Research Article | Open Access

Le Thi Phuong Ngoc, Le Khanh Luan, Tran Minh Thuyet, Nguyen Thanh Long, "On a Nonlinear Wave Equation Associated with Dirichlet Conditions: Solvability and Asymptotic Expansion of Solutions in Many Small Parameters", *International Scholarly Research Notices*, vol. 2011, Article ID 625908, 33 pages, 2011. https://doi.org/10.5402/2011/625908

# On a Nonlinear Wave Equation Associated with Dirichlet Conditions: Solvability and Asymptotic Expansion of Solutions in Many Small Parameters

**Academic Editor:**F. Jauberteau

#### Abstract

A Dirichlet problem for a nonlinear wave equation is investigated. Under suitable assumptions, we prove the solvability and the uniqueness of a weak solution of the above problem. On the other hand, a high-order asymptotic expansion of a weak solution in many small parameters is studied. Our approach is based on the Faedo-Galerkin method, the compact imbedding theorems, and the Taylor expansion of a function.

#### 1. Introduction

In this paper, we consider the following Dirichlet problem: where , and are given functions satisfying conditions specified later.

In the special cases, when the function is independent of , or , and the nonlinear term has the simple forms, the problem (1.1), with various initial-boundary conditions, has been studied by many authors, for example, Ortiz and Dinh [1], Dinh and Long [2, 3], Long and Diem [4], Long et al. [5], Long and Truong [6, 7], Long et al. [8], Ngoc et al. [9], and the references therein.

Ficken and Fleishman [10] and Rabinowitz [11] studied the periodic-Dirichlet problem for hyperbolic equations containing a small parameter, in particular, the differential equation

In [12], Kiguradze has established the existence and uniqueness of a classical solution of the periodic-Dirichlet problem for the following nonlinear wave equation: under the assumption that and are continuously differentiable functions (these conditions are sharp and cannot be weakened). Moreover, it is shown that the same results are valid for the equation with sufficiently small and continuously differentiable .

In [13], a unified approach to the previous cases was presented discussing the existence unique and asymptotic stability of classical solutions for a class of nonlinear continuous dynamical systems.

In [8], Long et al. have studied the linear recursive schemes and asymptotic expansion for the nonlinear wave equation with the mixed nonhomogeneous conditions

In the case of , and some other conditions, an asymptotic expansion of the weak solution of order in is considered.

This paper consists of four sections. In Section 2, we present some preliminaries. Using the Faedo-Galerkin method and the compact imbedding theorems, in Section 3, we prove the solvability and the uniqueness of a weak solution of the problem (1.1)–(1.3). In Section 4, based on the ideals and the techniques used in the above-mentioned papers, we study a high-order asymptotic expansion of a weak solution for the problem (1.1)–(1.3), where (1.1) has the form of a linear wave equation with nonlinear perturbations containing many small parameters*. *In order to avoid making the treatment too complicated without losing of generality, at first, an asymptotic expansion of a weak solution of order in two small parameters for the following equation:
associated with (1.2), (1.3), with , for all , and is established. Next, we note that the same results are valid for the equation in small parameters as follows
associated with (1.2), (1.3). The result obtained here is a relative generalization of [5–7, 14], where asymptotic expansion of a weak solution in two or three small parameters is given.

#### 2. Preliminaries

Put . Let us omit the definitions of usual function spaces that will be used in what follows such as . The norm in is denoted by . We denote by the scalar product in or a pair of dual products of continuous linear functional with an element of a function space. We denote by the norm of a Banach space and by the dual space of . We denote , the Banach space of real functions measurable, such that , with

Let denote , respectively. With , we put and ; , , .

Similarly, with , we put and .

On , we will use the following norms:

Then, we have the following lemma.

Lemma 2.1. *The imbedding is compact and
*

The proof of Lemma 2.1 is easy, hence we omit the details.

*Remark 2.2. *On and are two equivalent norms. Furthermore, we have the following inequalities:

*Remark 2.3. *(i) Let us note more that a unique weak solution of the problem (1.1)–(1.3) will be obtained in Section 3 (Theorem 3.2) in the following manner.

Find such that verifies the following variational equation:
and the initial conditions

(ii) With the regularity obtained by , it also follows from Theorem 3.2 that the problem (1.1)–(1.3) has a unique strong solution that satisfies

On the other hand, by , we can see that .

Also, if , then the weak solution of the problem (1.1)–(1.3) belongs to . So, the solution is almost classical which is rather natural, since the initial data do not belong necessarily to .

#### 3. The Existence and the Uniqueness of a Weak Solution

We make the following assumptions: (), (), ().

With and satisfying the assumptions and , respectively, for each and are given, we put the following constants: where and .

For each and , we get where .

We choose the first term . Suppose that

The problem (1.1)–(1.3) is associated with the following variational problem.

Find such that where

Then, we have the following theorem.

Theorem 3.1. *Let ( )–( ) hold. Then, there exist two constants and the linear recurrent sequence defined by (3.6)–(3.8).*

*Proof. *The proof consists of three steps.*Step 1. **The Faedo-Galerkin approximation* (introduced by Lions [15]).

