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Research Article | Open Access

Volume 2011 |Article ID 625908 | https://doi.org/10.5402/2011/625908

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
Received09 Mar 2011
Accepted12 Apr 2011
Published23 Jun 2011

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: π‘’π‘‘π‘‘βˆ’πœ•ξ€·πœ•π‘₯πœ‡(π‘₯,𝑑,𝑒)𝑒π‘₯ξ€Έξ€·=𝑓π‘₯,𝑑,𝑒,𝑒π‘₯,𝑒𝑑,0<π‘₯<1,0<𝑑<𝑇,(1.1)𝑒(0,𝑑)=𝑒(1,𝑑)=0,(1.2)𝑒(π‘₯,0)=̃𝑒0(π‘₯),𝑒𝑑(π‘₯,0)=̃𝑒1(π‘₯),(1.3) where ̃𝑒0,̃𝑒1,πœ‡, and 𝑓 are given functions satisfying conditions specified later.

In the special cases, when the function πœ‡(π‘₯,𝑑,𝑒) is independent of 𝑒,πœ‡(π‘₯,𝑑,𝑒)≑1, or πœ‡(π‘₯,𝑑,𝑒)=πœ‡(π‘₯,𝑑), and the nonlinear term 𝑓 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 parameterπœ€, in particular, the differential equationπ‘’π‘‘π‘‘βˆ’π‘’π‘₯π‘₯=2𝛼𝑒𝑑+πœ€π‘“π‘‘,π‘₯,𝑒,𝑒𝑑,𝑒π‘₯ξ€Έ.(1.4)

In [12], Kiguradze has established the existence and uniqueness of a classical solution π‘’βˆˆπΆ2([0,π‘Ž]×ℝ𝑛) of the periodic-Dirichlet problem for the following nonlinear wave equation:π‘’π‘‘π‘‘βˆ’π‘’π‘₯π‘₯=𝑔(𝑑,π‘₯,𝑒)+𝑔1(𝑒)𝑒𝑑,(1.5) under the assumption that 𝑔 and 𝑔1 are continuously differentiable functions (these conditions are sharp and cannot be weakened). Moreover, it is shown that the same results are valid for the equationπ‘’π‘‘π‘‘βˆ’π‘’π‘₯π‘₯=𝑔(𝑑,π‘₯,𝑒)+𝑔1(𝑒)𝑒𝑑+πœ€π‘žπ‘‘,π‘₯,𝑒,𝑒𝑑,𝑒π‘₯ξ€Έ,(1.6) with sufficiently small πœ€ and continuously differentiable π‘ž.

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 equationπ‘’π‘‘π‘‘βˆ’π‘’π‘₯π‘₯ξ€·=𝑓π‘₯,𝑑,𝑒,𝑒π‘₯,𝑒𝑑+πœ€π‘“1ξ€·π‘₯,𝑑,𝑒,𝑒π‘₯,𝑒𝑑,(1.7) with the mixed nonhomogeneous conditions𝑒π‘₯(0,𝑑)βˆ’β„Ž0𝑒(0,𝑑)=𝑔0(𝑑),𝑒(1,𝑑)=𝑔1(𝑑).(1.8)

In the case of 𝑔0,𝑔1∈𝐢3(ℝ+),π‘“βˆˆπΆπ‘+1([0,1]×ℝ+×ℝ3),𝑓1βˆˆπΆπ‘([0,1]×ℝ+×ℝ3), and some other conditions, an asymptotic expansion of the weak solution π‘’πœ€ of order 𝑁+1 in πœ€ 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 𝑒=π‘’πœ€1,πœ€2(π‘₯,𝑑) of order 𝑁+1 in two small parameters πœ€1,πœ€2 for the following equation:π‘’π‘‘π‘‘βˆ’πœ•πœ‡πœ•π‘₯ξ€·ξ€Ί0(π‘₯,𝑑)+πœ€1πœ‡1𝑒(π‘₯,𝑑,𝑒)π‘₯ξ€Έ=𝑓0(π‘₯,𝑑)+πœ€2𝑓1ξ€·π‘₯,𝑑,𝑒,𝑒π‘₯,𝑒𝑑,(1.9) associated with (1.2), (1.3), with πœ‡0∈𝐢2([0,1]×ℝ+),πœ‡1βˆˆπΆπ‘+1([0,1]×ℝ+×ℝ),πœ‡0(π‘₯,𝑑)β‰₯πœ‡βˆ—>0,πœ‡1(π‘₯,𝑑,𝑧)β‰₯0, for all (π‘₯,𝑑,𝑧)∈[0,1]×ℝ+×ℝ,𝑓0∈𝐢1([0,1]×ℝ+), and 𝑓1βˆˆπΆπ‘([0,1]×ℝ+×ℝ3) is established. Next, we note that the same results are valid for the equation in 𝑝 small parameters πœ€1,…,πœ€π‘ as followsπ‘’π‘‘π‘‘βˆ’πœ•πœ‡πœ•π‘₯0(π‘₯,𝑑)+𝑝𝑖=1πœ€π‘–πœ‡π‘–ξƒͺ𝑒(π‘₯,𝑑,𝑒)π‘₯ξƒ­=𝑓0(π‘₯,𝑑)+𝑝𝑖=1πœ€π‘–π‘“π‘–ξ€·π‘₯,𝑑,𝑒,𝑒π‘₯,𝑒𝑑,(1.10) 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 Ξ©=(0,1). Let us omit the definitions of usual function spaces that will be used in what follows such as 𝐿𝑝=𝐿𝑝(Ξ©),π»π‘š=π»π‘š(Ξ©),π»π‘š0=π»π‘š0(Ξ©). The norm in 𝐿2 is denoted by β€–β‹…β€–. We denote by βŸ¨β‹…,β‹…βŸ© the scalar product in 𝐿2 or a pair of dual products of continuous linear functional with an element of a function space. We denote by ‖⋅‖𝑋 the norm of a Banach space 𝑋 and by π‘‹ξ…ž the dual space of 𝑋. We denote 𝐿𝑝(0,𝑇;𝑋),1β‰€π‘β‰€βˆž, the Banach space of real functions π‘’βˆΆ(0,𝑇)→𝑋 measurable, such that ‖𝑒‖𝐿𝑝(0,𝑇;𝑋)<+∞, with ‖𝑒‖𝐿𝑝(0,𝑇;𝑋)=⎧βŽͺ⎨βŽͺβŽ©ξ‚΅ξ€œπ‘‡0‖𝑒(𝑑)‖𝑝𝑋𝑑𝑑1/𝑝,if1≀𝑝<∞,esssup0<𝑑<𝑇‖𝑒(𝑑)‖𝑋,if𝑝=∞.(2.1)

Let 𝑒(𝑑),π‘’ξ…ž(𝑑)=𝑒𝑑(𝑑)=̇𝑒(𝑑),π‘’ξ…žξ…ž(𝑑)=𝑒𝑑𝑑(𝑑)=Μˆπ‘’(𝑑),𝑒π‘₯(𝑑)=βˆ‡π‘’(𝑑),𝑒π‘₯π‘₯(𝑑)=Δ𝑒(𝑑) denote 𝑒(π‘₯,𝑑),πœ•π‘’/πœ•π‘‘(π‘₯,𝑑),πœ•2𝑒/πœ•π‘‘2(π‘₯,𝑑),πœ•π‘’/πœ•π‘₯(π‘₯,𝑑),πœ•2𝑒/πœ•π‘₯2(π‘₯,𝑑), respectively. With π‘“βˆˆπΆπ‘˜([0,1]×ℝ+×ℝ3),𝑓=𝑓(π‘₯,𝑑,𝑒,𝑣,𝑀), we put 𝐷1𝑓=πœ•π‘“/πœ•π‘₯,𝐷2𝑓=πœ•π‘“/πœ•π‘‘,𝐷3𝑓=πœ•π‘“/πœ•π‘’,𝐷4𝑓=πœ•π‘“/πœ•π‘£,𝐷5𝑓=πœ•π‘“/πœ•π‘€ and 𝐷𝛼𝑓=𝐷𝛼11𝐷𝛼22𝐷𝛼33𝐷𝛼44𝐷𝛼55𝑓; 𝛼=(𝛼1,𝛼2,𝛼3,𝛼4,𝛼5)βˆˆβ„€5+, |𝛼|=𝛼1+𝛼2+𝛼3+𝛼4+𝛼5=π‘˜, 𝐷(0,0,…,0)𝑓=𝑓.

Similarly, with πœ‡βˆˆπΆπ‘˜([0,1]×ℝ+×ℝ),πœ‡=πœ‡(π‘₯,𝑑,𝑧), we put 𝐷1πœ‡=πœ•πœ‡/πœ•π‘₯,𝐷2πœ‡=πœ•πœ‡/πœ•π‘‘,𝐷3πœ‡=πœ•πœ‡/πœ•π‘§ and π·π›½πœ‡=𝐷𝛽11𝐷𝛽22𝐷𝛽33,𝛽=(𝛽1,𝛽2,𝛽3)βˆˆβ„€3+,|𝛽|=𝛽1+𝛽2+𝛽3=π‘˜.

