Linear Approximation and Asymptotic Expansion associated to the Robin-Dirichlet Problem for a Kirchhoﬀ-Carrier Equation with a Viscoelastic Term

Le Thi Phuong Ngoc; Doan Thi Nhu Quynh; Nguyen Anh Triet; Nguyen Thanh Long∗

doi:10.5666/KMJ.2019.59.4.735

Article

KYUNGPOOK Math. J. 2019; 59(4): 735-769

Published online December 23, 2019

Linear Approximation and Asymptotic Expansion associated to the Robin-Dirichlet Problem for a Kirchhoﬀ-Carrier Equation with a Viscoelastic Term

Le Thi Phuong Ngoc, Doan Thi Nhu Quynh, Nguyen Anh Triet, Nguyen Thanh Long∗

University of Khanh Hoa, 01 Nguyen Chanh Str., Nha Trang City, Vietnam
e-mail : ngoc1966@gmail.com
Department of Fundamental sciences, Ho Chi Minh City University of Food Industry, 140 Le Trong Tan Str., Tay Thanh Ward, Tan Phu Dist., Ho Chi Minh City, Vietnam
Department of Mathematics and Computer Science, VNUHCM - University of Science, 227 Nguyen Van Cu Str., Dist. 5, Ho Chi Minh City, Vietnam
e-mail : doanquynh260919@yahoo.com
Department of Mathematics, University of Architecture of Ho Chi Minh City, 196 Pasteur Str., Dist. 3, Ho Chi Minh City, Vietnam
e-mail : triet.nguyenanh@uah.edu.vn
Department of Mathematics and Computer Science, VNUHCM - University of Science, 227 Nguyen Van Cu Str., Dist. 5, Ho Chi Minh City, Vietnam
e-mail : longnt2@gmail.com

Received: September 11, 2018; Accepted: March 4, 2019

Abstract

Go to

Abstract
1. Introduction
2. Preliminaries
3. The Existence and Uniqueness Theorem
Lemma 3.2
Lemma 3.3
Theorem 3.4
4. Asymptotic Expansion of the Solution with respect to a Small Parameter
Acknowledgements
References

In this paper, we consider the Robin-Dirichlet problem for a nonlinear wave equation of Kirchhoff-Carrier type with a viscoelastic term. 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.

Keywords: Faedo-Galerkin method, linear recurrent sequence, RobinDirichlet conditions, asymptotic expansion in a small parameter.

1. Introduction

Go to

Abstract
1. Introduction
2. Preliminaries
3. The Existence and Uniqueness Theorem
Lemma 3.2
Lemma 3.3
Theorem 3.4
4. Asymptotic Expansion of the Solution with respect to a Small Parameter
Acknowledgements
References

In this paper, we consider the following nonlinear Kirchhoff-Carrier wave equation with a viscoelastic term

(1.1)utt-λuxxt-∂∂x[μ1(x,t,u(x,t),‖u(t)‖2,‖ux(t)‖2)ux]+∫0tg(t-s)∂∂x[μ2(x,s,u(x,s),‖u(s)‖2,‖ux(s)‖2)ux(x,s)]ds=f(x,t,u,ux,ut,‖u(t)‖2,‖ux(t)‖2),0<x<1,0<t<T,

associated with Robin-Dirichlet conditions

(1.2)ux(0,t)-h0u(0,t)=u(1,t)=0,

and initial conditions

(1.3)u(x,0)=u˜0(x),ut(x,0)=u˜1(x),

where λ > 0, h₀ ≥ 0 are given constants, ũ₀, ũ₁, f, g, μ₁, μ₂ are given functions satisfying conditions, which will be specified later, ‖u(t)‖2=∫01u2(x,t)dx,‖ux(t)‖2=∫01ux2(x,t)dx.

When λ = 0, g = 0, f = 0, Eq. (1.1) is related to the Kirchhoff equation

(1.4)ρhutt=(P0+Eh2L∫0Lux2(y,t)dy)uxx,

presented by Kirchhoff in 1876 (see [9]). This equation is an extension of the classical D’Alembert wave equation which considers the effects of the changes in the length of the string during the vibrations. The parameters in (1.4) have the following meanings: u is the lateral deflection, L is the length of the string, h is the area of the cross - section, E is the Young modulus of the material, ρ is the mass density, and P₀ is the initial tension.

In [1], Carrier has also established the equation which models vibrations of an elastic string when changes in tension are not small

(1.5)ρutt-(1+EALT0∫0Lu2dx)uxx=0,

where u(x, t) is the x - derivative of the deformation, T₀ is the tension in the rest position, E is the Young modulus, A is the cross - section of a string, L is the length of a string and ρ is the density of a material.

One of the early classical studies dedicated to Kirchhoff equations was given by Pohozaev [29]. After the work of Lions, for example see [13], Kirchhoff equations as well as Kirchhoff-Carrier equations of the form Eq. (1.1) received much attention (see [2, 3, 4, 6, 7, 10, 11, 18, 23, 21, 28, 27, 30, 31] and references therein). A survey of the results about the mathematical aspects of Kirchhoff model can be found in Medeiros, Limaco and Menezes [19, 20].

It is also well known that, the study of the asymptotic behavior of nonlinear equations with a viscoelastic term has attracted lots of interest of researchers (for example, see [8, 22, 26] and references therein). In [8, 22], the viscoelastic wave equation of Kirchhoff type of the form

(1.6)utt-M(‖∇u‖2)Δu+∫0tg(t-s)Δu(s)ds+hut=|u|q-1u,

has been studied and the results of existence and blow up were proved. In [26], the author considered a viscoelastic plate equation with p-Laplacian, by introducing suitable energy and Lyapunov functionals, a general decay estimate for the energy was established.

The paper consists of four sections. Section 2 is devoted to some preliminaries. We begin Section 3 by establishing a sequence of approximate solutions of Prob. (1.1) – (1.3) based on the Faedo-Galerkin’s method. Thanks to a priori estimates, we first prove that this sequence is bounded in an appropriate space, by using compact imbedding theorems, we next show that this sequence converges and the existence of Prob. (1.1) – (1.3) follows. By Gronwall’s Lemma, the uniqueness of a weak solution is proved. In Section 4, we establish an asymptotic expansion of a weak solution u = u_ɛ of order N + 1 in a small parameter ɛ for the equation

(1.7)utt-λuxxt-∂∂x[μ1(x,t,u(x,t),‖u(t)‖2,‖ux(t)‖2)ux]+∫0tg(t-s)∂∂x[μ2(x,s,u(x,s),‖u(s)‖2,‖ux(s)‖2)ux(x,s)] ds=f(x,t,u,ux,ut,‖u(t)‖2,‖ux(t)‖2)+ɛf1(x,t,u,ux,ut,‖u(t)‖2,‖ux(t)‖2),

0 < x < 1, 0 < t < T, associated to (1.2), (1.3), with μ₁, μ2∈CN+1([0,1]×[0,T*]×ℝ×ℝ+2), μ₁(x, t, z₁, z₂, z₃) ≥ μ_* > 0, for all (x,t,z1,z2,z3)∈[0,1]×[0,T*]×ℝ×ℝ+2,f∈CN+1([0,1]×[0,T*]×ℝ3×ℝ+2),f1∈CN([0,1]×[0,T*]×ℝ3×ℝ+2). This result is a relative generalization of [14, 15, 16, 17, 24, 25].

