the Cauchy problem (1.1), (1.2) has a unique global solution, we shall consider the following auxiliary problem
(1.6)
(1.7)
First of all, we shall prove that the periodic boundary value problem of Equation (1.6) has a unique global solution by the Galerkin method. Next, we prove that the Cauchy problem (1.6), (1.7) has a unique global solution by constructing a sequence of periodic boundary value problem of Equation (1.6). Then, we can obtain a unique global solution of the Cauchy problem (1.1), (1.2) from (1.6), (1.7) by setting , and .
2. 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
(2.1)
(2.2)
(2.3)
where and are given functions defined on and satisfy (2.2).
Let be the orthogonal
base in composed of the eigenfunctions of the eigenvalue problem
corresponding to eigenvalue . Let
be the Galerkin approximate solution of the problem (2.1)-(2.3), where are the undermined functions, is a natural number.
Substituting and the approximations into the problem (2.1)-(2.3), we get
(2.4)
(2.5)
(2.6)
Multiplying both sides of (2.4) and (2.6) by , summing up for and integrating on , we have
(2.7)
(2.8)
Lemma 2.1. (Adams [7]) There exist constants and such that for any integers and , the following inequality holds
Lemma 2.2. Assume that is bounded blow, i.e., there is a constant such that for any . Then for any , the problem (2.7), (2.8) has a global classical solution . Moreover, we have the following estimate
(2.9)
where and in the sequel are constants which depend on , but not on and .
Proof: Let , then , and is a monotonically increasing function, and thus
.
Obviously, system (2.7) is equivalent to the following system
(2.10)
Multiplying both sides of (2.10) by , summing up for , and integrating by parts, we obtain
(2.11)
When , by virtue of Lemma 2.1, there is constants and such that
(2.12)
(2.13)
Adding to the both sides of (2.11), integrating in , making use of (2.12), (2.13), we get
(2.14)
Applying the Gronwall inequality to (2.14), we can obtain (2.9). When , adding to the both sides of (2.11), integrating the product over , making use of (2.13) and the Cauchy inequality and Gronwall inequality, we get (2.9) immediately.
Using (2.9) and the Leray-Schauder fixed point theorem [8], we can prove that the problem (2.7), (2.8) has a solution . Lemma 2.2 is proved.
Lemma 2.3. (Zhou and Fu [9]) Assume that is a k-times continuously differentiable function with respect to variables and . Then
where is a positive constant depending only on
and .
Lemma 2.4. Assume that the assumption of Lemma 2.2 hold, , , then there exist the estimates
(2.15)
(2.16)
Proof: We apply the mathematical induction to prove the estimate (2.15). The estimate (2.9) is the estimate (2.15) when . Suppose that when , the estimate (2.15) holds. We shall prove that, when , the estimate (2.15) holds too.
Multiply both sides of (2.7) by , summing up for , integrating by parts, we obtain
(2.17)
By lemma 2.1, there is a constant , such that
(2.18)
Adding to the both sides of (2.17), integrating the product over , Cauchy inequality, Lemma 2.3, (2.9) and (2.18), we have
(2.19)
where . It follows from (2.9), (2.19) and the Gronwall
inequality, we get
(2.20)
Multiply both sides of (2.7) by , summing up for , integrating by parts, Holder inequality, (2.15), we get
(2.21)
Theorem 2.1. Under the assumptions of Lemma 2.4, if , then the problem (2.1) - (2.3) has a unique generalized global solution , which has continuous derivatives and generalized derivatives , and .
Proof: First we give the definition of the generalized solution, which satisfies the identity
(2.22)
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 , we know
We can select a subsequence of and a function and , the subsequence uniformly converges to the limiting function in . In fact, is the uniformly bound in . Meanwhile
(2.23)
where and are the change vectors. Therefore, is equicontinuous in .
According to Ascoli-Arzela, we can select a subsequence of , still denoted by , such that there exists a function and , the subsequence uniformly converges to the limiting function in . The corresponding subsequences also uniformly converges to in , respectively.
Making use of the weakly compact theorem of the space , we know that the subsequences , and weakly converge to , and in , respectively.
In fact,
(2.24)
where . So the identity (2.22) holds. Obviously satisfies the periodic boundary value conditions (2.2), (2.3) in the classical sense. Therefore when , there exists a generalized global solution of the problem (2.1) - (2.3).
It is easy that we can get the uniqueness of the solution of the periodic boundary value problem (2.1) - (2.3). The Theorem 2.1 is proved.
Theorem 2.2. Under the assumptions of Lemma 2.4, if , then the periodic boundary value problem (2.1) - (2.3) has a unique global classical solution .
Proof: Differentiating (2.7) with respect to , we have
(2.25)
Multiplying both sides of (2.25) by , summing up for , integrating by parts and using the Holder inequality, combining (2.15), we conclude
(2.26)
Combining (2.26) and Sobolev embedding theorem, we know that
Using the method of Theorem 2.1, when , the periodic boundary value problem (2.1) - (2.3) has a global classical solution . It is easy to prove the uniqueness of solution for the problem (2.1) - (2.3).
3. Cauchy Problem (1.6), (1.7)
Theorem 3.1. Suppose that is bounded blow, , . If , then the Cauchy problem (1.6), (1.7) has a unique global generalized solution .
Proof: Let us take a real sequence such that as . For every , let us construct periodic functions and with period such that
i) ;
ii) for and
we consider the following periodic boundary value problem
(3.1)
(3.2)
(3.3)
Let be the orthogonal
base in composed of eigenfunctions of eigenvalue problem
corresponding to eigenvalue , where .
Suppose that the Galerkin approximate solution of (3.1)-(3.3) is
where are the undermined functions.