On 𝐻1, we will use the following norms:‖𝑣‖𝐻1=‖𝑣‖2+‖𝑣π‘₯β€–2ξ€Έ1/2.(2.2)

Then, we have the following lemma.

Lemma 2.1. The imbedding 𝐻1β†ͺ𝐢0(Ξ©) is compact and ‖𝑣‖𝐢0(Ξ©)β‰€βˆš2‖𝑣‖𝐻1βˆ€π‘£βˆˆπ»1.(2.3)

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

Remark 2.2. On 𝐻10,𝑣↦‖𝑣‖𝐻1 and 𝑣↦‖𝑣π‘₯β€– are two equivalent norms. Furthermore, we have the following inequalities: ‖𝑣‖𝐢0(Ξ©)≀‖𝑣π‘₯β€–βˆ€π‘£βˆˆπ»10.(2.4)

Remark 2.3. (i) Let us note more that a unique weak solution 𝑒 of the problem (1.1)–(1.3) will be obtained in Section 3 (Theorem 3.2) in the following manner.
Find ξ‚‹π‘’βˆˆπ‘Š={π‘’βˆˆπΏβˆž(0,𝑇;𝐻10∩𝐻2)βˆΆπ‘’ξ…žβˆˆπΏβˆž(0,𝑇;𝐻10),π‘’ξ…žξ…žβˆˆπΏβˆž(0,𝑇;𝐿2)} such that 𝑒 verifies the following variational equation: ξ«π‘’ξ…žξ…žξ¬(𝑑),𝑀+βŸ¨πœ‡(β‹…,𝑑,𝑒(𝑑))𝑒π‘₯(𝑑),𝑀π‘₯ξ«π‘“ξ€·βŸ©=β‹…,𝑑,𝑒(𝑑),𝑒π‘₯(𝑑),π‘’ξ…žξ€Έξ¬(𝑑),𝑀,βˆ€π‘€βˆˆπ»10,(2.5) and the initial conditions 𝑒(0)=̃𝑒0,π‘’ξ…ž(0)=̃𝑒1.(2.6)
(ii) With the regularity obtained by ξ‚‹π‘Šπ‘’βˆˆ, it also follows from Theorem 3.2 that the problem (1.1)–(1.3) has a unique strong solution 𝑒 that satisfies π‘’βˆˆπΆ0ξ€·0,𝑇;𝐻1ξ€Έβˆ©πΆ1ξ€·0,𝑇;𝐿2ξ€Έβˆ©πΏβˆžξ€·0,𝑇;𝐻2ξ€Έ,π‘’π‘‘βˆˆπΏβˆžξ€·0,𝑇;𝐻1ξ€Έ,π‘’π‘‘π‘‘βˆˆπΏβˆžξ€·0,𝑇;𝐿2ξ€Έ.(2.7)

On the other hand, by ξ‚‹π‘Šπ‘’βˆˆ, we can see that 𝑒,𝑒π‘₯,𝑒𝑑,𝑒π‘₯π‘₯,𝑒π‘₯𝑑,π‘’π‘‘π‘‘βˆˆπΏβˆž(0,𝑇;𝐿2)βŠ‚πΏ2(𝑄𝑇).

Also, if (𝑒0,𝑒1)∈(𝐻10∩𝐻2)×𝐻10, then the weak solution 𝑒 of the problem (1.1)–(1.3) belongs to 𝐻2(𝑄𝑇). So, the solution is almost classical which is rather natural, since the initial data (𝑒0,𝑒1) do not belong necessarily to 𝐢2(Ξ©)×𝐢1(Ξ©).

3. The Existence and the Uniqueness of a Weak Solution

We make the following assumptions: (𝐻1)̃𝑒0∈𝐻10∩𝐻2,̃𝑒1∈𝐻10, (𝐻2)πœ‡βˆˆπΆ2([0,1]×ℝ+×ℝ),πœ‡(π‘₯,𝑑,𝑧)β‰₯πœ‡βˆ—>0,forall(π‘₯,𝑑,𝑧)∈[0,1]×ℝ+×ℝ, (𝐻3)π‘“βˆˆπΆ1(Ω×ℝ+×ℝ3).

With πœ‡ and 𝑓 satisfying the assumptions (𝐻2) and (𝐻3), respectively, for each π‘‡βˆ—>0 and 𝑀>0 are given, we put the following constants:𝐾𝑀(πœ‡)=β€–πœ‡β€–πΆ2(ξ‚π·βˆ—π‘€),𝐾(3.1)𝑀(𝑓)=‖𝑓‖𝐢1(π·βˆ—π‘€),(3.2) where ξ‚π·βˆ—π‘€={(π‘₯,𝑑,𝑧)∢0≀π‘₯≀1,0β‰€π‘‘β‰€π‘‡βˆ—,|𝑧|≀𝑀} and π·βˆ—π‘€={(π‘₯,𝑑,𝑒,𝑣,𝑀)βˆˆβ„+×ℝ+×ℝ3∢0≀π‘₯≀1,0β‰€π‘‘β‰€π‘‡βˆ—,|𝑒|,|𝑣|,|𝑀|≀𝑀}.

For each π‘‡βˆˆ(0,π‘‡βˆ—] and 𝑀>0, we getξ€½π‘Š(𝑀,𝑇)=π‘£βˆˆπΏβˆžξ€·0,𝑇;𝐻10∩𝐻2ξ€ΈβˆΆπ‘£π‘‘βˆˆπΏβˆžξ€·0,𝑇;𝐻10ξ€Έ,π‘£π‘‘π‘‘βˆˆπΏ2𝑄𝑇,withβ€–π‘£β€–πΏβˆž(0,𝑇;𝐻10∩𝐻2),β€–π‘£π‘‘β€–πΏβˆž(0,𝑇;𝐻10),‖𝑣𝑑𝑑‖𝐿2(𝑄𝑇),π‘Šβ‰€π‘€(3.3)1ξ€½(𝑀,𝑇)=π‘£βˆˆπ‘Š(𝑀,𝑇)βˆΆπ‘£π‘‘π‘‘βˆˆπΏβˆžξ€·0,𝑇;𝐿2,ξ€Έξ€Ύ(3.4) where 𝑄𝑇=Ω×(0,𝑇).

We choose the first term 𝑒0≑̃𝑒0βˆˆπ‘Š1(𝑀,𝑇). Suppose thatπ‘’π‘šβˆ’1βˆˆπ‘Š1(𝑀,𝑇),π‘šβ‰₯1.(3.5)

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

Find π‘’π‘šβˆˆπ‘Š1(𝑀,𝑇) such thatξ«π‘’π‘šξ…žξ…žξ¬(𝑑),𝑣+βŸ¨πœ‡π‘š(𝑑)βˆ‡π‘’π‘š(𝑑),βˆ‡π‘£βŸ©=βŸ¨πΉπ‘š(𝑑),π‘£βŸ©,βˆ€π‘£βˆˆπ»10,𝑒(3.6)π‘š(0)=̃𝑒0,π‘’ξ…žπ‘š(0)=̃𝑒1,(3.7) whereπœ‡π‘šξ€·(π‘₯,𝑑)=πœ‡π‘₯,𝑑,π‘’π‘šβˆ’1ξ€Έ(𝑑),πΉπ‘šξ€·(π‘₯,𝑑)=𝑓π‘₯,𝑑,π‘’π‘šβˆ’1(π‘₯,𝑑),βˆ‡π‘’π‘šβˆ’1(π‘₯,𝑑),π‘’ξ…žπ‘šβˆ’1ξ€Έ(π‘₯,𝑑).(3.8)

Then, we have the following theorem.

