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: (),
(). We consider the following perturbed problem, where are two small parameters such that :
By Theorem 3.2, the problem () has a unique weak solution depending on . When , () is denoted by (). We will study the asymptotic expansion of with respect to . We use the following notations. For a multi-index , and , we put
We first note the following lemma.
Lemma 4.1. Let and . Then,
where the coefficients depending on are defined by the recurrent formulas
Let be a unique weak solution of the problem () corresponding to that is,
Let us consider the sequence of weak solutions , defined by the following problems:
where are defined by the recurrent formulas as follows:
with , , , , defined by
where , , , , ,
Then, we have the following lemma.
Lemma 4.2. Let , , be the functions defined by (4.5) and (4.7). Put , then one has
where , with is a constant depending only on .
Proof. (i) In the case of , the proof of (4.9) is easy, hence we omit the details. We only prove with . We write . Using Taylor's expansion of the function around the point up to order , we obtain from (4.2) that
where
We note that
On the other hand, if we put
then by the boundedness of the functions in the function space , we obtain from (4.3), (4.12), and (4.14) that , with and is a constant depending only on . Therefore, we obtain from (4.5), (4.11), (4.13), and (4.14) that
Hence, (4.9) in Lemma 4.2 is proved. (ii) We also only prove (4.10) with . Using Taylor's expansion of the function around the point up to order , we obtain from (4.2) that
where
We also note that
Similarly,
where , with is a constant depending only on . Then, (4.10) holds. Lemma 4.2 is proved.
Remark 4.3. Lemma 4.2 is a generalization of the formula given in [17, page 262, formula (4.38)], and it is useful to obtain Lemma 4.4 below. These lemmas are the key to the asymptotic expansion of a weak solution of order in two small parameters . By as a unique weak solution of , satisfies the problem
where
Lemma 4.4. Let and hold. Then
where is a constant depending only on .
Proof. We only need prove with . Using (4.9) for the function , we obtain
By (4.6), (4.8), we write
On the other hand, from (4.24), we compute
where
Hence,
Similarly, we write
where is bounded in the function space by a constant depending only on . Combining (4.4), (4.21), (4.27), and (4.28) yields
By the boundedness of the functions in the function space , we obtain from (4.26) and (4.29) that
where is a constant depending only on . The proof of Lemma 4.4 is complete.
Now, we consider the sequence of functions defined by
With , we have the problem
Multiplying two sides of (4.32) by , we compute without difficulty from (4.22) that
where . By
we get
with .
We will prove that there exists a constant , independent of and , such that
Multiplying two sides of (4.31) with and after integrating in , we obtain without difficulty from (4.22) that
where . We will estimate the integrals on the right-hand side of (4.39) as follows.
Using Gronwall's lemma, we deduce from (4.50) that
where
We can assume that
with sufficiently small .
Lemma 4.5. Let the sequence satisfy
where are the given constants. Then,
This lemma is useful, as it will be said below, and it is easy to prove.
Applying Lemma 4.5 with , it follows from (4.55) that
On the other hand, the linear recurrent sequence defined by (4.31) converges strongly in the space to the solution of the problem (4.20). Hence, letting in (4.56) yields
it meansthat
Consequently, we obtain the following theorem.
Theorem 4.6. Let and hold. Then there exist constants and such that, for every , with , the problem () has a unique weak solution satisfying an asymptotic expansion up to order as in (4.58), where the functions are the weak solutions of the problems (), (), , respectively.
The Problem with Many Small Parameters Next, we note that the results as above still holdfor the problem in small parameters as follows:
For more detail, we also make the following assumptions: ,
. For a multi-index , and , we also put
Let be a unique weak solution of the problem (), which is () corresponding to Let the sequence of weak solutions be defined by the problems (), in which , are defined by suitable recurrent formulas. Then, the following similar theorem holds.
Theorem 4.7. Let , and hold. Then there exist constants and such that, for every , with , the problem has a unique weak solution satisfying an asymptotic estimation up to order as follows:
The proof of Theorem 4.7 is similar the one as above let us omit it.
Remark 4.8. Typical examples about asymptotic expansion of solutions in a small parameter can be found in the research of many authors such as [1, 3, 4, 8, 9, 17–19]. However, to our knowledge, in the case of asymptotic expansion in many small parameters, there is only partial results, for example, [5–7, 14], concerning asymptotic expansion of solutions in two or three small parameters.
