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Mathematical Problems in Engineering
Volume 2018 (2018), Article ID 3626543, 18 pages
https://doi.org/10.1155/2018/3626543
Research Article

On a Nonlinear Wave Equation of Kirchhoff-Carrier Type: Linear Approximation and Asymptotic Expansion of Solution in a Small Parameter

1Dong Nai University, 4 Le Quy Don Str., Tan Hiep District, Bien Hoa City, Vietnam
2Department of Mathematics and Computer Science, VNUHCM-University of Science, 227 Nguyen Van Cu Str., Dist. 5, Ho Chi Minh City, Vietnam
3University of Khanh Hoa, 01 Nguyen Chanh Str., Nha Trang City, Vietnam

Correspondence should be addressed to Nguyen Thanh Long; moc.liamg@2tngnol

Received 25 June 2017; Accepted 10 December 2017; Published 22 January 2018

Academic Editor: Filippo Cacace

Copyright © 2018 Nguyen Huu Nhan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the Robin-Dirichlet problem for a nonlinear wave equation of Kirchhoff-Carrier type. Using the Faedo-Galerkin method and the linearization method for nonlinear terms, the existence and uniqueness of a weak solution are proved. An asymptotic expansion of high order in a small parameter of a weak solution is also discussed.

1. Introduction

In this paper, we consider the following Robin-Dirichlet problem for a nonlinear wave equation of Kirchhoff-Carrier type:where , , , , are given functions and is a given constant.

Equation (1) can be considered as a general equation containing relatively some classical equations; for example, when , (1) has a relation to the Kirchhoff wave equation:(see [1]). This equation is a generalization of the well-known D’Alembert’s wave equation for free vibrations of elastic strings. Kirchhoff’s model takes into account the changes in length of the string produced by transverse vibrations. The parameters in (4) have the following meanings: is the lateral deflection, is the length of the string, is the area of the cross section, is the Young modulus of the material, is the mass density, and is the initial tension.

In another case, with , (1) contains the form of Carrier equation. In [2], Carrier established the equation modeling the vibration of an elastic string when the changes in tension are not small:where is the -derivative of the deformation, is the tension in the rest position, is the Young modulus, is the cross section of a string, is the length of a string, is the density of a material. Therefore, it is clear that (1) considered here contains (4) and (5) as special cases.

Moreover, with various boundary conditions, the particular forms of (1) have been extensively studied by many authors; for example, we refer to [315] and the references given therein. In these works, many interesting results about existence, regularity, asymptotic behavior, asymptotic expansion, and decay of solutions were obtained.

Cavalcanti et al., in [47], investigated a series of four papers in which the results of existence, global existence, exponential or uniform decay rates, and asymptotic behavior for Kirchhoff-Carrier models are considered.

In [10], the unique existence and asymptotic expansion of solutions of (1) with associated with the boundary conditionsand the initial conditions are also studied.

In [15], de Lima Santos studied the asymptotic behavior of solutions of (1) with , , associated with the Dirichlet boundary condition at and a boundary condition of memory type at ; that is, .

In [3], Beilin investigated the existence and uniqueness of a generalized solution for the following wave equation with an integral nonlocal conditionwhere is a bounded domain in with a smooth boundary, is the unit outward normal on , and , , , are given functions. Nonlocal conditions come up when values of the function on the boundary are connected to values inside the domain. There are various types of nonlocal boundary conditions of integral form for hyperbolic, parabolic, or elliptic equations; the ones were introduced in [3].

The well-posedness and optimal decay rate estimates of the energy associated with the Kirchhoff-Carrier problem with memorywhere is a bounded domain in , with a smooth boundary , are proved in [8].

In [11], the following nonlinear wave equation with initial conditions and boundary conditions of two-point type has been investigated:

In [12], by combining the linearization method for the nonlinear term, the Faedo-Galerkin method, and the weak compact method, the existence of a unique weak solution of an initial and boundary value problem for nonlinear wave equation with the nonhomogeneous boundary conditions is proved.

Very recently, in [13, 14], with the same method used in [12], the authors proved the results of existence and uniqueness for the wave equations with nonlinear sources containing the nonlocal terms. In [13], the linearization method together with Taylor’s expansion is used for both of the source term and the nonlinear integral in it. These techniques have not been used before.