Theorem 3.1. Let ( 𝐻1)–( 𝐻3) hold. Then, there exist two constants 𝑀>0,𝑇>0 and the linear recurrent sequence {π‘’π‘š}βŠ‚π‘Š1(𝑀,𝑇) 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 {𝑀𝑗} on 𝐻10βˆΆπ‘€π‘—βˆš(π‘₯)=2sin(π‘—πœ‹π‘₯),π‘—βˆˆβ„•, formed by the eigenfunctions of the Laplacian βˆ’Ξ”=βˆ’πœ•2/πœ•π‘₯2. Put π‘’π‘š(π‘˜)(𝑑)=π‘˜ξ“π‘—=1𝑐(π‘˜)π‘šπ‘—(𝑑)𝑀𝑗,(3.9) where the coefficients 𝑐(π‘˜)π‘šπ‘— satisfy the system of linear differential equations ξ‚¬Μˆπ‘’π‘š(π‘˜)(𝑑),𝑀𝑗+ξ‚¬πœ‡π‘š(𝑑)βˆ‡π‘’π‘š(π‘˜)(𝑑),βˆ‡π‘€π‘—ξ‚­=ξ«πΉπ‘š(𝑑),𝑀𝑗𝑒,1β‰€π‘—β‰€π‘˜,(3.10)π‘š(π‘˜)(0)=̃𝑒0π‘˜,Μ‡π‘’π‘š(π‘˜)(0)=̃𝑒1π‘˜,(3.11) where ̃𝑒0π‘˜=π‘˜ξ“π‘—=1𝛼𝑗(π‘˜)π‘€π‘—βŸΆΜƒπ‘’0stronglyin𝐻10∩𝐻2,̃𝑒1π‘˜=π‘˜ξ“π‘—=1𝛽𝑗(π‘˜)π‘€π‘—βŸΆΜƒπ‘’1stronglyin𝐻10.(3.12)
Note that by (3.5), it is not difficult to prove that the system (3.10), (3.11) has a unique solution π‘’π‘š(π‘˜)(𝑑) on interval [0,𝑇], so let us omit the details.
Step 2. A priori estimates. At first, put π‘ π‘š(π‘˜)(𝑑)=π‘π‘š(π‘˜)(𝑑)+π‘žπ‘š(π‘˜)(ξ€œπ‘‘)+𝑑0β€–β€–Μˆπ‘’π‘š(π‘˜)(‖‖𝑠)2𝑝𝑑𝑠,π‘š(π‘˜)β€–β€–(𝑑)=Μ‡π‘’π‘š(π‘˜)β€–β€–(𝑑)2+β€–β€–βˆšπœ‡π‘š(𝑑)βˆ‡π‘’π‘š(π‘˜)β€–β€–(𝑑)2,π‘žπ‘š(π‘˜)β€–β€–(𝑑)=βˆ‡Μ‡π‘’π‘š(π‘˜)β€–β€–(𝑑)2+β€–β€–βˆšπœ‡π‘š(𝑑)Ξ”π‘’π‘š(π‘˜)β€–β€–(𝑑)2.(3.13)
Then, it follows from (3.9)–(3.11), (3.13) that π‘ π‘š(π‘˜)(𝑑)=π‘ π‘š(π‘˜)(0)+2βŸ¨βˆ‡πœ‡π‘š(0)βˆ‡Μƒπ‘’0π‘˜,Δ̃𝑒0π‘˜βŸ©+2βŸ¨πΉπ‘š(0),Δ̃𝑒0π‘˜βŸ©+ξ€œπ‘‘0ξ€œπ‘‘π‘ 10πœ‡ξ…žπ‘šξ‚€||(π‘₯,𝑠)βˆ‡π‘’π‘š(π‘˜)||(π‘₯,𝑠)2+||Ξ”π‘’π‘š(π‘˜)||(π‘₯,𝑠)2ξ‚ξ€œπ‘‘π‘₯+2𝑑0ξ‚¬πΉπ‘š(𝑠),Μ‡π‘’π‘š(π‘˜)ξ‚­ξ€œ(𝑠)𝑑𝑠+2𝑑0ξ‚¬πœ•ξ‚€πœ•π‘ βˆ‡πœ‡π‘š(𝑠)βˆ‡π‘’π‘š(π‘˜)(𝑠),Ξ”π‘’π‘š(π‘˜)(𝑠)π‘‘π‘ βˆ’2βˆ‡πœ‡π‘š(𝑑)βˆ‡π‘’π‘š(π‘˜)(𝑑),Ξ”π‘’π‘š(π‘˜)𝐹(𝑑)βˆ’2π‘š(𝑑),Ξ”π‘’π‘š(π‘˜)ξ‚­ξ€œ(𝑑)+2𝑑0ξƒ‘πœ•πΉπ‘šπœ•π‘‘(𝑠),Ξ”π‘’π‘š(π‘˜)ξƒ’ξ€œ(𝑠)𝑑𝑠+𝑑0β€–β€–Μˆπ‘’π‘š(π‘˜)β€–β€–(𝑠)2𝑑𝑠=π‘žπ‘š(π‘˜)(0)+2βŸ¨βˆ‡πœ‡π‘š(0)βˆ‡Μƒπ‘’0π‘˜,Δ̃𝑒0π‘˜βŸ©+2βŸ¨πΉπ‘š(0),Δ̃𝑒0π‘˜βŸ©+7𝑗=1𝐼𝑗.(3.14)
Next, we will estimate the terms 𝐼𝑗,𝑗=1,2,…,7 on the right-hand side of (3.14) as follows.
First Term 𝐼1
We have πœ‡ξ…žπ‘š(𝑑)=𝐷2πœ‡ξ€·π‘₯,𝑑,π‘’π‘šβˆ’1ξ€Έ(𝑑)+𝐷3πœ‡ξ€·π‘₯,𝑑,π‘’π‘šβˆ’1𝑒(𝑑)ξ…žπ‘šβˆ’1(𝑑).(3.15)

From (3.1), (3.5), and (3.8), we have ||πœ‡ξ…žπ‘š||≀𝐾(π‘₯,𝑑)(1+𝑀)𝑀(πœ‡).(3.16)
Hence, 𝐼1=ξ€œπ‘‘0ξ€œπ‘‘π‘ 10πœ‡ξ…žπ‘šξ‚€||(π‘₯,𝑠)βˆ‡π‘’π‘š(π‘˜)||(π‘₯,𝑠)2+||Ξ”π‘’π‘š(π‘˜)||(π‘₯,𝑠)2𝑑π‘₯≀1+π‘€πœ‡βˆ—ξ‚πΎπ‘€ξ€œ(πœ‡)𝑑0π‘ π‘š(π‘˜)(𝑠)𝑑𝑠.(3.17)
Second Term
By using (𝐻3), we obtain from (3.2), (3.5), and (3.13)2 that 𝐼2ξ€œ=2𝑑0ξ‚¬πΉπ‘š(𝑠),Μ‡π‘’π‘š(π‘˜)(𝑠)𝑑𝑠≀𝑇𝐾2𝑀(ξ€œπ‘“)+𝑑0π‘π‘š(π‘˜)(𝑠)𝑑𝑠.(3.18)
Third Term
The Cauchy-Schwartz inequality yields ||𝐼3||||||ξ€œ=2𝑑0ξ‚¬πœ•ξ‚€πœ•π‘ βˆ‡πœ‡π‘š(𝑠)βˆ‡π‘’π‘š(π‘˜)(𝑠),Ξ”π‘’π‘š(π‘˜)ξ‚­||||≀2(𝑠)π‘‘π‘ βˆšπœ‡βˆ—ξ€œπ‘‘0π‘Ÿπ‘š(π‘˜)(𝑠)π‘žπ‘š(π‘˜)(𝑠)𝑑𝑠,(3.19) where π‘Ÿπ‘š(π‘˜)(𝑠)=β€–πœ•/πœ•π‘ (βˆ‡πœ‡π‘š(𝑠)βˆ‡π‘’π‘š(π‘˜)(𝑠))β€–.