Consider a special basis on , formed by the eigenfunctions of the Laplacian . Put
where the coefficients satisfy the system of linear differential equations
where

Note that by (3.5), it is not difficult to prove that the system (3.10), (3.11) has a unique solution on interval , so let us omit the details.*Step 2. **A priori estimates*. At first, put

Then, it follows from (3.9)–(3.11), (3.13) that

Next, we will estimate the terms on the right-hand side of (3.14) as follows.*First Term *

We have

From (3.1), (3.5), and (3.8), we have

Hence,
*Second Term*

By using , we obtain from (3.2), (3.5), and (3.13) that
*Third Term*

The Cauchy-Schwartz inequality yields
where .

We note

On the other hand, by , it is implies that

Similarly, the following equality
gives

It follows from (3.20)–(3.23) that

Hence, we obtain from (3.19) and (3.24) that
*Fourth Term *

By the Cauchy-Schwartz inequality, we have
for all . On the other hand

Hence, we obtain from (3.26), (3.27) that
for all .*Fifth Term *

By (3.5), (3.8), and (3.13), we obtain

Note that
where we use the notation . By (3.2), (3.5), and (3.30), we obtain

Hence, we deduce from (3.29) and (3.31) that
*Sixth Term *

By (3.2), (3.5), (3.13), and (3.31), we get
*Seventh Term *

Equation (3.10) is rewritten as follows:

Hence, by replacing with and integrating
we need, estimate .

Combining (3.1), (3.5), and (3.13) yields

Therefore, from (3.35) and (3.36), we obtain

Choosing , with , it follows from (3.13), (3.14), (3.17), (3.18), (3.25), (3.28), (3.32), (3.33), and (3.37) that
where

By , we deduce from (3.12), (3.39) that there exists independent of and , such that

Notice that by , we deduce from (3.39) that

So, from (3.39) and (3.41), we can choose such that

Finally, it follows from (3.38), (3.40), and (3.42) that

By using Gronwall's lemma, we deduce from (3.44) that

Therefore, we have
*Step 3. *Limiting process.

From (3.46), we can extract from a subsequence still denoted by such that
as , and

Based on (3.47), passing to limit in (3.10), (3.11) as , we have satisfying (3.6)–(3.8). On the other hand, it follows from (3.5), (3.6), and (3.47) that

Hence, , and the proof of Theorem 3.1 is complete.

Theorem 3.2. *Let ( )–( ) hold. Then, there exist and satisfying (3.40), (3.42), and (3.43) such that the problem (1.1)–(1.3) has a unique weak solution . **Furthermore, the linear recurrent sequence defined by (3.6)–(3.8) converges to the solution strongly in the space
**
with the following estimation:
**
where as in (3.43) and is a constant depending only on and .*

*Proof. *(i) *The existence*. First, we note that is a Banach space with respect to the norm (see Lions [15])

Next, we prove that is a Cauchy sequence in . Let . Then, satisfies the variational problem

Taking in (3.53), after integrating in , we get
in which
and all integrals on the right-hand side of (3.54) are estimated as follows.*First Integral*

By (3.16), we obtain
*Second Integral*

By ,
so
*Third Integral*

Using again, we get

Note that

Hence,

We also note that
where we use the notation . Therefore, it implies from (3.61) and (3.62) that

Hence,

Combining (3.54)–(3.56), (3.58), and (3.64) yields

Using Gronwall's lemma, (3.65) gives
where as in (3.43).

Hence, we obtain from (3.66) that

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

On the other hand, from (3.48), we deduce the existence of a subsequence of such that

Note that

Hence, from (3.68) and (3.71), we obtain

Finally, passing to limit in (3.6)–(3.8) as , it implies from (3.68), (3.69), and (3.72) that there exists satisfying the equation

On the other hand, by , we obtain from (3.70), (3.72), and (3.73) that
thus , and Step 1 follows.

(ii) *The uniqueness of the solution*.

Let be two weak solutions of the problem (1.1)–(1.3). Then, satisfies the variational problem

We take in (3.75) and integrate in to get
where

We now estimate the terms on the right-hand side of (3.76) as follows:

On the other hand

Hence,

It follows from (3.80), (3.82) that

Combining (3.76)–(3.79) and (3.83) yields

Using Gronwall's lemma, it follows from (3.84) that that is, .

Theorem 3.2 is proved completely.

*Remark 3.3. *(i) In the case of and the boundary condition in [4] standing for (1.2), we obtained some similar results in [4].

(ii) In the case of , and the boundary condition in [8] standing for (1.2), some results as above were given in [8].

*Remark 3.4. *By Galerkin method, as in Remark 2.3, the local existence of a strong solution of the problem (1.1)–(1.3) is proved.

In the case of and , obviously, the problem (1.1)–(1.3) is linear. Then, by the same method and applying Banach's theorem [16, Chapter 5, Theorem 17.1], it is not difficult to prove that the problem (1.1)–(1.3) is global solvability. To strengthen some hypotheses, it is possible to prove existence of a classical solution .

#### 4. Asymptotic Expansion of a Weak Solution in Many Small Parameters

In this section, we will study a high-order asymptotic expansion of a weak solution for the problem (1.1)–(1.3), in which (1.1) has the form of a linear wave equation with nonlinear perturbations containing many small parameters*. *

*The Problem with Two Small Parameters*

At first, we consider the case of the nonlinear perturbations containing two small parameters*. *

Let hold. We make the following assumptions: (),
(