2. Preliminaries

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Abstract
1. Introduction
2. Preliminaries
3. The Existence and Uniqueness Theorem
Lemma 3.2
Lemma 3.3
Theorem 3.4
4. Asymptotic Expansion of the Solution with respect to a Small Parameter
Acknowledgements
References

The notation we use in this paper is standard and can be found in Lion’s book [12], with Ω = (0, 1), Q_T = Ω × (0, T), T > 0 and ||·|| is the norm in L². For a Banach space X, ||·||_X denotes the norm of X. We denote L^p(0, T; X), 1 ≤ p ≤ ∞ the Banach space of real functions u: (0, T) → X measurable, such that ||u||_L^p(0,T;X) < + ∞, with

‖u‖Lp(0,T;X)={(∫0T‖u(t)‖Xpdt)1/p,if 1≤p<∞,esssup0<t<T‖u(t)‖X,if p=∞.

With f∈Ck([0,1]×[0,T*]×ℝ3×ℝ+2), f = f(x, t, y₁, y₂, y₃, y₄, y₅), (x, t) ∈ [0, 1] × [0, T^*], (y1,y2,y3,y4,y5)∈ℝ3×ℝ+2, we put D1f=∂f∂x,D2f=∂f∂t,Di+2f=∂f∂yi, i = 1, …, 5 and Dαf=D1α1…D7α7f,α=(α1,…,α7)∈ℤ+7, |α| = α₁ + … + α₇ ≤ k, D^(0,…,0)f = f.

Similarly, with μ∈Ck([0,1]×[0,T*]×ℝ×ℝ+2), μ = μ_i (x, t, z₁, z₂, z₃), (x, t) ∈ [0, 1] × [0, T^*], (z1,z2,z3)∈ℝ×ℝ+2, we put D1μ=∂μ∂x,D2μ=∂μ∂y,Dj+2μ=∂μ∂yj, j = 1, 2, 3, and Dβμ=D1β1…D5β5μ,β=(β1,…,β5)∈ℤ+5, |β| = β₁ + … + β₅ ≤ k, D^(0,…,0)μ = μ.

On H¹ ≡ H¹ (Ω), we shall use the following norm

(2.1)‖v‖H1=(‖v‖2+‖vx‖2)12.

We set

(2.2)V={v∈H1(0,1):v(1)=0},

and

(2.3)a(u,v)=〈ux,vx〉+h0u(0)v(0),for all u,v∈V,

Then, V is a closed subspace of H¹ and three norms v ↦ ||v||_H_¹, v ↦ ||v_x|| and v↦‖v‖a=a(v,v) are equivalent on V. On the other hand, V is continuously and densely embedded in L². Identifying L² with (L²)′ (the dual of L²), we have V ↪ L² = (L²)′ ↪ V′. We remark that the notation 〈·, ·〉 is also used for the pairing between V and V′.

We have the following lemmas involving known properties.

Lemma 2.1

The embedding H¹ ↪ C⁰ (Ω̄) is compact and

(2.4)‖v‖C0(Ω¯)≤2‖v‖H1 for all v∈H1.

Lemma 2.2

Let h₀ ≥ 0. Then the embedding V ↪ C⁰ (Ω̄) is compact and

(2.5){‖v‖C0(Ω¯)≤‖vx‖≤‖v‖afor allv∈V,12‖v‖H1≤‖vx‖≤‖v‖a≤1+h0‖v‖H1for allv∈V.

Lemma 2.3

Let h₀ ≥ 0. Then the symmetric bilinear form a (·, ·) defined by (2.3) is continuous on V × V and coercive on V, i.e.,

(2.6)(i)|a(u,v)|≤(1+h0)‖ux‖ ‖vx‖,for allu,v∈V,(ii)a(v,v)≥‖vx‖2,for allv∈V.

Lemma 2.4

Let h₀ ≥ 0. There exists the Hilbert orthonormal base {w_j} of the space L²consisting of eigenfunctions w_j corresponding to eigenvalues λ_j such that

(2.7)(i)0<λ1≤λ2≤⋯≤λj≤λj+1≤⋯,limj→+∞λj=+∞,(ii)a(wj,v)=λj〈wj,v〉 for all v∈V, j=1,2,….

Furthermore, the sequence {wj/λj } is also the Hilbert orthonormal base of V with respect to the scalar product a (·, ·). On the other hand, w_j satisfies the following boundary value problem

(2.8){-Δwj=λjwj, in Ω,wjx(0)-h0wj(0)=wj(1)=0,wj∈V∩C∞(Ω¯).

The proof of Lemma 2.4 can be found in [32, p.87, Theorem 7.7], with H = L² and V, a(·, ·) as defined by (2.3).

3. The Existence and Uniqueness Theorem

Go to

Abstract
1. Introduction
2. Preliminaries
3. The Existence and Uniqueness Theorem
Lemma 3.2
Lemma 3.3
Theorem 3.4
4. Asymptotic Expansion of the Solution with respect to a Small Parameter
Acknowledgements
References

In this section, we consider the local existence for Problem (1.1)–(1.3), with h₀, λ ∈ ℝ, h₀ ≥ 0, λ > 0. Here, it is said that u is a weak solution of Problem (1.1)–(1.3) if

(3.1)u∈V˜T={v∈L∞(0,T;V∩H2):v′∈L∞(0,T;V∩H2),v″∈L2(0,T;V)∩L∞(0,T;L2)},

and u satisfies the following variational equation

(3.2)〈u″(t),v〉+λa(u′(t),v)+a1[u] (t;u(t),v)=〈f [u] (t),v〉+∫0tg(t-s)a2[u] (s;u(s),v) ds,

for all v ∈ V, and a.e., t ∈ (0, T), together with the initial conditions

(3.3)u(0)=u˜0, u′(0)=u˜1,

where, for each ū ∈ Ṽ_T and i = 1, 2, {a_i[ū](t; ·, ·)}_0≤_t_≤_T is the family of symmetric bilinear forms on V × V defined by

(3.4)ai[u¯] (t;u,v)=〈μi[u¯] (t)ux,vx〉+h0μi[u¯] (0,t)u(0)v(0),

for all u, v ∈ V, 0 ≤ t ≤ T, with h₀ ≥ 0 is given constant, and

(3.5)μi[u¯] (x,t)=μi (x,t,u¯(x,t),‖u¯(t)‖2,‖u¯x(t)‖2),i=1,2,f[u] (x,t)=f(x,t,u(x,t),ux(x,t),u′(x,t),‖u(t)‖2,‖ux(t)‖2).

