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

Nguyen Huu Nhan,1,2 Le Thi Phuong Ngoc,³ and Nguyen Thanh Long²

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; longnt2@gmail.com

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 [3–15] 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 [4–7], 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 conditions

and 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 condition

where 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 memory

where 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 [10–14], 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 and

where (see [16]).

Lemma 2. Let . The imbedding is compact and

for 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 conditions

where, for each , is the family of symmetric bilinear forms on defined by

for 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 that

we associate problem (1) with the following problem.

Find satisfying the linear variational problem

where

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 form

where the coefficients satisfy the system of linear differential equations

where

The system of (33) can be rewritten in form

where

By (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 has

The proof of Lemma is easy; hence we omit the details.
Next, we put

where

Then, it follows from (33), (37), (38), (39) that

We shall estimate the terms on the right-hand side of (40) as follows.
First Term . We note that

where we use the notations

So, by (24), (25), and (41), we obtain

Hence,

Second Term . By Lemma  7 (ii) and (iv), we have

Third Term . The Cauchy-Schwartz inequality leads to

where .
We shall estimate the term as follows.
By , we have

On the other hand, by , it implies that

Similarly, from the following equality

we obtain that

By (48) and (50), it follows from (47) that

where

Therefore, from (46) and (51), we obtain

Fourth Term . Applying the Cauchy-Schwartz inequality again, we have

for all . On the other hand, it follows from (51) that

Hence, we obtain from (54) and (55) that

Fifth Term

Sixth Term . Similarly, we obtain

Seventh Term . We have

Eighth Term . We note that (33)₁ can be rewritten as follows:

Hence, it follows after replacing with and integrating that

We estimate the term
By (48), we obtain

Therefore, by Lemma  7 (ii), (61) and (62), we obtain

where

Choosing , with , it follows from (40), (44), (45), (53), (56)–(59), and (63) that

where

By means of the convergences in (34), we can deduce the existence of a constant independent of and such that

So, from (66)₂, we can choose , such that

Finally, it follows from (65), (67), and (68) that

By using Gronwall’s Lemma, we deduce from (70) that

for all , for all and . Therefore, we have

Step  3 (limiting process). From (72), we deduce the existence of a subsequence of still so denoted, such that

Passing to limit in (33), we have satisfying (30), (31) in On the other hand, it follows from (30)₁ and (73)₄ that ; hence and the proof of Theorem 6 is complete.

We note that is a Banach space with respect to the norm (see Lions [18]).

We use the result given in Theorem 6 and the compact imbedding theorems to prove the existence and uniqueness of a weak solution of prob. (1)–(3). Hence, we get the main result in this section as follows.

Theorem 8. Let – hold. Then one has the following.
(i) Prob. (1)–(3) has a unique weak solution , where the constants and are chosen as in Theorem 6.
(ii) Furthermore, the recurrent sequence defined by (29)-(30) converges to the solution of prob. (1)–(3) strongly in .
And one has the estimate

where the constant is defined as in (69) and is a constant depending only on , , , , , , , and .

Proof.
(a) Existence of the Solution. We shall prove that is a Cauchy sequence in . Let . Then satisfies the variational problem

Note that

Taking in (76)₁, after integrating in , we get

where

and the integrals on the right-hand side of (78) are estimated as follows.
First Integral . By (37) and (79), we have

Second Integral . By the inequalities

and from the equation

we obtain that

This implies that

Combining (78), (80), and (84), we obtain

Using Gronwall’s Lemma, we deduce from (85) that

where is defined as in (69), which implies that

It follows that is a Cauchy sequence in . Then there exists such that

Note that ; then there exists a subsequence of such that

We also note that

Hence, from (88) and (90), we obtain

On the other hand, for all , we have

Hence

Finally, passing to limit in (30)-(31) as , it is implied from (88), (89)1,3, and (93) that there exists satisfying the equation

for all and the initial conditions

On the other hand, from the assumptions – we obtain from (89)₄, (93), and (94) that

and thus we have . The existence result follows.
(b) Uniqueness of the Solution. Let , be two weak solutions of prob. (1)–(3). Then satisfies the variational problem

where

We take in (97)₁ and integrate in to get

where
Put , and then it follows from (99) that

By Gronwall’s Lemma, we deduce ; that is, . Theorem 8 is proved completely.