We note π‘Ÿπ‘š(π‘˜)β€–β€–β€–(𝑠)=βˆ‡πœ‡π‘š(𝑠)βˆ‡Μ‡π‘’π‘š(π‘˜)πœ•(𝑠)+ξ€·πœ•π‘ βˆ‡πœ‡π‘šξ€Έ(𝑠)βˆ‡π‘’π‘š(π‘˜)‖‖‖≀‖‖(𝑠)βˆ‡πœ‡π‘šβ€–β€–(𝑠)𝐢0(Ξ©)+1βˆšπœ‡βˆ—β€–β€–β€–πœ•πœ•π‘ βˆ‡πœ‡π‘šβ€–β€–β€–ξƒͺ(𝑠)π‘ π‘š(π‘˜)(𝑠).(3.20)
On the other hand, by βˆ‡πœ‡π‘š(π‘₯,𝑠)=𝐷1πœ‡(π‘₯,𝑠,π‘’π‘šβˆ’1(π‘₯,𝑠))+𝐷3πœ‡(π‘₯,𝑠,π‘’π‘šβˆ’1(π‘₯,𝑠))βˆ‡π‘’π‘šβˆ’1(π‘₯,𝑠), it is implies that β€–β€–βˆ‡πœ‡π‘šβ€–β€–(𝑠)𝐢0(Ξ©)≀𝐾𝑀‖‖(πœ‡)1+βˆ‡π‘’π‘šβˆ’1β€–β€–(𝑠)𝐢0(Ξ©)𝐾≀2(1+𝑀)𝑀(πœ‡).(3.21)
Similarly, the following equality πœ•πœ•π‘ βˆ‡πœ‡π‘š(π‘₯,𝑠)=𝐷1𝐷1πœ‡ξ€·π‘₯,𝑠,π‘’π‘šβˆ’1ξ€Έ(π‘₯,𝑠)+𝐷3𝐷1πœ‡ξ€·π‘₯,𝑠,π‘’π‘šβˆ’1𝑒(π‘₯,𝑠)ξ…žπ‘šβˆ’1+𝐷(π‘₯,𝑠)1𝐷3πœ‡ξ€·π‘₯,𝑠,π‘’π‘šβˆ’1ξ€Έ(π‘₯,𝑠)+𝐷3𝐷3πœ‡ξ€·π‘₯,𝑠,π‘’π‘šβˆ’1𝑒(π‘₯,𝑠)ξ…žπ‘šβˆ’1ξ€»(π‘₯,𝑠)βˆ‡π‘’π‘šβˆ’1(π‘₯,𝑠)+𝐷3πœ‡ξ€·π‘₯,𝑠,π‘’π‘šβˆ’1(ξ€Έπ‘₯,𝑠)βˆ‡π‘’ξ…žπ‘šβˆ’1(π‘₯,𝑠)(3.22) gives β€–β€–β€–πœ•πœ•π‘ βˆ‡πœ‡π‘šβ€–β€–β€–β‰€ξ€·(𝑠)1+3𝑀+𝑀2𝐾𝑀(πœ‡).(3.23)
It follows from (3.20)–(3.23) that π‘Ÿπ‘š(π‘˜)(𝑠)≀2(1+𝑀)+1+3𝑀+𝑀2βˆšπœ‡βˆ—ξƒ­ξ‚πΎπ‘€ξ”(πœ‡)π‘ π‘š(π‘˜)(𝑠).(3.24)
Hence, we obtain from (3.19) and (3.24) that ||𝐼3||≀2βˆšπœ‡βˆ—ξƒ¬2(1+𝑀)+1+3𝑀+𝑀2βˆšπœ‡βˆ—ξƒ­ξ‚πΎπ‘€ξ€œ(πœ‡)𝑑0π‘ π‘š(π‘˜)(𝑠)𝑑𝑠.(3.25)
Fourth Term 𝐼4
By the Cauchy-Schwartz inequality, we have ||𝐼4||=|||ξ‚¬βˆ’2βˆ‡πœ‡π‘š(𝑑)βˆ‡π‘’π‘š(π‘˜)(𝑑),Ξ”π‘’π‘š(π‘˜)ξ‚­|||≀1(𝑑)π›½β€–β€–βˆ‡πœ‡π‘š(𝑑)βˆ‡π‘’π‘š(π‘˜)β€–β€–(𝑑)2β€–β€–+π›½Ξ”π‘’π‘š(π‘˜)β€–β€–(𝑑)2,(3.26) for all 𝛽>0. On the other hand β€–β€–βˆ‡πœ‡π‘š(𝑑)βˆ‡π‘’π‘š(π‘˜)(β€–β€–=‖‖‖𝑑)βˆ‡πœ‡π‘š(0)βˆ‡Μƒπ‘’0π‘˜+ξ€œπ‘‘0πœ•ξ‚€πœ•π‘ βˆ‡πœ‡π‘š(𝑠)βˆ‡π‘’π‘š(π‘˜)(‖‖‖≀‖‖𝑠)π‘‘π‘ βˆ‡πœ‡π‘šβ€–β€–(0)𝐢0(Ξ©)β€–β€–βˆ‡Μƒπ‘’0π‘˜β€–β€–+ξ€œπ‘‘0π‘Ÿπ‘š(π‘˜)(𝑠)𝑑𝑠.(3.27)

Hence, we obtain from (3.26), (3.27) that ||𝐼4||β‰€π›½πœ‡βˆ—π‘žπ‘š(π‘˜)2(𝑑)+π›½β€–βˆ‡πœ‡π‘š(0)β€–2𝐢0ξ‚€Ξ©ξ‚β€–βˆ‡Μƒπ‘’0π‘˜β€–2+2𝛽𝑇2(1+𝑀)+1+3𝑀+𝑀2βˆšπœ‡βˆ—ξƒ­2𝐾2π‘€ξ€œ(πœ‡)𝑑0π‘ π‘š(π‘˜)(𝑠)𝑑𝑠,(3.28) for all 𝛽>0.
Fifth Term 𝐼5
By (3.5), (3.8), and (3.13), we obtain ||𝐼5||=|||ξ‚¬πΉβˆ’2π‘š(𝑑),Ξ”π‘’π‘š(π‘˜)ξ‚­|||≀1(𝑑)π›½β€–β€–πΉπ‘šβ€–β€–(𝑑)2β€–β€–+π›½Ξ”π‘’π‘š(π‘˜)β€–β€–(𝑑)2≀2π›½β€–β€–πΉπ‘šβ€–β€–(0)2+2π›½π‘‡ξ€œπ‘‡0β€–β€–β€–πœ•πΉπ‘šβ€–β€–β€–πœ•π‘ (𝑠)2𝛽𝑑𝑠+πœ‡βˆ—π‘ π‘š(π‘˜)(𝑑),βˆ€π›½>0.(3.29)

Note that πœ•πΉπ‘šπœ•π‘‘(𝑑)=𝐷2π‘“ξ€Ίπ‘’π‘šβˆ’1ξ€»+𝐷3π‘“ξ€Ίπ‘’π‘šβˆ’1ξ€»π‘’ξ…žπ‘šβˆ’1(𝑑)+𝐷4π‘“ξ€Ίπ‘’π‘šβˆ’1ξ€»βˆ‡π‘’ξ…žπ‘šβˆ’1(𝑑)+𝐷5π‘“ξ€Ίπ‘’π‘šβˆ’1ξ€»π‘’ξ…žξ…žπ‘šβˆ’1(𝑑),(3.30) where we use the notation 𝐷𝑖𝑓[π‘’π‘šβˆ’1]=𝐷𝑖𝑓(π‘₯,𝑑,π‘’π‘šβˆ’1(π‘₯,𝑑),βˆ‡π‘’π‘šβˆ’1(π‘₯,𝑑),π‘’ξ…žπ‘šβˆ’1(π‘₯,𝑑)),𝑖=2,…,5. By (3.2), (3.5), and (3.30), we obtain β€–β€–β€–πœ•πΉπ‘šβ€–β€–β€–πœ•π‘‘(𝑑)≀𝐾𝑀‖‖𝑒(𝑓)1+2𝑀+ξ…žξ…žπ‘šβˆ’1β€–β€–ξ€Έ.(𝑑)(3.31)
Hence, we deduce from (3.29) and (3.31) that ||𝐼5||≀2π›½β€–πΉπ‘š(0)β€–2+4𝛽𝑇𝐾2𝑀(𝑓)(1+2𝑀)2𝑇+𝑀2ξ€»+π›½πœ‡βˆ—π‘ π‘š(π‘˜)(𝑑),βˆ€π›½>0.(3.32)
Sixth Term 𝐼6
By (3.2), (3.5), (3.13)3, and (3.31), we get ||𝐼6||||||ξ€œ=2𝑑0ξƒ‘πœ•πΉπ‘šπœ•π‘‘(𝑠),Ξ”π‘’π‘š(π‘˜)ξƒ’||||β‰€ξ€œ(𝑠)𝑑𝑠𝑑0β€–β€–β€–πœ•πΉπ‘šβ€–β€–β€–ξ€œπœ•π‘‘(𝑠)𝑑𝑠+𝑑0β€–β€–β€–πœ•πΉπ‘šβ€–β€–β€–β€–β€–πœ•π‘‘(𝑠)Ξ”π‘’π‘š(π‘˜)β€–β€–(𝑠)2π‘‘π‘ β‰€πΎπ‘€ξƒ¬βˆš(𝑓)(1+2𝑀)𝑇+π‘‡ξ‚΅ξ€œπ‘‡0β€–β€–π‘’ξ…žξ…žπ‘šβˆ’1β€–β€–(𝑠)2𝑑𝑠1/2ξƒ­+1πœ‡βˆ—πΎπ‘€ξ€œ(𝑓)𝑑0‖‖𝑒1+2𝑀+ξ…žξ…žπ‘šβˆ’1β€–β€–ξ€Έπ‘ž(𝑠)π‘š(π‘˜)(𝑠)π‘‘π‘ β‰€πΎπ‘€ξ‚ƒβˆš(𝑓)(1+2𝑀)𝑇+ξ‚„+1π‘‡π‘€πœ‡βˆ—πΎπ‘€ξ€œ(𝑓)𝑑0‖‖𝑒1+2𝑀+ξ…žξ…žπ‘šβˆ’1β€–β€–ξ€Έπ‘ž(𝑠)π‘š(π‘˜)(𝑠)𝑑𝑠.(3.33)
Seventh Term 𝐼7
Equation (3.10) is rewritten as follows: ξ‚¬Μˆπ‘’π‘š(π‘˜)(𝑑),π‘€π‘—ξ‚­βˆ’ξ‚¬πœ•ξ‚€πœ‡πœ•π‘₯π‘š(𝑑)βˆ‡π‘’π‘š(π‘˜)(𝑑),𝑀𝑗=ξ«πΉπ‘š(𝑑),𝑀𝑗,1β‰€π‘—β‰€π‘˜.(3.34)