Consider T^* > 0 fixed, we make the following assumptions:

(H₁) ũ₀, ũ₁ ∈ V ∩ H²;

(H₂) g ∈ H¹ (0, T^*);

(H₃) μ1∈C2([0,1]×[0,T*]×ℝ×ℝ+2),μ1 (x,t,y1,y2,y3)≥μ*>0,∀(x,t,y1,y2,y3)∈[0,1]×[0,T*]×ℝ×ℝ+2;

(H₄) μ2∈C1([0,1]×[0,T*]×ℝ×ℝ+2);

(H₅) f∈C1([0,1]×[0,T*]×ℝ3×ℝ+2).

For each M > 0 given, we set the constants K_M (f), K̃_M (μ₁), K̃_M (μ₂) as follows

(3.6)KM(f)=‖f‖C1(AM)=‖f‖C0(AM)+∑i=17‖Dif‖C0(AM),K˜M(μi)=‖μi‖C2(A˜M)=∑|β|≤2‖Dβμi‖C0(A˜M),i=1,2,

with

(3.7){‖f‖C0(AM)=sup(x,t,y1,…,y5)∈AM∣f(x,t,y1,…,y5)∣,‖μi‖C0(A˜M)=sup(x,t,z1,…,z3)∈A˜M∣μi(x,t,z1,…,z3)∣,i=1,2,AM=(x,t,y1,…,y5):{0≤x≤1,0≤t≤T*,∣y1∣,∣y2∣,∣y3∣≤2M,0≤y4,y5≤M2},A˜M={(x,t,z1,…,z3):0≤x≤1,0≤t≤T*,∣z1∣≤M,0≤z2,z3≤M2}.

For each T ∈ (0, T^*], we denote

(3.8)VT={v∈L∞(0,T;V∩H2):v′∈L∞(0,T;V∩H2),v″∈L2(0,T;V)},

it is a Banach space with respect to the norm

(3.9)‖v‖VT=max{‖v‖L∞(0,T;V∩H2);‖v′‖L∞(0,T;V∩H2);‖v″‖L2(0,T;V)}.

For every M >0, we put

(3.10)W(M,T)={v∈VT:‖v‖VT≤M},W1(M,T)={v∈W(M,T):v″∈L∞(0,T;L2)}.

Next, we will establish the recurrent sequence {u_m}. The first term is chosen as u₀ ≡ 0, suppose that

(3.11)um-1∈W1(M,T),

based on the associate problem (3.2), we find u_m ∈ W₁(M, T) (m ≥ 1) satisfying the linear variational problem

(3.12){〈um″(t),v〉+λa(um′(t),v)+a1(m)(t;um(t),v)=∫0tg(t-s)a2(m)(s;um(s),v)ds+〈Fm(t),v〉,∀v∈V,um(0)=u˜0,um′(0)=u˜1,

where

(3.13){ai(m)(t;u,v)=ai[um-1] (t;u,v)=〈μi(m)(t)ux,vx〉+h0μi(m)(0,t)u(0)v(0),∀u,v∈V;Fm(x,t)=f[um-1] (x,t)=f(x,t,um-1 (x,t),∇um-1(x,t),um-1′(x,t),‖um-1(t)‖2,‖∇um-1(t)‖2);μi(m)(x,t)=μi[um-1] (x,t)=μi(x,t,um-1(x,t),‖um-1(t)‖2,‖∇um-1(t)‖2),i=1,2.

Then we have the following theorem.

Theorem 3.1

Let (H₁)–(H₅) hold. Then there exist constants M, T > 0 such that, for u₀ ≡ 0, there exists a recurrent sequence {u_m} ⊂ W₁(M, T) defined by (3.11)–(3.13).

Proof

The proof consists of three steps.

Step 1. The Faedo-Galerkin approximation

Consider the basis {w_j} for V given by Lemma 2.4. Put

(3.14)um(k)(t)=∑j=1kcmj(k)(t)wj,

where the coefficients cmj(k) satisfy the system of linear integrodifferential equations

(3.15){〈u¨m(k)(t),wj〉+λa(u˙m(k)(t),wj)+a1(m)(t;um(k)(t),wj)=〈Fm(t),wj〉+∫0tg(t-s)a2(m) (s;um(k)(s),wj) ds,1≤j≤k,um(k)(0)=u˜0k,u˙m(k)(0)=u˜1k,

in which

(3.16){u˜0k=∑j=1kαj(k)wj→u˜0 strongly in V∩H2,u˜1k=∑j=1kβj(k)wj→u˜1 strongly in V∩H2.

The system (3.15), (3.16) can be written in the form

(3.17){c¨mj(k)(t)+λλjc˙mj(k)(t)+∑i=1ka1ij(m)(t)cmi(k)(t)=∑i=1k∫0tg(t-s)a2ij(m)(s)cmi(k)(s)ds+fmj(t),cmj(k)(0)=αj(k),c˙mj(k)(0)=βj(k),1≤j≤k,

where

(3.18)aγij(m)(t)=aγ(m)(t;wi,wj),fmj(t)=〈Fm(t),wj〉,γ=1,2,1≤i,j≤k.

Note that by (3.11), using standard methods in ordinary differential equations (see [5]), the system (3.17) has a unique solution cmj(k)(t), 1 ≤ j ≤ k on interval [0,Tm(k)]⊂[0,T].

Step 2. A priori estimate

First, we put

(3.19)Sm(k)(t)=pm(k)(t)+qm(k)(t)+rm(k)(t),

where

(3.20){pm(k)(t)=‖u˙m(k)(t)‖2+a1(m) (t;um(k)(t),um(k)(t))+2λ∫0t‖u˙m(k)(s)‖a2ds,qm(k)(t)=‖u˙m(k)(t)‖a2+‖μ1(m)(t)Δum(k)(t)‖2+2λ∫0t‖Δu˙m(k)(s)‖2ds,rm(k)(t)=λ‖Δu˙m(k)(t)‖2+2∫0t‖u¨m(k)(s)‖a2ds.