Acknowledgments
The authors wish to express their sincere thanks to the referees for the suggestions and valuable comments. The authors are also extremely grateful for the support given by Vietnam's National Foundation for Science and Technology Development (NAFOSTED) under Project no. 101.01-2010.15.
References
E. L. Ortiz and A. P. N. Dinh, “Linear recursive schemes associated with some nonlinear partial differential equations in one dimension and the tau method,” SIAM Journal on Mathematical Analysis, vol. 18, no. 2, pp. 452–464, 1987.
A. P. N. Dinh and N. T. Long, “Linear approximation and asymptotic expansion associated to the nonlinear wave equation in one dimension,” Demonstratio Mathematica, vol. 19, no. 1, pp. 45–63, 1986.
N. T. Long and T. N. Diem, “On the nonlinear wave equation associated with the mixed homogeneous conditions,” Nonlinear Analysis, vol. 29, no. 11, pp. 1217–1230, 1997.
N. T. Long, A. P. N. Dinh, and T. N. Diem, “On a shock problem involving a nonlinear viscoelastic bar,” Boundary Value Problems, no. 3, pp. 337–358, 2005.
N. T. Long and L. X. Truong, “Existence and asymptotic expansion for a viscoelastic problem with a mixed nonhomogeneous condition,” Nonlinear Analysis, Theory, Methods & Applications. Series A: Theory and Methods, vol. 67, no. 3, pp. 842–864, 2007.
N. T. Long and L. X. Truong, “Existence and asymptotic expansion of solutions to a nonlinear wave equation with a memory condition at the boundary,” Electronic Journal of Differential Equations, vol. 2007, no. 48, pp. 1–19, 2007.
N. T. Long, N. C. Tam, and N. T. T. Truc, “On the nonlinear wave equation with the mixed nonhomogeneous conditions: linear approximation and asymptotic expansion of solutions,” Demonstratio Mathematica, vol. 38, no. 2, pp. 365–386, 2005.
L. T. P. Ngoc, L. N. K. Hang, and N. T. Long, “On a nonlinear wave equation associated with the boundary conditions involving convolution,” Nonlinear Analysis, Theory, Methods & Applications. Series A: Theory and Methods, vol. 70, no. 11, pp. 3943–3965, 2009.
F. A. Ficken and B. A. Fleishman, “Initial value problems and time-periodic solutions for a nonlinear wave equation,” Communications on Pure and Applied Mathematics, vol. 10, pp. 331–356, 1957.
P. H. Rabinowitz, “Periodic solutions of nonlinear hyperbolic partial differential equations,” Communications on Pure and Applied Mathematics, vol. 20, pp. 145–205, 1967.
T. Kiguradze, “On bounded and time-periodic solutions of nonlinear wave equations,” Journal of Mathematical Analysis and Applications, vol. 259, no. 1, pp. 253–276, 2001.
T. Caughey and J. Ellison, “Existence, uniqueness and stability of solutions of a class of nonlinear partial differential equations,” Journal of Mathematical Analysis and Applications, vol. 51, pp. 1–32, 1975.
L. T. P. Ngoc, L. K. Luan, T. M. Thuyet, and N. T. Long, “On the nonlinear wave equation with the mixed nonhomogeneous conditions: linear approximation and asymptotic expansion of solutions,” Nonlinear Analysis, Theory, Methods & Applications. Series A: Theory and Methods, vol. 71, no. 11, pp. 5799–5819, 2009.
J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non-Linéaires, Dunod-Gauthier Villars, Paris, France, 1969.
K. Deimling, Nonlinear Functional Analysis, Springer, NewYork, NY, USA, 1985.
N. T. Long, “On the nonlinear wave equation associated with the mixed homogeneous conditions,” Journal of Mathematical Analysis and Applications, vol. 306, no. 1, pp. 243–268, 2005.
J. Boujot, A. P. N. Dinh, and J. P. Veyrier, “Oscillateurs harmoniques faiblement perturbés: L'algorithme numérique des “par de géants”,” RAIRO Analyse Numérique, vol. 14, no. 1, pp. 3–23, 1980.
N. T. Long, A. P. N. Dinh, and T. N. Diem, “Linear recursive schemes and asymptotic expansion associated with the Kirchoff—Carrier operator,” Journal of Mathematical Analysis and Applications, vol. 267, no. 1, pp. 116–134, 2002.
