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
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.
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 , Dinh and Long [2, 3], Long and Diem , Long et al. , Long and Truong [6, 7], Long et al. , Ngoc et al. , and the references therein.
In , 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 , 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 , 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.
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 .
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
Find such that where
Then, we have the following theorem.
Proof. The proof consists of three steps.
Step 1. The Faedo-Galerkin approximation (introduced by Lions ).
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.
From (3.1), (3.5), and (3.8), we have
By using , we obtain from (3.2), (3.5), and (3.13) that Third Term
The Cauchy-Schwartz inequality yields where .
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
By the Cauchy-Schwartz inequality, we have for all . On the other hand
Hence, we obtain from (3.26), (3.27) that for all .
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
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 )
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.
By (3.16), we obtain Second Integral
By , so Third Integral
Using again, we get
We also note that where we use the notation . Therefore, it implies from (3.61) and (3.62) that
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
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
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  standing for (1.2), we obtained some similar results in .
(ii) In the case of , and the boundary condition in  standing for (1.2), some results as above were given in .
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: (), (