Then, it follows from (3.15), (3.19), (3.20) that

(3.21)Sm(k)(t)=Sm(k)(0)+2〈J1(m,k)(0),Δu˜1k〉+2∫0t〈Fm(s),u˙m(k)(s)〉ds+2∫0ta(Fm(s),u˙m(k)(s)+u¨m(k)(s)) ds+∫0ta˙1(m)(s;um(k)(s),um(k)(s)) ds+∫0tds∫01μ˙1(m)(x,s)|Δum(k)(x,s)|2ds-2∫0t〈μ1x(m)(s)umx(k)(s),Δu˙m(k)(s)〉ds+2∫0t〈J˙1(m,k)(s),Δu˙m(k)(s)〉 ds-2〈J1(m,k)(t),Δu˙m(k)(t)〉+2∫0tdτ∫0τg(τ-s)a2(m) (s;um(k)(s),u˙m(k)(τ)) ds+2∫0tdτ∫0τg(τ-s) 〈J2(m,k)(s),Δu˙m(k)(τ)〉 ds+2∫0tg(t-s) 〈J2(m,k)(s),Δu˙m(k)(t)〉 ds-2g(0)∫0t〈J2(m,k)(τ),Δu˙m(k)(τ)〉 dτ-2∫0tdτ∫0τg′(τ-s) 〈J2(m,k)(s),Δu˙m(k)(τ)〉 ds≡sm(k)(0)+2〈J1(m,k)(0),Δu˜1k〉+∑j=112Ij,

where

(3.22)Ji(m,k)(x,t)=∂∂x (μi(m)(x,t) umx(k)(x,t)), i=1,2.

Next, we need the following lemmas.

Lemma 3.2

Go to

Abstract
1. Introduction
2. Preliminaries
3. The Existence and Uniqueness Theorem
Lemma 3.2
Lemma 3.3
Theorem 3.4
4. Asymptotic Expansion of the Solution with respect to a Small Parameter
Acknowledgements
References

Put

(3.23)a˙i(m)(t;u,v)=〈μ˙i(m)(t)ux,vx〉+h0μ˙i(m) (0,t) u(0) v(0), for all u,v∈V.

Then we have

(3.24)(i)ddtai(m) (t;um(k)(t),um(k)(t))=2ai(m) (t;um(k)(t),u˙m(k)(t))+a˙i(m)(t;um(k)(t),um(k)(t)),(ii)|a˙i(m)(t;u,v)|≤γMK˜M(μi) ‖u‖a ‖v‖a, for all u,v∈V,(iii)|ai(m)(t;u,v)|≤K˜M(μi) ‖u‖a ‖v‖a, for all u,v∈V,∀t∈[0,T*],(iv)a1(m)(t;v,v)≥μ*‖v‖a2, for all v∈V,

where γ_M = 1 + M + 4M², i = 1, 2.

Lemma 3.3

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Abstract
1. Introduction
2. Preliminaries
3. The Existence and Uniqueness Theorem
Lemma 3.2
Lemma 3.3
Theorem 3.4
4. Asymptotic Expansion of the Solution with respect to a Small Parameter
Acknowledgements
References

We have

(3.25)(i)‖Ji(m,k)(t)‖≤d0(M)K˜M(μi)Sm(k)(t),(ii)‖J˙i(m,k)(t)‖≤d1(M)K˜M(μi)Sm(k)(t),

whered0(M)=2+2Mμ*,d1(M)=1+1λ+2M+(5+2M)γMμ*, i = 1, 2.

Proof

The proof of Lemmas 3.2, 3.3 are easy, hence we omit the details.

We shall estimate the terms I_j on the right-hand side of (3.21) as follows.

Estimation of I₁

The Cauchy-Schwartz inequality leads to

(3.26)I1=2∫0t〈Fm(s),u˙m(k)(s)〉 ds≤2∫0t‖Fm(s)‖ ‖u˙m(k)(s)‖ ds ≤TKM2(f)+∫0tSm(k)(s) ds.

Estimation of the terms I₂ and I₃

We note that, by (3.13)₂ we have

(3.27)Fmx(x,t)=D1f[um-1] (x,t)+D3f[um-1] (x,t)∇um-1(x,t) +D4f[um-1] (x,t)Δum-1(x,t) +D5f[um-1] (x,t)∇um-1′(x,t),

where we use the notations

Dif[um-1] (x,t)=Dif(x,t,um-1(x,t),∇um-1(x,t),um-1′) (x,t),‖um-1(t)‖2,‖∇um-1(t)‖2),

i = 1, 2, …, 7, so, by (3.6), (3.11) and (3.27), we obtain

(3.28)‖Fmx(t)‖≤(1+3M) KM(f)≤2γMKM(f),‖Fm(t)‖a2=‖Fmx(t)‖2+h0Fm2(0,t)≤(4γM2+h0) KM2(f).

By Lemma 3.2, (ii), (iv) and the following inequalities

(3.29)2ab≤βa2+1βb2, ∀a,b∈ℝ, ∀β>0,

and

Sm(k)(t)≥2λ∫0t‖u˙m(k)(s)‖a2ds+2∫0t‖u¨m(k)(s)‖a2ds ≥min{1,λ}∫0t(‖u˙m(k)(s)‖a+‖u¨m(k)(s)‖a)2 ds,

we shall estimate respectively the following terms I₂, I₃ on the right-hand side of (3.21) as follows

(3.30)I2=2∫0ta(Fm(s),u˙m(k)(s)+u¨m(k)(s)) ds ≤2∫0t‖Fm(s)‖a (‖u˙m(k)(s)‖a+‖u¨m(k)(s)‖a)ds ≤1β min{1,λ}∫0t‖Fm(s)‖a2ds+β min{1,λ}∫0t(‖u˙m(k)(s)‖a+‖u¨m(k)(s)‖a)2 ds ≤T(4γM2+h0)β min{1,λ}KM2(f)+βSm(k)(t), ∀β>0;I3=∫0ta˙1(m) (s;um(k)(s),um(k)(s)) ds≤γMK˜M(μ1)∫0t‖um(k)(s)‖a2ds ≤γMK˜M(μ1)μ*∫0tSm(k)(s) ds.

Estimation of I₄

On the other hand, by (3.13)₃, we have

(3.31)μ˙1(m)(x,t)=D2μ1[um-1] (x,t)+D3μ1[um-1] (x,t) um-1′(x,t) +2D4μ1[um-1] (x,t) 〈um-1(t),um-1′(t)〉 +2D5μ1[um-1] (x,t) 〈∇um-1(t),∇um-1′(t)〉,

where we use the notations

Diμ1 [um-1] (x,t)=Diμ1 (x,t,um-1 (x,t),‖um-1(t)‖2,‖∇um-1(t)‖2),

i = 1, 2, …, 5, it implies that

(3.32)|μ˙1(m)(x,t)|≤(1+M+4M2)K˜M(μ1)=γMK˜M(μ1).

Hence, we deduce from (3.19), (3.20) and (3.32), that

(3.33)I4=∫0tds∫01μ˙1(m)(x,s)|Δum(k)(x,s)|2ds ≤γMK˜M(μ1)∫0t‖Δum(k)(s)‖2ds ≤γMK˜M(μ1)μ*∫0tSm(k)(s)ds.