Hence, by replacing 𝑀𝑗 with Μˆπ‘’π‘š(π‘˜)(𝑑) and integrating 𝐼7=ξ€œπ‘‘0β€–β€–Μˆπ‘’π‘š(π‘˜)(‖‖𝑠)2ξ€œπ‘‘π‘ β‰€2𝑑0β€–β€–β€–πœ•ξ‚€πœ‡πœ•π‘₯π‘š(𝑠)βˆ‡π‘’π‘š(π‘˜)(‖‖‖𝑠)2ξ€œπ‘‘π‘ +2𝑑0β€–β€–πΉπ‘š(‖‖𝑠)2ξ€œπ‘‘π‘ β‰€2𝑑0β€–β€–β€–πœ•ξ‚€πœ‡πœ•π‘₯π‘š(𝑠)βˆ‡π‘’π‘š(π‘˜)‖‖‖(𝑠)2𝑑𝑠+2𝑇𝐾2𝑀(𝑓),(3.35) we need, estimate β€–πœ•/πœ•π‘₯(πœ‡π‘š(𝑠)βˆ‡π‘£π‘š(π‘˜)(𝑠))β€–.
Combining (3.1), (3.5), and (3.13) yields β€–β€–β€–πœ•ξ‚€πœ‡πœ•π‘₯π‘š(𝑠)βˆ‡π‘’π‘š(π‘˜)‖‖‖=β€–β€–(𝑠)βˆ‡πœ‡π‘š(𝑠)βˆ‡π‘’π‘š(π‘˜)(𝑠)+πœ‡π‘š(𝑠)Ξ”π‘’π‘š(π‘˜)‖‖≀‖‖(𝑠)βˆ‡πœ‡π‘šβ€–β€–(𝑠)𝐢0(Ξ©)β€–β€–βˆ‡π‘’π‘š(π‘˜)β€–β€–+β€–β€–πœ‡(𝑠)π‘šβ€–β€–(𝑠)𝐢0(Ξ©)β€–β€–Ξ”π‘’π‘š(π‘˜)‖‖≀2(𝑠)βˆšπœ‡βˆ—ξ‚πΎ(1+𝑀)𝑀(πœ‡)π‘π‘š(π‘˜)1(𝑠)+βˆšπœ‡βˆ—ξ‚πΎπ‘€ξ”(πœ‡)π‘žπ‘š(π‘˜)≀3(𝑠)βˆšπœ‡βˆ—ξ‚πΎ(1+𝑀)𝑀(πœ‡)π‘ π‘š(π‘˜)(𝑠).(3.36)
Therefore, from (3.35) and (3.36), we obtain 𝐼7≀2𝑇𝐾2𝑀(𝑓)+18πœ‡βˆ—(1+𝑀)2𝐾2𝑀(ξ€œπœ‡)𝑑0π‘ π‘š(π‘˜)(𝑠)𝑑𝑠.(3.37)
Choosing 𝛽>0, with 2𝛽/πœ‡βˆ—β‰€1/2, it follows from (3.13), (3.14), (3.17), (3.18), (3.25), (3.28), (3.32), (3.33), and (3.37) that π‘ π‘š(π‘˜)(𝐢𝑑)≀0π‘˜+𝐢1(ξ€œπ‘€,𝑇)+𝑑0𝐢2(2𝑀,𝑇)+πœ‡βˆ—πΎπ‘€(‖‖𝑒𝑓)ξ…žξ…žπ‘šβˆ’1(‖‖𝑠𝑠)π‘š(π‘˜)(𝑠)𝑑𝑠,(3.38) where 𝐢0π‘˜=𝐢0π‘˜ξ€·π›½,𝑓,πœ‡,̃𝑒0,̃𝑒1,̃𝑒0π‘˜,̃𝑒1π‘˜ξ€Έ=2π‘ π‘š(π‘˜)(0)+4βŸ¨βˆ‡πœ‡π‘š(0)βˆ‡Μƒπ‘’0π‘˜,Δ̃𝑒0π‘˜βŸ©+4βŸ¨πΉπ‘š(0),Δ̃𝑒0π‘˜βŸ©+4π›½β€–βˆ‡πœ‡π‘š(0)β€–2𝐢0ξ‚€Ξ©ξ‚β€–βˆ‡Μƒπ‘’0π‘˜β€–2+4π›½β€–πΉπ‘š(0)β€–2,𝐢1𝐢(𝑀,𝑇)=1ξ‚΅4(𝛽,𝑓,𝑀,𝑇)=23+𝛽(1+2𝑀)2𝑇+𝑀2𝑇𝐾2𝑀(ξ‚ƒβˆšπ‘“)+2𝑀+(1+2𝑀)π‘‡ξ‚„βˆšπ‘‡πΎπ‘€(𝐢𝑓),2𝐢(𝑀,𝑇)=22(𝛽,𝑓,πœ‡,𝑀,𝑇)=2+πœ‡0(1+2𝑀)𝐾𝑀+2(𝑓)πœ‡βˆ—βˆšξ€Ίξ€·1+4πœ‡βˆ—ξ€Έξ€·(1+𝑀)+21+3𝑀+𝑀2𝐾𝑀+4(πœ‡)πœ‡βˆ—ξ‚Έ1π›½π‘‡ξ€·βˆš2(1+𝑀)πœ‡βˆ—+1+3𝑀+𝑀2ξ€Έ2+9(1+𝑀)2𝐾2𝑀(πœ‡).(3.39)
By (𝐻1), we deduce from (3.12), (3.39)1 that there exists 𝑀>0 independent of π‘š and π‘˜, such that 𝐢0π‘˜β‰€12𝑀2.(3.40)
Notice that by (𝐻3), we deduce from (3.39)2,3 that lim𝑇→0+𝐢1(𝑀,𝑇)=lim𝑇→0+𝑇𝐢2(𝑀,𝑇)=0.(3.41)
So, from (3.39) and (3.41), we can choose 𝑇>0 such that ξ‚€12𝑀2+𝐢1𝑇𝐢(𝑀,𝑇)exp22(𝑀,𝑇)+πœ‡0πΎπ‘€βˆš(𝑓)𝑇𝑀≀𝑀2π‘˜,(3.42)𝑇=11+βˆšπœ‡βˆ—ξƒͺβˆšπ‘‡ξ”4𝐾2𝑀(𝑓)+(4+𝑀)2𝑀2𝐾2𝑀(πœ‡)𝑒𝑇[1+((1+𝑀)/2πœ‡βˆ—)𝐾𝑀(πœ‡)]<1.(3.43)
Finally, it follows from (3.38), (3.40), and (3.42) that π‘ π‘š(π‘˜)(𝑑)≀𝑀2𝐢expβˆ’π‘‡22(𝑀,𝑇)βˆ’πœ‡0πΎπ‘€βˆš(𝑓)ξ‚Ά+ξ€œπ‘‡π‘€π‘‘0𝐢22(𝑀,𝑇)+πœ‡0𝐾𝑀‖‖𝑒(𝑓)ξ…žξ…žπ‘šβˆ’1‖‖𝑠(𝑠)π‘š(π‘˜)(𝑠)𝑑𝑠.(3.44)
By using Gronwall's lemma, we deduce from (3.44) that π‘ π‘š(π‘˜)(𝑑)≀𝑀2𝐢expβˆ’π‘‡22(𝑀,𝑇)βˆ’πœ‡0πΎπ‘€βˆš(𝑓)ξ‚Άξ‚Έξ€œπ‘‡π‘€Γ—exp𝑇0𝐢22(𝑀,𝑇)+πœ‡0𝐾𝑀‖‖𝑒(𝑓)ξ…žξ…žπ‘šβˆ’1β€–β€–ξ‚Άξ‚Ή(𝑠)𝑑𝑠≀𝑀2𝐢expβˆ’π‘‡22(𝑀,𝑇)βˆ’πœ‡0πΎπ‘€βˆš(𝑓)𝑇𝐢𝑇𝑀×exp22(𝑀,𝑇)+πœ‡0πΎπ‘€βˆš(𝑓)π‘‡β€–β€–π‘’ξ…žξ…žπ‘šβˆ’1‖‖𝐿2(𝑄𝑇)≀𝑀2.(3.45)
Therefore, we have π‘’π‘š(π‘˜)βˆˆπ‘Š(𝑀,𝑇),βˆ€π‘š,π‘˜βˆˆβ„•.(3.46)
Step 3. Limiting process.
From (3.46), we can extract from {π‘’π‘š(π‘˜)} a subsequence still denoted by {π‘’π‘š(π‘˜)} such that π‘’π‘š(π‘˜)βŸΆπ‘’π‘šinπΏβˆžξ€·0,𝑇;𝐻10∩𝐻2ξ€Έweakβˆ—,Μ‡π‘’π‘š(π‘˜)βŸΆπ‘’ξ…žπ‘šinπΏβˆžξ€·0,𝑇;𝐻10ξ€Έweakβˆ—,Μˆπ‘’π‘š(π‘˜)βŸΆπ‘’π‘šξ…žξ…žin𝐿2𝑄𝑇weak,(3.47) as π‘˜β†’βˆž, and π‘’π‘šβˆˆπ‘Š(𝑀,𝑇).(3.48)
Based on (3.47), passing to limit in (3.10), (3.11) as π‘˜β†’βˆž, we have π‘’π‘š satisfying (3.6)–(3.8). On the other hand, it follows from (3.5), (3.6), and (3.47) that π‘’π‘šξ…žξ…ž=βˆ‡πœ‡π‘šβˆ‡π‘’π‘š+πœ‡π‘šΞ”π‘’π‘šξ€·+𝑓π‘₯,𝑑,π‘’π‘šβˆ’1,βˆ‡π‘’π‘šβˆ’1,π‘’ξ…žπ‘šβˆ’1ξ€ΈβˆˆπΏβˆžξ€·0,𝑇;𝐿2ξ€Έ.(3.49)
Hence, π‘’π‘šβˆˆπ‘Š1(𝑀,𝑇), and the proof of Theorem 3.1 is complete.