Let satisfy the following equation and conditions
(3.4)
(3.5)
(3.6)
By the same method as in the estimates (2.15), (2.16), we have
(3.7)
where and in the sequel are constants independent of and . By the Sobolev imbedding theorem when , we get
(3.8)
(3.9)
By virtue of (3.9) and Ascoli-Arzela theorem, we can select from a subsequence, still denoted by , such that when ,
uniformly converge to limiting functions in , respectively.
The estimates (3.8) still holds for the above subsequence . Hence, we can select from a subsequence, still denoted by , such
that when , the subsequences , and weakly converge to limiting functions , and in ,
respectively.
From the corollary of the resonance theorem [10], it follows that the estimates (3.8), (3.9) still hold for , which is the generalized solution of the problem (3.1)-(3.3). Using Ascoli-Arzela theorem, we can select from a subsequence still denoted by , such that when , the subsequences uniformly converge to limiting functions in any domain , respectively.
It follows from (3.8) that we can select from a subsequence, still denoted by , such that when , in , the subsequences , and weakly converge to limiting functions , and , respectively. The obtained limiting function is just the global generalized solution of the auxiliary problem (1.6), (1.7).
Clearly, the generalized solution of the auxiliary problem (1.6), (1.7) is also unique. Therefore when , 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 , then the Cauchy problem (1.6), (1.7) has a unique global classical solution .
Proof: Combining the estimates (2.15), (2.16) with (2.35), we obtain
(3.10)
By the Sobolev imbedding theorem when , we get
Using the method of Theorem 3.1, when , the Cauchy problem (1.6), (1.7) has a global classical solution. It is easy to prove the uniqueness of solution for the problem (1.6), (1.7).
4. Cauchy Problem (1.1), (1.2)
Lemma 4.1. [11] Suppose that , then
may be embedded into , and for any , we have
where is a set of nonnegative integers.
Theorem 4.1. Suppose that , is bounded blow, , . If , then the Cauchy problem (1.1), (1.2) has a unique global generalized solution .
Proof: Differentiating (3.4) and (3.6) with respect to , we have
(4.1)
(4.2)
Let
(4.3)
Substituting (4.3) into (4.1), (3.5) and (4.2), we obtain
(4.4)
(4.5)
(4.6)
By using the change (4.3), it follows from (3.7) that
(4.7)
From (4.7) and the Sobolev imbedding theorem, we know that
(4.8)
By using the same method as in Section 3, it follows from (4.7) and (4.8) that, when , the Cauchy problem (1.1), (1.2) has a generalized global solution .
It is easy that, we prove the uniqueness of solution for the problem (1.1), (1.2). Hence, Theorem 4.1 is proved.
Theorem 4.2. Assume that the assumptions of Theorem 4.1 hold, If Math_277#, then the Cauchy problem (1.1), (1.2) has a unique global classical solution .
Proof: By virtue of Theorem 3.2, when , the problem (1.6), (1.7) has a unique global classical solution . Differentiating Equation (1.6), (1.7) with and substituting into this equation and into the obtained initial value condition, we get is the classical global solution of (1.1), (1.2). The proof is completed.
Acknowledgements
This work is supported by the National Natural Science Foundation of China (NSFC) under Grant No. 11671367 and Natural Science Foundation of Henan Province, China under Grant No. 152300410227.
Cite this paper
Li, H., Song, C.M. and Chen, L. (2017) Cauchy Problem of the Singularly Perturbed Sixth Order Boussinesq Type Equation. Journal of Applied Mathematics and Physics, 5, 1648-1657. https://doi.org/10.4236/jamp.2017.59137
References
- 1. Darapi, P. and Hua, W. (1999) A Numerical Method for Solving an Ill-Posed Bous-sinesq Equation Arising in Water Waves and Nonlinear Lattices. Applied Mathematics and Computation, 101, 159-207. https://doi.org/10.1016/S0096-3003(98)10070-X
- 2. Dash, R.K. and Daripa, P. (2002) Analytical and Numerical Studies of a Singularly Perturbed Boussinesq Equa-tion. Applied Mathematics and Computation, 126, 1-30. https://doi.org/10.1016/S0096-3003(01)00166-7
- 3. Darapi, P. and Hua, W. (2001) Weakly Non-Local Solitary Wave Solutions of a Singularly Perturbed Boussinesq Equation. Mathematics and Computers in Simulation, 55, 393-405. https://doi.org/10.1016/S0378-4754(00)00288-3
- 4. Feng, Z.S. (2003) Traveling Solitary Wave Solutions to the Generalized Boussinesq Equation. Wave Motion, 37, 17-23. https://doi.org/10.1016/S0165-2125(02)00019-7
- 5. Song, C., Li, H. and Li, J. (2013) Initial Boundary Value Problem for the Singularly Perturbed Bous-sineaq-Type Equation. Discrete and Continuous Dynamical Systems, 709-717.
- 6. Song, C., Li, J. and Gao, R. (2014) Nonexistence of Global Solutions to the Initial Boundary Value Problem for the Singularly Perturbed Sixth Order Bous-sinesq Equation. Hindawi Publishing Corporation, Journal of Applied Mathemat-ics.
- 7. Admas, R.A. (1975) Sobolev Space. Academic Press, New York.
- 8. Friedman, A. (1964) Partial Differential Equation of Parabolic Type. Pren-tice Hall, Eagliweed Cliffs.
- 9. Zhou, Y.L. and Fu, H.Y. (1983) Nonlinear Hyper-bolic Systems of Higher Order Generalized Sine-Gordon Type. Acta Mathematica Sinica, 26, 234-249.
- 10. Yosida, K. (1980) Functional Analysis. 6th edition, Springer Verlag, Berlin.
- 11. Wang, Y.D. (1989) L2 Theory of Partial Differential Equations. Peking University Press, Beijing.