Estimation of I₅

Similarly, by the following formula

(3.34)μ1x(m)(x,t)=D1μ1 [um-1] (x,t)+D3μ1 [um-1] (x,t)∇um-1(x,t),

and by (3.6), (3.11) and (3.34), we obtain

(3.35)|μ1x(m)(x,t)|≤(1+2M)K˜M(μ1)≤2γMK˜M(μ1).

Using the inequality (3.35) and the following inequalities

Sm(k)(t)≥a1(m) (t;um(k)(t),um(k)(t))+λ‖Δu˙m(k)(t)‖2 ≥μ*‖umx(k)(t)‖2+λ‖Δu˙m(k)(t)‖2 ≥min{μ*,λ} (‖umx(k)(t)‖2+‖Δu˙m(k)(t)‖2),

we shall estimate the following term I₅ as follows

(3.36)I5=-2∫0t〈μ1x(m)(s) umx(k)(s),Δu˙m(k)(s)〉 ds ≤4γMK˜M(μ1)∫0t‖umx(k)(s)‖ ‖Δu˙m(k)(s)‖ ds ≤2γMK˜M(μ1)min{μ*,λ}∫0tSm(k)(s)ds.

Estimation of I₆

By Lemma 3.3, (ii) and the inequality Sm(k)(t)≥λ‖Δu˙m(k)(t)‖2, we have

(3.37)I6=2∫0t〈J˙1(m,k)(s),Δu˙m(k)(s)〉 ds ≤2∫0t‖J˙1(m,k)(s)‖ ‖Δu˙m(k)(s)‖ ds ≤2λd1(M)K˜M(μ1)∫0tSm(k)(s)ds.

Estimation of I₇

By Lemma 3.3, (ii), we have

(3.38)‖J1(m,k)(t)‖2=(‖J1(m,k)(0)+∫0tJ˙1(m,k)(s) ds‖)2 ≤2‖J1(m,k)(0)‖2+2T∫0t‖J˙1(m,k)(s)‖2 ds ≤2‖J1(m,k)(0)‖2+2Td12(M)K˜M2(μ1)∫0tSm(k)(s) ds.

We deduce from the inequalities (3.38) and Sm(k)(t)≥λ‖Δu˙m(k)(t)‖2 that

(3.39)I7=-2〈J1(m,k)(t),Δu˙m(k)(t)〉≤2λ‖J1(m,k)(t)‖Sm(k)(t) ≤1βλ‖J1(m,k)(t)‖2+βSm(k)(t) ≤2βλ‖J1(m,k)(0)‖2+2βλTd12(M)K˜M2(μ1)∫0tSm(k)(s) ds+βSm(k)(t),

for all β > 0.

Estimation of I₈

By Lemma 3.3, (ii) and the inequalities

Sm(k)(t)≥a1(m)(t;um(k)(t),um(k)(t))+‖u˙m(k)(t)‖a2 ≥μ*‖um(k)(t)‖a2+‖u˙m(k)(t)‖a2≥2μ*‖um(k)(t)‖a‖u˙m(k)(t)‖a,

we have

(3.40)I8=2∫0tdτ∫0τg(τ-s)a2(m) (s;um(k)(s),u˙m(k)(τ)) ds ≤2K˜M(μ2)∫0tdτ∫0τ∣g(τ-s)∣Sm(k)(s)μ*Sm(k)(τ)ds ≤2μ*K˜M(μ2)∫0tdτ∫0τ∣g(τ-s)∣Sm(k)(s)Sm(k)(τ)ds ≤2μ*K˜M(μ2)T‖g‖L2(0,T*)∫0tSm(k)(s) ds.

Estimation of I₉, I₁₀, I₁₁, I₁₂

By Lemma 3.3, (ii) and the inequalities

Sm(k)(t)≥a1(m) (t;um(k)(t),um(k)(t))+‖u˙m(k)(t)‖a2≥μ*‖um(k)(t)‖a2+‖u˙m(k)(t)‖a2 ≥2μ*‖um(k)(t)‖a‖u˙m(k)(t)‖a,Sm(k)(t)≥λ‖Δu˙m(k)(t)‖2,

it is not difficult to estimate the following terms

(3.41)I9=2∫0tdτ∫0τg(τ-s) 〈J2(m,k)(s),Δu˙m(k)(τ)〉 ds ≤2∫0tdτ∫0τ∣g(τ-s)∣ ‖J2(m,k)(s)‖ ‖Δu˙m(k)(τ)‖ ds ≤2λd0(M)K˜M(μ2)∫0tdτ∫0τ∣g(τ-s)∣Sm(k)(s)Sm(k)(τ)ds ≤2λd0(M)K˜M(μ2)T‖g‖L2(0,T*)∫0tSm(k)(s) ds;I10=2∫0tg(t-s) 〈J2(m,k)(s),Δu˙m(k)(t)〉 ds ≤2λd0(M)K˜M(μ2)∫0t∣g(t-s)∣Sm(k)(s)Sm(k)(t)ds ≤1βλd02(M)K˜M2(μ2)‖g‖L2(0,T*)2∫0tSm(k)(s) ds+βSm(k)(t),∀β>0;I11=-2g(0)∫0t〈J2(m,k)(τ),Δu˙m(k)(τ)〉 dτ ≤2∣g(0)∣∫0t‖J2(m,k)(τ)‖ ‖Δu˙m(k)(τ)‖ dτ ≤2λ∣g(0)∣d0(M)K˜M(μ2)∫0tSm(k)(τ) dτ;I12=-2∫0tdτ∫0τg′(τ-s) 〈J2(m,k)(s),Δu˙m(k)(τ)〉 ds ≤2∫0tdτ∫0τ∣g′(τ-s)∣ ‖J2(m,k)(s)‖ ‖Δu˙m(k)(τ)‖ ds ≤2λd0(M)K˜M(μ2)∫0tdτ∫0τ∣g′(τ-s)∣Sm(k)(s)Sm(k)(τ)ds ≤2λd0(M)K˜M(μ2)T‖g′‖L2(0,T*)∫0tSm(k)(s) ds.

Choosing β=16 it follows from (3.21), (3.26), (3.30), (3.33), (3.36), (3.37), (3.39)–(3.41), that

(3.42)Sm(k)(t)≤S¯m,k+DM(1)TKM2(f)+DM(2)(T)∫0tSm(k)(s) ds,

where

(3.43)S¯m,k=2Sm(k)(0)+4〈J1(m,k)(0),Δu˜1k〉+24λ‖J1(m,k)(0)‖2,DM(1)=2+12(14γM2+h0)min{1,λ},DM(2)(T)=2+4(1μ*+1min{μ*,λ}) γMK˜M(μ1) +4λ(∣g(0)∣+T‖g′‖L2(0,T*)+T‖g‖L2(0,T*)) d0(M)K˜M(μ2) +4(1λ+6λTd1(M)K˜M(μ1)) d1(M)K˜M(μ1) +4(Tμ*+3λd02(M)K˜M(μ2) ‖g‖L2(0,T*)) K˜M(μ2) ‖g‖L2(0,T*).