Theorem 3.2. Let ( 𝐻1)–( 𝐻3) hold. Then, there exist 𝑀>0 and 𝑇>0 satisfying (3.40), (3.42), and (3.43) such that the problem (1.1)–(1.3) has a unique weak solution π‘’βˆˆπ‘Š1(𝑀,𝑇).
Furthermore, the linear recurrent sequence {π‘’π‘š} defined by (3.6)–(3.8) converges to the solution 𝑒 strongly in the space π‘Š1ξ€½(𝑇)=π‘€βˆˆπΏβˆžξ€·0,𝑇;𝐻10ξ€ΈβˆΆπ‘€ξ…žβˆˆπΏβˆžξ€·0,𝑇;𝐿2,ξ€Έξ€Ύ(3.50) with the following estimation: β€–β€–π‘’π‘šβ€–β€–βˆ’π‘’πΏβˆž(0,𝑇;𝐻10)+β€–β€–π‘’ξ…žπ‘šβˆ’π‘’ξ…žβ€–β€–πΏβˆž(0,𝑇;𝐿2)β‰€πΆπ‘˜π‘šπ‘‡,βˆ€π‘šβˆˆβ„•,(3.51) where π‘˜π‘‡<1 as in (3.43) and 𝐢 is a constant depending only on 𝑇,̃𝑒0,̃𝑒1 and π‘˜π‘‡.

Proof. (i) The existence. First, we note that π‘Š1(𝑇) is a Banach space with respect to the norm (see Lions [15]) β€–π‘€β€–π‘Š1(𝑇)=β€–π‘€β€–πΏβˆž(0,𝑇;𝐻10)+β€–β€–π‘€ξ…žβ€–β€–πΏβˆž(0,𝑇;𝐿2).(3.52)
Next, we prove that {π‘’π‘š} is a Cauchy sequence in π‘Š1(𝑇). Let π‘£π‘š=π‘’π‘š+1βˆ’π‘’π‘š. Then, π‘£π‘š satisfies the variational problem ξ«π‘£π‘šξ…žξ…žξ¬+ξ«πœ‡(𝑑),π‘€π‘š+1(𝑑)βˆ‡π‘£π‘šξ¬=ξ‚¬πœ•(𝑑),βˆ‡π‘€πœ‡πœ•π‘₯ξ€Ίξ€·π‘š+1(𝑑)βˆ’πœ‡π‘šξ€Έ(𝑑)βˆ‡π‘’π‘šξ€»ξ‚­+𝐹(𝑑),π‘€π‘š+1(𝑑)βˆ’πΉπ‘šξ¬(𝑑),𝑀,βˆ€π‘€βˆˆπ»10,π‘£π‘š(0)=π‘£ξ…žπ‘š(0)=0.(3.53)
Taking 𝑀=π‘£ξ…žπ‘š in (3.53)1, after integrating in 𝑑, we get π‘π‘šξ€œ(𝑑)=𝑑0ξ€œπ‘‘π‘ 10πœ‡ξ…žπ‘š+1||(π‘₯,𝑠)βˆ‡π‘£π‘š||(𝑠)2ξ€œπ‘‘π‘₯+2𝑑0ξ«πΉπ‘š+1(𝑠)βˆ’πΉπ‘š(𝑠),π‘£ξ…žπ‘šξ¬ξ€œ(𝑠)𝑑𝑠+2𝑑0ξ‚¬πœ•πœ‡πœ•π‘₯ξ€Ίξ€·π‘š+1(𝑠)βˆ’πœ‡π‘šξ€Έ(𝑠)βˆ‡π‘’π‘šξ€»(𝑠),π‘£ξ…žπ‘šξ‚­(𝑠)𝑑𝑠=3𝑖=1𝐽𝑖,(3.54) in which π‘π‘šβ€–β€–π‘£(𝑑)=ξ…žπ‘šβ€–β€–(𝑑)2+β€–β€–βˆšπœ‡π‘š+1(𝑑)βˆ‡π‘£π‘šβ€–β€–(𝑑)2,(3.55) and all integrals on the right-hand side of (3.54) are estimated as follows.
First Integral
By (3.16), we obtain ||𝐽1||≀||||ξ€œπ‘‘0ξ€œπ‘‘π‘ 10πœ‡ξ…žπ‘š+1||(π‘₯,𝑠)βˆ‡π‘£π‘š||(𝑠)2||||≀𝑑π‘₯1+π‘€πœ‡βˆ—ξ‚πΎπ‘€ξ€œ(πœ‡)𝑑0π‘π‘š(𝑠)𝑑𝑠.(3.56)
Second Integral
By (𝐻3), β€–β€–πΉπ‘š+1(𝑑)βˆ’πΉπ‘šβ€–β€–(𝑑)≀2𝐾𝑀‖‖(𝑓)βˆ‡π‘£π‘šβˆ’1β€–β€–+‖‖𝑣(𝑑)ξ…žπ‘šβˆ’1β€–β€–ξ€»(𝑑)≀2𝐾𝑀‖‖𝑣(𝑓)π‘šβˆ’1β€–β€–π‘Š1(𝑇),(3.57) so ||𝐽2||||||ξ€œβ‰€2𝑑0ξ«πΉπ‘š+1(𝑠)βˆ’πΉπ‘š(𝑠),π‘£ξ…žπ‘šξ¬||||(𝑠)𝑑𝑠≀4𝐾𝑀‖‖𝑣(𝑓)π‘šβˆ’1β€–β€–π‘Š1(𝑇)ξ€œπ‘‘0β€–β€–π‘£ξ…žπ‘šβ€–β€–(𝑠)𝑑𝑠≀4𝑇𝐾2𝑀‖‖𝑣(𝑓)π‘šβˆ’1β€–β€–2π‘Š1(𝑇)+ξ€œπ‘‘0π‘π‘š(𝑠)𝑑𝑠.(3.58)
Third Integral
Using (𝐻2) again, we get ||𝐽3||||||ξ€œ=2𝑑0ξ‚¬πœ•πœ‡πœ•π‘₯ξ€Ίξ€·π‘š+1(𝑠)βˆ’πœ‡π‘šξ€Έ(𝑠)βˆ‡π‘’π‘šξ€»(𝑠),π‘£ξ…žπ‘šξ‚­||||β‰€ξ€œ(𝑠)𝑑𝑠𝑑0β€–β€–β€–πœ•πœ‡πœ•π‘₯ξ€Ίξ€·π‘š+1(𝑠)βˆ’πœ‡π‘šξ€Έ(𝑠)βˆ‡π‘’π‘šξ€»β€–β€–β€–(𝑠)2ξ€œπ‘‘π‘ +𝑑0π‘π‘š(𝑠)𝑑𝑠.(3.59)

