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

Nhan, Nguyen Huu; Ngoc, Le Thi Phuong; Long, Nguyen Thanh

doi:https://doi.org/10.1155/2018/3626543

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2018 | Article ID 3626543 | https://doi.org/10.1155/2018/3626543

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

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

Academic Editor: Filippo Cacace

Received25 Jun 2017

Accepted10 Dec 2017

Published22 Jan 2018

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 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 [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 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 thatPassing 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 estimatewhere 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 problemNote thatTaking in (76)₁, after integrating in , we getwhereand the integrals on the right-hand side of (78) are estimated as follows.
First Integral . By (37) and (79), we haveSecond Integral . By the inequalitiesand from the equationwe obtain thatThis implies thatCombining (78), (80), and (84), we obtainUsing Gronwall’s Lemma, we deduce from (85) thatwhere is defined as in (69), which implies thatIt follows that is a Cauchy sequence in . Then there exists such thatNote that ; then there exists a subsequence of such thatWe also note thatHence, from (88) and (90), we obtainOn the other hand, for all , we have HenceFinally, passing to limit in (30)-(31) as , it is implied from (88), (89)_1,3, and (93) that there exists satisfying the equationfor all and the initial conditionsOn the other hand, from the assumptions – we obtain from (89)₄, (93), and (94) thatand 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 problemwhereWe take in (97)₁ and integrate in to getwhere
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 , . Thenwhere 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 formulaswith , which are defined by the formulaswhere andwith , 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 haswith , 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 obtainwhereSimilarly, with , by using Taylor’s expansion of the function around the point up to order , we obtainwhere, , , ,
Note, by formula (103), we getwhere .
Similarly, with , we also havewhere .
Hence, we deduce from (116), (117) thatwhereHence, it follows from (114), (118) thatwhereThereforewithOn the other hand, we also havewherewith , which are defined by (108).
Decompose the sum into the sum of two sums and ; therefore, we deduce from (124) thatwhereHence, it follows from (112) and (124) thatwhere , , are defined by (106)–(108) andBy 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 problemwhere

Then, we have the following lemma.

Lemma 13. Let , , , and hold. Then there exists a constant such thatwhere 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 obtainwhere , with being a constant depending only on , , , , , .
By (133), we rewrite as follows:Hence, we deduce from (110) and (134) thatwhereWe decompose the sum into the sum of two sums and ; therefore, we deduce from (135) thatwhereCombining (105), (131), and (137) leads toBy the boundedness of the functions , , , in the function space , we obtain from (110), (136), (138), and (139) thatwhere is a constant depending only on , , , , , , ,
Lemma 13 is proved.

Proof of Theorem 10. Consider the sequence defined byBy multiplying two sides of (141) with and after integration in , we havewhereBy using Lemma 13, we deduce from (142) thatWe estimate the integrals on the right-hand side of (144) as follows.
Estimating . We note that and we havewith ,
It follows from (146) thatEstimating . First, we need an estimation .
We also note thatFrom the equationit follows thatwhere .
Next, by , it follows thatwhere .
By (151), we obtainCombining (144), (147), and (152), this leads towhere .
By using Gronwall’s Lemma, we deduce from (153) thatwhere ,
We can assume thatWe require the following lemma whose proof is immediate.
Lemma 14. Let the sequence satisfywhere , are the given constants. ThenApplying Lemma 14 with , , , it follows from (154) thatwhere 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 getThis 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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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.

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