Estimation of S̄_m,k

Notice that the formula J1(m,k)(0)=∂∂x[μ1(x,0,u˜0(x),‖u˜0‖2,‖u˜0x‖2)u˜0kx] independent of m. By means of the convergences in (3.16), we can deduce the existence of a constant M >0 independent of k and m such that

(3.44)S¯m,k=2Sm(k)(0)+4〈J1(m,k)(0),Δu˜1k〉+24λ‖J1(m,k)(0)‖2 =2a1(m)(0;u˜0k,u˜0k)+2‖μ1(m)(0)Δu˜0k‖2 +2‖u˜1k‖2+ 2‖u˜1k‖a2+2λ‖Δu˜1k‖2 +4〈∂∂x(μ1(m)(·,0) u˜0kx),Δu˜1k〉 +24λ‖∂∂x(μ1(m)(·,0) u˜0k)‖2 ≤12M2 for all m,k∈ℕ.

Hence, from (3.43)_2,3, we can choose T ∈ (0, T^*], such that

(3.45)(M22+DM(1)TKM2(f)) eTDM(2)(T)≤M2,

and

(3.46)kT=(1+12λ+1μ*)TD¯1(M)exp(TD¯2(M))<1,

where

(3.47)D¯1(M)=8(1+2M)2KM2(f) +6(1+4M)2M2λ(K˜M2 (μ1)+K˜M2(μ2)‖g‖L1(0,T*)2),D¯2(M)=1+γMK˜M (μ1)μ*+3K˜M2 (μ2)T*‖g‖L2(0,T*)2λμ*.

Finally, it follows from (3.42), (3.44) and (3.45), that

(3.48)Sm(k) (t)≤M2e-TDM(2)(T)+DM(2)(T)∫0tSm(k) (s) ds.

By using Gronwall’s Lemma, we deduce from (3.48) that

(3.49)Sm(k) (t)≤M2e-TDM(2)(T)etDM(2)(T)≤M2,

for all t ∈ [0, T], for all m and k ∈ ℕ.

Therefore, we have

(3.50)um(k)∈W(M,T), for all m and k.

Step 3. Limiting process

From (3.50), we deduce the existence of a subsequence of {um(k) } still so denoted, such that

(3.51){um(k)→uminL∞(0,T;V∩H2) weak*,u˙m(k)→um′inL∞(0,T;V∩H2) weak*,u¨m(k)→um″inL2(0,T;V) weak,um∈W(M,T).

Passing to limit in (3.15), we have u_m satisfying (3.12) in L²(0, T).

The proof of Theorem 3.1 is complete.

We note that W₁(T) = {v ∈ L^∞(0, T; V): v′ ∈ L^∞(0, T; L²) ∩ L²(0, T; V)} is a Banach space with respect to the norm (see Lions [12]).

‖v‖W1(T)= ‖v‖L∞(0,T;V)+‖v′‖L∞(0,T;L2)+‖v′‖L∞(0,T;V).

We use the result given in Theorem 3.1 to prove the existence and uniqueness of a weak solution of Prob. (1.1) – (1.3). Hence, we get the main result in this section as follows.

Theorem 3.4

Go to

Abstract
1. Introduction
2. Preliminaries
3. The Existence and Uniqueness Theorem
Lemma 3.2
Lemma 3.3
Theorem 3.4
4. Asymptotic Expansion of the Solution with respect to a Small Parameter
Acknowledgements
References

Let (H₁) – (H₅) hold. Then

(i) Prob. (1.1) – (1.3) has a unique weak solution u ∈ W₁(M, T), where M >0 and T > 0 are chosen constants as in Theorem 3.1.

Furthermore,

(ii) The recurrent sequence {u_m} defined by (3.11) – (3.13) converges to the solution u of Prob. (1.1) – (1.3) strongly in W₁(T).

And we have the estimate

(3.52)‖um-u‖W1(T)≤CTkTm, for all m∈ℕ,

where the constant k_T ∈ [0, 1) is defined as in (3.46) and C_T is a constant depending only on T, h₀, f, g, μ₁, μ₂, ũ₀, ũ₁and k_T.

Proof

(a) Existence of the solution

We shall prove that {u_m} is a Cauchy sequence in W₁(T). Let w_m = u_m₊₁ − u_m. Then w_m satisfies the variational problem

(3.53){〈wm″(t),v〉+λa(wm′(t),v)+a1(m+1)(t;wm (t),v)=〈Fm+1 (t)-Fm (t),v〉+∫0tg (t-s) a2(m+1)(s;wm (s),v) ds-a¯1(m)(t;um (t),v)+∫0tg (t-s) a¯2(m)(s;um (s),v) ds, ∀v∈V,wm(0)=wm′(0)=0,

where

(3.54)a¯i(m)(t;u,φ)=ai(m+1)(t;u,φ)-ai(m)(t;u,φ) =〈(μi(m+1)(t)-μi(m)(t)) ux,φx〉 +h0 (μi(m+1)(0,t)-μi(m)(0,t)) u(0) φ(0),

∀u, φ ∈ V, with i = 1, 2.

Note that

(3.55)|a¯i(m)(t;u,φ)|≤‖μi(m+1)(t)-μi(m)(t)‖C0(Ω¯)‖u‖a‖φ‖a, ∀u, φ∈V,ddta1(m+1)(t;wm(t),wm(t))=2a1(m+1)(t;wm (t),wm′(t))+a˙1(m+1)(t;wm (t),wm (t)).

Taking w=wm′(t) in (3.53)₁, after integrating in t, we get

(3.56)Zm(t)=∫0ta˙1(m+1)(s;wm(s),wm(s))ds +2∫0t〈Fm+1(s)-Fm(s),wm′(s)〉 ds +2∫0tdτ∫0τg (τ-s) a2(m+1)(s;wm(s),wm′(τ)) ds -2∫0ta¯1(m)(s;um (s),wm′(s)) ds +2∫0tdτ∫0τg (τ-s) a¯2(m)(s;um (s),wm′(τ)) ds ≡J1+J2+J3+J4+J5,

where

(3.57)Zm(t)=‖wm′(t)‖2+ a1(m+1)(t;wm(t),wm(t))+2λ∫0t‖wm′(s)‖a2ds ≥‖wm′(t)‖2+μ*‖wm(t)‖a2+ 2λ∫0t‖wm′(s)‖a2ds,

and the integrals on the right - hand side of (3.56) are estimated as follows.

First integral J₁

By Lemma 3.2, (ii) and (3.57), we have

(3.58)∣J1∣ ≤∫0t|a˙1(m+1)(s;wm(s),wm(s))|ds≤γMK˜M(μ1)μ*∫0tZm(s)ds.