Note that πœ•πœ‡πœ•π‘₯ξ€Ίξ€·π‘š+1(𝑠)βˆ’πœ‡π‘šξ€Έ(𝑠)βˆ‡π‘’π‘šξ€»=ξ€·πœ‡(𝑠)π‘š+1(𝑠)βˆ’πœ‡π‘šξ€Έ(𝑠)Ξ”π‘’π‘š+𝐷(𝑠)1πœ‡ξ€Ίπ‘’π‘šξ€»βˆ’π·1πœ‡ξ€Ίπ‘’π‘šβˆ’1ξ€»ξ€Έβˆ‡π‘’π‘šξ€·π·(𝑠)+3πœ‡ξ€Ίπ‘’π‘šξ€»βˆ’π·3πœ‡ξ€Ίπ‘’π‘šβˆ’1||ξ€»ξ€Έβˆ‡π‘’π‘š||(𝑠)2+𝐷3πœ‡ξ€Ίπ‘’π‘šβˆ’1ξ€»βˆ‡π‘£π‘šβˆ’1(𝑠)βˆ‡π‘’π‘š(𝑠).(3.60)
Hence, β€–β€–β€–πœ•πœ‡πœ•π‘₯ξ€Ίξ€·π‘š+1(𝑠)βˆ’πœ‡π‘šξ€Έ(𝑠)βˆ‡π‘’π‘šξ€»β€–β€–β€–β‰€β€–β€–πœ‡(𝑠)π‘š+1(𝑠)βˆ’πœ‡π‘šβ€–β€–(𝑠)𝐢0(Ξ©)β€–β€–Ξ”π‘’π‘šβ€–β€–+‖‖𝐷(𝑠)1πœ‡ξ€Ίπ‘’π‘šξ€»βˆ’π·1πœ‡ξ€Ίπ‘’π‘šβˆ’1‖‖𝐢0(Ξ©)β€–β€–βˆ‡π‘’π‘šβ€–β€–+‖‖𝐷(𝑠)1πœ‡ξ€Ίπ‘’π‘šξ€»βˆ’π·1πœ‡ξ€Ίπ‘’π‘šβˆ’1‖‖𝐢0(Ξ©)β€–β€–βˆ‡π‘’π‘š(‖‖𝑑)2𝐢0Ω+‖‖𝐷3πœ‡ξ€Ίπ‘’π‘šβˆ’1‖‖𝐢0(Ξ©)β€–β€–βˆ‡π‘’π‘šβ€–β€–(𝑠)𝐢0(Ξ©)β€–β€–βˆ‡π‘£π‘šβˆ’1β€–β€–.(𝑠)(3.61)
We also note that β€–β€–πœ‡π‘š+1(𝑠)βˆ’πœ‡π‘šβ€–β€–(𝑠)𝐢0(Ξ©)≀𝐾𝑀‖‖𝑀(πœ‡)π‘šβˆ’1β€–β€–π‘Š1(𝑇),β€–β€–π·π‘–πœ‡[π‘’π‘š]βˆ’π·π‘–πœ‡[π‘’π‘šβˆ’1]‖‖𝐢0(Ξ©)≀𝐾𝑀‖‖𝑀(πœ‡)π‘šβˆ’1β€–β€–π‘Š1(T)β€–β€–,𝑖=1,3,βˆ‡π‘’π‘šβ€–β€–(𝑠)𝐢0(Ξ©)β‰€βˆš2β€–β€–βˆ‡π‘’π‘šβ€–β€–(𝑠)𝐻1β‰€βˆš2ξ”β€–βˆ‡π‘’π‘š(𝑠)β€–2+β€–β€–Ξ”π‘’π‘šβ€–β€–(𝑠)2‖‖𝐷≀2𝑀,3πœ‡[π‘’π‘š]‖‖𝐢0(Ξ©)≀𝐾𝑀(πœ‡),(3.62) where we use the notation π·π‘–πœ‡[π‘’π‘šβˆ’1]=π·π‘–πœ‡(π‘₯,𝑑,π‘’π‘š(π‘₯,𝑑)),𝑖=1,2,3. Therefore, it implies from (3.61) and (3.62) that β€–β€–β€–πœ•πœ‡πœ•π‘₯ξ€Ίξ€·π‘š+1(𝑠)βˆ’πœ‡π‘šξ€Έ(𝑠)βˆ‡π‘’π‘šξ€»β€–β€–β€–ξ‚πΎ(𝑠)≀(4+𝑀)𝑀𝑀(πœ‡)β€–π‘£π‘šβˆ’1β€–π‘Š1(𝑇).(3.63)
Hence, ||𝐽3||≀(4+𝑀)2𝑀2𝑇𝐾2𝑀(πœ‡)β€–π‘£π‘šβˆ’1β€–2π‘Š1(𝑇)+ξ€œπ‘‘0π‘π‘š(𝑠)𝑑𝑠.(3.64)
Combining (3.54)–(3.56), (3.58), and (3.64) yields π‘π‘š(𝑑)≀𝑇4𝐾2𝑀(𝑓)+(4+𝑀)2𝑀2𝐾2𝑀(ξ‚„πœ‡)β€–π‘£π‘šβˆ’1β€–2π‘Š1(𝑇)+ξ‚΅2+1+π‘€πœ‡βˆ—ξ‚πΎπ‘€(ξ‚Άξ€œπœ‡)𝑑0π‘π‘š(𝑠)𝑑𝑠.(3.65)
Using Gronwall's lemma, (3.65) gives β€–β€–π‘£π‘šβ€–β€–π‘Š1(𝑇)β‰€π‘˜π‘‡β€–β€–π‘£π‘šβˆ’1β€–β€–π‘Š1(𝑇)βˆ€π‘šβˆˆβ„•,(3.66) where π‘˜π‘‡<1 as in (3.43).
Hence, we obtain from (3.66) that β€–β€–π‘’π‘š+π‘βˆ’π‘’π‘šβ€–β€–π‘Š1(𝑇)β‰€π‘˜π‘šπ‘‡1βˆ’π‘˜π‘‡β€–β€–π‘’1βˆ’π‘’0β€–β€–π‘Š1(𝑇)βˆ€π‘š,π‘βˆˆβ„•,(3.67)
It follows that {π‘’π‘š} is a Cauchy sequence in π‘Š1(𝑇). Then, there exists π‘’βˆˆπ‘Š1(𝑇) such that π‘’π‘šβŸΆπ‘’stronglyinπ‘Š1(𝑇).(3.68)
On the other hand, from (3.48), we deduce the existence of a subsequence {π‘’π‘šπ‘—} of {π‘’π‘š} such that π‘’π‘šπ‘—βŸΆπ‘’inπΏβˆžξ€·0,𝑇;𝐻10∩𝐻2ξ€Έweakβˆ—,π‘’ξ…žπ‘šπ‘—βŸΆπ‘’ξ…žinπΏβˆžξ€·0,𝑇;𝐻10ξ€Έweakβˆ—,π‘’π‘šξ…žξ…žπ‘—βŸΆπ‘’ξ…žξ…žin𝐿2𝑄𝑇weak,(3.69)π‘’βˆˆπ‘Š(𝑀,𝑇).(3.70)
Note that ||πœ‡π‘š||≀𝐾(π‘₯,𝑑)βˆ’πœ‡(π‘₯,𝑑,𝑒(π‘₯,𝑑))𝑀‖‖𝑒(πœ‡)π‘šβˆ’1β€–β€–βˆ’π‘’π‘Š1(𝑇),β€–β€–πΉπ‘šξ€·(𝑑)βˆ’π‘“β‹…,𝑑,𝑒(𝑑),𝑒π‘₯(𝑑),π‘’ξ…žξ€Έβ€–β€–(𝑑)≀2𝐾𝑀‖‖𝑒(𝑓)π‘šβˆ’1β€–β€–βˆ’π‘’π‘Š1(𝑇).(3.71)
Hence, from (3.68) and (3.71), we obtain πœ‡π‘šβŸΆπœ‡(β‹…,β‹…,𝑒)stronglyinπΏβˆžξ€·π‘„π‘‡ξ€Έ,πΉπ‘šξ€·βŸΆπ‘“β‹…,𝑑,𝑒(𝑑),𝑒π‘₯(𝑑),π‘’ξ…žξ€Έ(𝑑)stronglyinπΏβˆžξ€·0,𝑇;𝐿2ξ€Έ.(3.72)
Finally, passing to limit in (3.6)–(3.8) as π‘š=π‘šπ‘—β†’βˆž, it implies from (3.68), (3.69), and (3.72) that there exists π‘’βˆˆπ‘Š(𝑀,𝑇) satisfying the equation ξ«π‘’ξ…žξ…žξ¬(𝑑),𝑀+βŸ¨πœ‡(β‹…,𝑑,𝑒(𝑑))𝑒π‘₯𝑓(𝑑),βˆ‡π‘€βŸ©=β‹…,𝑑,𝑒(𝑑),𝑒π‘₯(𝑑),π‘’ξ…žξ€Έξ¬(𝑑),𝑀,βˆ€π‘€βˆˆπ»10,𝑒(0)=̃𝑒0,π‘’ξ…ž(0)=̃𝑒1.(3.73)