In the same spirit of [1014], we establish the local existence and uniqueness for prob. (1)–(3) by using the Faedo-Galerkin method and the weak compact method. These results are presented in Section 3. In Section 4, the perturbed solution is approximated by the polynomial of degree in a small parameter for the following perturbed equation:associated with (2), (3), where

2. Preliminaries

Put and denote the usual function spaces used in this paper by the notations , . Let be either the scalar product in or the dual pairing of a continuous linear functional and an element of a function space. The notation stands for the norm in , is the norm in the Banach space , and is the dual space of .

We denote for the Banach space of real functions measurable, such that

Let , , , , , denote , , , , , respectively.

With , , we put , , , , with and ; , .

Similarly, with , we also put

We shall use the following norm on :

We put

is a closed subspace of and on three norms , , and are equivalent norms.

We have the following lemmas, the proofs of which are straightforward and hence we omit the details.

Lemma 1. The imbedding is compact andwhere (see [16]).

Lemma 2. Let . The imbedding is compact andfor all .

Lemma 3. Let . Then the symmetric bilinear form defined by (15) is continuous on and coercive on .

Lemma 4. Let . Then there exists the Hilbert orthonormal base of consisting of the eigenfunctions corresponding to the eigenvalues such that Furthermore, the sequence is also a Hilbert orthonormal base of with respect to the scalar product .
On the other hand, we also have satisfying the following boundary value problem:

The proof of Lemma 4 can be found in ([17], p.87, Theorem ), with and as defined by (14), (15).

Remark 5. The weak formulation of the initial-boundary value problem (1)–(3) can be given in the following manner: Find , such that satisfies the following variational equation:for all , a.e., , together with the initial conditionswhere, for each , is the family of symmetric bilinear forms on defined byfor all , , with being given constant, and

3. The Existence and Uniqueness

Let . We make the following assumptions:() satisfying the condition .().() and there exists a constant such that , for all .().

For each given, we set the constants , , , , as follows:where

For every and , we put in which .

Then is a Banach space with respect to the norm (See Lions [18]) We also put

Now, we establish the recurrent sequence The first term is chosen as , and supposing thatwe associate problem (1) with the following problem.

Find satisfying the linear variational problemwhere

Theorem 6. Suppose that hold. Then, there exist positive constants such that the recurrent sequence is defined by (29)–(31).

Proof. The proof consists of several steps.
Step  1 (the Faedo-Galerkin approximation (introduced by Lions [18])). Consider the basis for as in Lemma 4. Approximate solution of (29)–(31) problem which will be found in formwhere the coefficients satisfy the system of linear differential equationswhereThe system of (33) can be rewritten in formwhereBy (29), it is not difficult to prove that system (35), (36) has a unique solution on interval , so let us omit the details (see [19]).
Step  2 (a priori estimates). First, we need the following lemma.
Lemma  7. Putting , one hasThe proof of Lemma is easy; hence we omit the details.
Next, we putwhereThen, it follows from (33), (37), (38), (39) thatWe shall estimate the terms on the right-hand side of (40) as follows.
First Term . We note thatwhere we use the notations So, by (24), (25), and (41), we obtain Hence,Second Term . By Lemma  7 (ii) and (iv), we haveThird Term . The Cauchy-Schwartz inequality leads towhere .
We shall estimate the term as follows.
By , we haveOn the other hand, by , it implies thatSimilarly, from the following equality we obtain thatBy (48) and (50), it follows from (47) thatwhereTherefore, from (46) and (51), we obtainFourth Term . Applying the Cauchy-Schwartz inequality again, we havefor all . On the other hand, it follows from (51) thatHence, we obtain from (54) and (55) thatFifth Term Sixth Term . Similarly, we obtainSeventh Term . We haveEighth Term . We note that (33)1 can be rewritten as follows:Hence, it follows after replacing with and integrating thatWe estimate the term
By (48), we obtainTherefore, by Lemma  7 (ii), (61) and (62), we obtainwhere Choosing , with , it follows from (40), (44), (45), (53), (56)–(59), and (63) thatwhereBy means of the convergences in (34), we can deduce the existence of a constant independent of and such thatSo, from (66)2, we can choose , such thatFinally, it follows from (65), (67), and (68) thatBy using Gronwall’s Lemma, we deduce from (70) thatfor all , for all and . Therefore, we haveStep  3 (limiting process). From (72), we deduce the existence of a subsequence of still so denoted, such that