Second integral J₂

By the following inequality

(3.59) ‖Fm+1(t)-Fm(t)‖≤2(1+2M)KM(f) [‖∇wm-1(t)‖+‖wm-1′(t)‖]≤2(1+2M)KM(f)‖wm-1‖W1(T),

we obtain

(3.60)|J2|≤2∫0t‖Fm+1(s)-Fm(s)‖ ‖wm′(s)‖ds ≤4(1+2M)KM(f)‖wm-1‖W1(T)∫0t‖wm′(s)‖ds ≤4T(1+2M)2KM2(f)‖wm-1‖W1(T)2+∫0tZm(s)ds.

Third integral J₃

By Lemma 3.2, (iii) and (3.57), we have

(3.61)∣J3∣≤2∫0tdτ∫0τ∣g (τ-s)∣|a2(m+1)(s;wm(s),wm′(τ))|ds≤2K˜M (μ2)∫0tdτ∫0τ∣g (τ-s)∣ ‖wm (s)‖a‖wm′(τ)‖ads≤2μ*K˜M (μ2)T*‖g‖L2(0,T*)(∫0t‖wm′(τ)‖a2dτ)1/2(∫0tZm(s)ds)1/2≤2βλ∫0t‖wm′(τ)‖a2dτ+12βλμ*K˜M2(μ2)T*‖g‖L2(0,T*)2∫0tZm(s)ds≤βZm(t)+12βλμ*K˜M2(μ2)T*‖g‖L2(0,T*)2∫0tZm(s)ds, ∀β>0.

Fourth integral J₄

By (3.55)₁ and the following inequality

(3.62)‖μi(m+1)(s)-μi(m)(s)‖C0(Ω¯)≤(1+4M)K˜M(μi)‖∇wm-1(s)‖≤(1+4M)K˜M(μi)‖wm-1‖W1(T), i=1,2,

we obtain that

(3.63)J4=-2∫0ta¯1(m)(s;um(s),wm′(s)) ds≤2∫0t‖μ1(m+1)(s)-μ1(m)(s)‖C0(Ω)‖um (s)‖a‖wm′(s)‖ads≤2(1+4M)K˜M(μ1)‖wm-1‖W1(T)∫0t‖um (s)‖a‖wm′(s)‖ads≤2(1+4M)MK˜M(μ1)‖wm-1‖W1(T)∫0t‖wm′(s)‖ads≤2βλ∫0t‖wm′(s)‖a2ds+12βλT(1+4M)2M2K˜M2 (μ1)‖wm-1‖W1(T)2≤βZm(t)+12βλT(1+4M)2M2K˜M2 (μ1)‖wm-1‖W1(T)2, ∀β>0.

Fifth term J₅

Similarly to (3.63), we have

(3.64)J5=2∫0tdτ∫0τg (τ-s) a¯2(m)(s;um(s),wm′(τ)) ds≤2∫0tdτ∫0τ∣g (τ-s)∣‖μ2(m+1)(s)-μ2(m)(s)‖C0(Ω¯)‖um (s)‖a‖wm′(τ)‖ads≤2(1+4M)MK˜M(μ2)‖wm-1‖W1(T)∫0tdτ∫0τ∣g (τ-s)∣ ‖wm′(τ)‖ads≤2(1+4M)MK˜M(μ2)‖wm-1‖W1(T)‖g‖L1(0,T*)∫0t‖wm′(τ)‖adτ≤2βλ∫0t‖wm′(s)‖a2ds+12βλT(1+4M)2M2K˜M2 (μ2)‖g‖L1(0,T*)2‖wm-1‖W1(T)2≤βZm(t)+12βλT(1+4M)2M2K˜M2 (μ2)‖g‖L1(0,T*)2‖wm-1‖W1(T)2, ∀β>0.

Choosing β=16, it follows from (3.56), (3.58), (3.60), (3.61), (3.63), (3.64) that

(3.65)Zm(t)≤TD¯1(M)‖wm-1‖W1(T)2+ 2D¯2(M)∫0tZm(s)ds,

where D̄₁(M) and D̄₂(M) are the constants as in (3.47).

Using Gronwall’s Lemma, we deduce from (3.65) that

(3.66)‖wm‖W1(T)≤kT‖wm-1‖W1(T), ∀m∈ℕ,

where k_T ∈ (0, 1) is defined as in (3.46), which implies that

(3.67)‖um-um+p‖W1(T)≤ ‖u0-u1‖W1(T)(1-kT)-1kTm, ∀m,p∈ℕ.

It follows that {u_m} is a Cauchy sequence in W₁(T). Then there exists u ∈ W₁(T) such that

(3.68)um→u strongly in W1(T).

Note that u_m ∈ W(M, T), then there exists a subsequence {u_{m_j}} of {u_m} such that

(3.69){umj→uinL∞(0,T;V∩H2) weak*,umj′→u′inL∞(0,T;V∩H2) weak*,umj″→u″inL2(0,T;V) weak,um∈W(M,T).

We note that

(3.70)‖Fm-f[u]‖L∞(0,T;L2)≤2(1+2M)KM(f)‖um-1-u‖W1(T).

Hence, we deduce from (3.68) and (3.70) that

(3.71)Fm→f[u] strongly in L∞(0,T;L2).

We also note that

(3.72)‖μi(m)(t)-μi[u] (t)‖C0(Ω¯)≤(1+4M)K˜M(μi)‖um-1-u‖W1(T), a.e. t∈(0,T), i=1,2.

On the other hand, for all v ∈ V, we have

(3.73)|a1(m)(t;um(t),v)-a1[u](t;u(t),v)|≤|a1(m)(t;um(t),v)-a1[u](t;um(t),v)|+∣a1[u](t;um(t)-u(t),v)∣≤‖μ1(m)(t)-μ1[u](t)‖C0(Ω¯)‖um(t)‖a‖v‖a+K˜M(μ1)‖um(t)-u(t)‖a‖v‖a≤K˜M(μ1)[M(1+4M)‖um-1-u‖W1(T)+‖um-u‖W1(T)]‖v‖a.

Hence

(3.74)|∫0Ta1(m)(t;um(t),v)φ(t)dt-∫0Ta1[u](t;u(t),v)φ(t)dt|≤K˜M(μ1) [M(1+4M)‖um-1-u‖W1(T)+‖um-u‖W1(T)]‖v‖a‖φ‖L1(0,T)→0,∀v∈V,∀φ∈L1(0,T).

Similarly

(3.75)|∫0T(∫0tg (t-s) a2(m)(s;um(s),v) ds) φ(t)dt-∫0T(∫0tg (t-s) a2 [u](s;u (s),v) ds) φ(t)dt|≤K˜M(μ2)‖g‖L1(0,T) [M(1+4M)‖um-1-u‖W1(T)+‖um-u‖W1(T)]‖v‖a‖φ‖L1(0,T)→0,

∀v ∈ V, ∀φ ∈ L¹(0, T).