On the other hand, by (𝐻2), we obtain from (3.70), (3.72)2, and (3.73)1 that π‘’ξ…žξ…ž=𝐷1πœ‡[𝑒]𝑒π‘₯+𝐷3πœ‡[𝑒]𝑒2π‘₯[𝑒]𝑒+πœ‡π‘₯π‘₯ξ€·+𝑓π‘₯,𝑑,𝑒,𝑒π‘₯,π‘’ξ…žξ€ΈβˆˆπΏβˆžξ€·0,𝑇;𝐿2ξ€Έ,(3.74) thus π‘’βˆˆπ‘Š1(𝑀,𝑇), and Step 1 follows.
(ii) The uniqueness of the solution.
Let 𝑒1,𝑒2βˆˆπ‘Š1(𝑀,𝑇) be two weak solutions of the problem (1.1)–(1.3). Then, 𝑒=𝑒1βˆ’π‘’2 satisfies the variational problem βŸ¨π‘’ξ…žξ…ž(𝑑),π‘€βŸ©+βŸ¨πœ‡1(𝑑)𝑒π‘₯(𝑑),𝑀π‘₯ξ‚¬πœ•βŸ©=πœ‡πœ•π‘₯ξ€·ξ€Ί1(𝑑)βˆ’πœ‡2𝑒(𝑑)2π‘₯ξ€Έξ‚­(𝑑),𝑀+⟨𝐹2(𝑑)βˆ’πΉ1(𝑑),π‘€βŸ©,βˆ€π‘€βˆˆπ»10,𝑒(0)=π‘’ξ…žπœ‡(0)=0,𝑖(𝑑)=πœ‡π‘₯,𝑑,𝑒𝑖𝑒(𝑑)β‰‘πœ‡π‘–ξ€»,𝐹𝑖(𝑑)=𝑓π‘₯,𝑑,𝑒𝑖(𝑑),𝑒𝑖π‘₯(𝑑),π‘’ξ…žπ‘–ξ€Έ(𝑑),𝑖=1,2.(3.75)
We take 𝑀=π‘’ξ…ž in (3.75)1 and integrate in 𝑑 to get ξ€œπœŒ(𝑑)=𝑑0ξ€œπ‘‘π‘ 10πœ‡ξ…ž1(π‘₯,𝑠)𝑒2π‘₯ξ€œ(π‘₯,𝑠)𝑑π‘₯+2𝑑0𝐹1(𝑠)βˆ’πΉ2(𝑠),π‘’ξ…žξ¬ξ€œ(𝑠)𝑑𝑠+2𝑑0ξ‚¬πœ•πœ‡πœ•π‘₯ξ€·ξ€Ί1(𝑠)βˆ’πœ‡2𝑒(𝑠)2π‘₯ξ€Έ(𝑠),π‘’ξ…žξ‚­π‘‘π‘ β‰‘3𝑖=1πœŒπ‘–(𝑑),(3.76) where β€–β€–π‘’πœŒ(𝑑)=ξ…žβ€–β€–(𝑑)2+β€–β€–βˆšπœ‡1(𝑑)𝑒π‘₯β€–β€–(𝑑)2.(3.77)
We now estimate the terms on the right-hand side of (3.76) as follows: 𝜌1ξ€œ(𝑑)=𝑑0ξ€œπ‘‘π‘ 10πœ‡ξ…ž1(π‘₯,𝑠)𝑒2π‘₯1(π‘₯,𝑠)𝑑π‘₯β‰€πœ‡βˆ—ξ‚πΎ(1+𝑀)π‘€ξ€œ(πœ‡)𝑑0𝜌(𝑠)π‘‘π‘ β‰‘πœŒπ‘€(1)ξ€œπ‘‘0𝜌𝜌(𝑠)𝑑𝑠,(3.78)2(ξ€œπ‘‘)=2𝑑0𝐹1(𝑠)βˆ’πΉ2(𝑠),π‘’ξ…ž(𝑠)𝑑𝑠≀4𝐾𝑀(ξ€œπ‘“)𝑑0‖‖𝑒π‘₯(β€–β€–+‖‖𝑒𝑠)ξ…ž(‖‖‖‖𝑒𝑠)ξ…ž(‖‖1𝑠)𝑑𝑠≀41+βˆšπœ‡βˆ—ξƒͺπΎπ‘€ξ€œ(𝑓)𝑑0𝜌(𝑠)π‘‘π‘ β‰‘πœŒπ‘€(2)ξ€œπ‘‘0𝜌𝜌(𝑠)𝑑𝑠,(3.79)3(ξ€œπ‘‘)=2𝑑0ξ‚¬πœ•πœ‡πœ•π‘₯ξ€·ξ€Ί1(𝑠)βˆ’πœ‡2(𝑒𝑠)2π‘₯(𝑠),π‘’ξ…žξ‚­ξ€œπ‘‘π‘ β‰€2𝑑0β€–β€–β€–πœ•πœ‡πœ•π‘₯ξ€·ξ€Ί1(𝑠)βˆ’πœ‡2(𝑒𝑠)2π‘₯(‖‖‖‖‖𝑒𝑠)ξ…ž(‖‖𝑠)𝑑𝑠.(3.80)
On the other hand πœ•πœ‡πœ•π‘₯ξ€·ξ€Ί1(𝑠)βˆ’πœ‡2𝑒(𝑠)2π‘₯ξ€Έ=ξ€Ίπœ‡(𝑠)1(𝑠)βˆ’πœ‡2𝑒(𝑠)2π‘₯π‘₯𝐷(𝑠)+1πœ‡ξ€Ίπ‘’1ξ€»βˆ’π·1πœ‡ξ€Ίπ‘’2𝑒2π‘₯+𝐷(𝑠)3πœ‡ξ€Ίπ‘’1ξ€»βˆ’π·3πœ‡ξ€Ίπ‘’2𝑒1π‘₯𝑒2π‘₯+D3πœ‡ξ€Ίπ‘’2𝑒π‘₯𝑒2π‘₯.(3.81)
Hence, β€–β€–β€–πœ•πœ‡πœ•π‘₯ξ€·ξ€Ί1(𝑠)βˆ’πœ‡2𝑒(𝑠)2π‘₯ξ€Έβ€–β€–β€–β‰€β€–β€–πœ‡(𝑠)1(𝑠)βˆ’πœ‡2β€–β€–(𝑠)𝐢0(Ξ©)‖‖𝑒2π‘₯π‘₯β€–β€–+‖‖𝐷(𝑠)1πœ‡[𝑒1]βˆ’π·1πœ‡[𝑒2]‖‖𝐢0(Ξ©)‖‖𝑒2π‘₯β€–β€–+‖‖𝐷(𝑠)3πœ‡[𝑒1]βˆ’π·3πœ‡[𝑒2]‖‖𝐢0(Ξ©)‖‖𝑒1π‘₯β€–β€–(𝑠)𝐢0(Ξ©)‖‖𝑒2π‘₯β€–β€–(𝑠)𝐢0(Ξ©)+‖‖𝐷3πœ‡[𝑒2]‖‖𝐢0(Ξ©)‖‖𝑒π‘₯‖‖‖‖𝑒(𝑠)2π‘₯β€–β€–(𝑠)𝐢0(Ξ©)𝐾≀(3+𝑀)𝑀𝑀‖‖𝑒(πœ‡)π‘₯β€–β€–.(𝑠)(3.82)
It follows from (3.80), (3.82) that 𝜌3(1𝑑)β‰€βˆšπœ‡βˆ—(𝐾3+𝑀)𝑀𝑀(ξ€œπœ‡)𝑑0𝜌(𝑠)π‘‘π‘ β‰‘πœŒπ‘€(3)ξ€œπ‘‘0𝜌(𝑠)𝑑𝑠.(3.83)
Combining (3.76)–(3.79) and (3.83) yields ξ‚€πœŒπœŒ(𝑑)≀𝑀(1)+πœŒπ‘€(2)+πœŒπ‘€(3)ξ‚ξ€œπ‘‘0𝜌(𝑠)𝑑𝑠.(3.84)
Using Gronwall's lemma, it follows from (3.84) that πœŒβ‰‘0 that is, 𝑒1≑𝑒2.
Theorem 3.2 is proved completely.

Remark 3.3. (i) In the case of πœ‡β‰‘1,π‘“βˆˆπΆ1(Ω×ℝ+×ℝ3) and the boundary condition in [4] standing for (1.2), we obtained some similar results in [4].
(ii) In the case of πœ‡β‰‘1,π‘“βˆˆπΆ1(Ω×ℝ+×ℝ3),𝑓(1,𝑑,𝑒,𝑣,𝑀)=0,forall𝑑β‰₯0,forall(𝑒,𝑣,𝑀)βˆˆβ„3, 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 π‘’βˆˆπ»2(𝑄𝑇) of the problem (1.1)–(1.3) is proved.

In the case of πœ‡=πœ‡(π‘₯,𝑑) and 𝑓=𝑓(π‘₯,𝑑), 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 π‘’βˆˆπΆ2(𝑄𝑇)∩𝐢1(𝑄𝑇).

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 (𝐻1) hold. We make the following assumptions: (𝐻4)πœ‡0∈𝐢2([0,1]×ℝ+),πœ‡1βˆˆπΆπ‘+1([0,1]×ℝ+×ℝ),πœ‡0β‰₯πœ‡βˆ—>0,πœ‡1β‰₯0, (