4. Asymptotic Expansion of the Solution with respect to a Small Parameter

In this section, let – hold. We make more the following assumptions:

().() and

We consider the following perturbed problem, where is a small parameter such that :

where

By Theorem 8, problem has a unique weak solution depending on , satisfying , in which are independent of ; these constants are chosen as in (67), (68), and (69), with , stand for , , respectively.

Moreover, we can prove that the limit in suitable function spaces of the family as is a unique weak solution of the problem (corresponding to ) also satisfying .

We shall study the asymptotic expansion of the solution of the problem with respect to a small parameter .

We use the following notations. For a multi-index and , we put

First, we need the following lemma.

Lemma 9. Let , and  , . Then

where the coefficients , depend on defined by the formulas

Proof of Lemma 9. The proof of Lemma 9 is easy; hence we omit the details.

Now, we assume that

(), ;(), ,  and

We use notations , .

Let be a unique weak solution of the problem corresponding to ; that is,

Let us consider the sequence of the weak solutions , defined by the following problems:

where , , are defined by the formulas

with , which are defined by the formulas

where and

with , are defined by

Then, we have the following theorem.

Theorem 10. Let , , , and hold. Then there exist constants and such that, for every , the problem has a unique weak solution satisfying the asymptotic estimation up to order as follows:

where the functions , , are the weak solutions of the problems , , , respectively, and is a constant depending only on , , , , , , , ,

In order to prove Theorem 10, we need the following lemmas.

Lemma 11. Let , , be the functions defined by the formulas (106)–(108). Putting , then one has

with , where is a constant depending only on , , , , , .

Proof of Lemma 11. In the case of , the proof of (110) is easy; hence we omit the details, which we only prove with . Putting , we rewrite as follows:

where and
By using Taylor’s expansion of the function around the point up to order , we obtain

where

Similarly, with , by using Taylor’s expansion of the function around the point up to order , we obtain

where

, , , ,
Note, by formula (103), we get

where .
Similarly, with , we also have

where .
Hence, we deduce from (116), (117) that

where

Hence, it follows from (114), (118) that

where

Therefore

with

On the other hand, we also have

where

with , which are defined by (108).
Decompose the sum into the sum of two sums and ; therefore, we deduce from (124) that

where

Hence, it follows from (112) and (124) that

where , , are defined by (106)–(108) and

By the boundedness of the functions , , , in the function space , we obtain from (113), (115), (121), (123)₂, (124), and (127) that , where is a constant depending only on , , , , . Thus, Lemma 11 is proved.

Remark 12. Lemma 11 is a generalization of the formula contained in ([9], p.262, formula (4.38)) and it is useful to obtain the following Lemma 13. These lemmas are the key to establish the asymptotic expansion of the weak solution of order in a small parameter as follows.
Let be the unique weak solution of problem . Then satisfies the problem

where

Then, we have the following lemma.

Lemma 13. Let , , , and hold. Then there exists a constant such that

where is a constant depending only on , , , , , ,

Proof of Lemma 13. In the case of , the proof of Lemma 13 is easy; hence we omit the details, which we only prove with .
By using formula (110) for the function we obtain

where , with being a constant depending only on , , , , , .
By (133), we rewrite as follows:

Hence, we deduce from (110) and (134) that

where

We decompose the sum into the sum of two sums and ; therefore, we deduce from (135) that

where

Combining (105), (131), and (137) leads to

By the boundedness of the functions , , , in the function space , we obtain from (110), (136), (138), and (139) that

where is a constant depending only on , , , , , , ,
Lemma 13 is proved.

Proof of Theorem 10. Consider the sequence defined by

By multiplying two sides of (141) with and after integration in , we have

where

By using Lemma 13, we deduce from (142) that

We estimate the integrals on the right-hand side of (144) as follows.
Estimating . We note that

and we have

with ,
It follows from (146) that

Estimating . First, we need an estimation .
We also note that

From the equation

it follows that

where .
Next, by , it follows that

where .
By (151), we obtain

Combining (144), (147), and (152), this leads to

where .
By using Gronwall’s Lemma, we deduce from (153) that

where ,
We can assume that

We require the following lemma whose proof is immediate.
Lemma 14. Let the sequence satisfy

where , are the given constants. Then

Applying Lemma 14 with , , , it follows from (154) that

where is a constant depending only on .
On the other hand, the linear recurrent sequence defined by (141) converges strongly in the space to the solution of problem (130). Hence, letting in (158), we get

This implies (109).
The proof of Theorem 10 is complete.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to this article. They read and approved the final manuscript.

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