Finally, passing to limit in (3.12), (3.13) as m = m_j → ∞, it implies from (3.68), (3.69), (3.71), (3.74) and (3.75) that there exists u ∈ W(M, T) satisfying the equation

(3.76)〈u″(t),w〉+λa(u′(t),w)+a1[u](t;u(t),w)=∫0tg (t-s) a2[u] (s;u (s),w) ds+〈f[u](t),w〉,

for all w ∈ V and the initial conditions

(3.77)u(0)=u˜0, u′(0)=u˜1.

On the other hand, we have from (H₂), (H₃), (H₆), and (3.69)_1,2,3, that

(3.78)utt=λuxxt+μ1[u]uxx+∂∂x(μ1[u]) ux -∫0tg (t-s) μ2[u] (x,s) uxx (x,s) ds -∫0tg (t-s)∂∂x(μ2[u]) (x,s) ux (x,s) ds+f[μ] ≡F˜∈L∞(0,T;L2).

Thus u ∈ W₁(M, T). The existence result follows.

(b) Uniqueness of the solution

Let u₁, u₂ ∈ W₁(M, T) be two weak solutions of Prob. (1.1) – (1.3). Then u = u₁ − u₂ satisfies the variational problem

(3.79){〈u″(t),w〉+λa(u′(t),w)+a1[u1](t;u(t),w)=〈f[u1](t)-f[u2](t),w〉+∫0tg (t-s) a2[u1] (s;u (s),w) ds-〈(μ1[u1](t)-μ1[u2](t)) u2x(t),wx〉-h0 (μ1[u1](0,t)-μ1[u2](0,t)) u2(0,t)w(0)+∫0tg (t-s) [〈(μ2[u1](s)-μ2[u2](s) ) u2x(s),wx〉+h0 (μ2[u1](0,s)-μ2[u2](0,s) ) u2 (0,s) w(0)] ds, ∀w∈V,u(0)=u′(0)=0,

where

(3.80)aj[ui](t;v,w)=〈μj[ui](t)vx,wx〉 +h0μj[ui](0,t)v(0)w(0), v, w∈V,μj[ui](x,t)=μj (x,t,ui (x,t),‖ui(t)‖2,‖uix(t)‖2), i,j=1,2.

We take w = u′ in (3.79)₁ and integrate in t to get

(3.81)Z(t)=∫0ta˙1[u1](s;u(s),u(s))ds+2∫0t〈f[u1](s)-f[u2](s),u′(s)〉 ds+2∫0tdτ∫0τg(τ-s) a2[u1] (s;u (s),u′(τ)) ds-2∫0t[〈(μ1[u1](s)-μ1[u2](s) ) u2x(s),ux′(s)〉+h0(μ1[u1](0,s)-μ1[u2](0,s) ) u2(0,s) u′(0,s)] ds+2∫0tdτ∫0τg(τ-s) [〈(μ2[u1](s)-μ2[u2](s) ) u2x(s),ux′(τ)〉+h0 (μ2[u1](0,s)-μ2[u2](0,s) ) u2 (0,s) u′(0,τ)] ds,

where

(3.82)Z(t)=‖u′(t)‖2+a1[u1](t;u(t),u(t))+2λ∫0t‖u′(s)‖a2ds.

Put

Z˜M=2γMK˜M(μ1)μ*+8 (1+1μ*) (1+M)KM(f)+6λμ*[K˜M2(μ2)T‖g‖L2(0,T*)2+(1+4M)2M2 (K˜M2(μ1)+K˜M2(μ2)T‖g‖L2(0,T*)2)],

then it follows from (3.81) that

Z(t)≤Z˜M∫0tZ(s)ds.

By Gronwall’s Lemma, we deduce Z(t) = 0, i.e., u₁ ≡ u₂. Theorem 3.4 is proved completely.

4. Asymptotic Expansion of the Solution with respect to a Small Parameter

Go to

Abstract
1. Introduction
2. Preliminaries
3. The Existence and Uniqueness Theorem
Lemma 3.2
Lemma 3.3
Theorem 3.4
4. Asymptotic Expansion of the Solution with respect to a Small Parameter
Acknowledgements
References

In this section, let (H₁) – (H₅) hold. We also make the following assumptions:

(H₆) f1∈C1([0,1]×[0,T*]×ℝ3×ℝ+2).

We consider the following perturbed problem, where ɛ is a small parameter, with |ɛ| < 1:

(Pɛ){utt-λuxxt-∂∂x[μ1[u](x,t)ux]+∫0tg (t-s)∂∂x[μ2[u](x,s)ux (x,s)] ds=Fɛ[u](x,t), 0<x<1,0<t<T,ux(0,t)-h0u(0,t)=u(1,t)=0,u(x,0)=u˜0(x), ut(x,0)=u˜1(x),μi[u](x,t)=μi (x,t,u(x,t),‖u(t)‖2,‖ux(t)‖2),i=1,2,Fɛ[u](x,t)=f[u](x,t)+ɛf1[u](x,t),f[u](x,t)=f(x,t,u,ux,ut,‖u(t)‖2,‖ux(t)‖2).

By the assumptions (H₁) – (H₆) and theorem 3.4, Prob. (P_ɛ) has a unique weak solution u depending on ɛ: u = u_ɛ. When ɛ = 0, (P_ɛ) is denoted by (P₀). We shall study the asymptotic expansion of the solution u_ɛ of Prob. (P_ɛ) with respect to a small parameter ɛ.

We use the following notations. For a multi-index α=(α1,…,αN)∈ℤ+N, and x = (x₁, …, x_N) ∈ ℝ^N, we put

{|α|=α1+…+αN, α!=α1!…αN!,xα=x1α1…xNαN,α, β∈ℤ+N, α≤β⇔αi≤βi ∀i=1,…,N.

First, we shall need the following lemma.

Lemma 4.1

Let m, N ∈ ℕ, x = (x₁, …, x_N) ∈ ℝ^N, and ɛ ∈ ℝ. Then

(4.1)(∑i=1Nxiɛi)m=∑k=mmNPN[m][x]kɛk,

where the coefficientsPN[m][x]k, m ≤ k ≤ mN depending on x = (x₁, …, x_N) are defined by the formula

(4.2)PN[m][x]k={xk,1≤k≤N, m=1,∑α∈Ak[m](N)m!α!xα,m≤k≤mN, m≥2,

withAk[m](N)={α∈ℤ+N:|α|=m, ∑i=1Niαi=k}.

Proof

The proof of this lemma is easy, hence we omit the details.

Now, we assume that

(H₇) μ1,μ2∈CN+1([0,1]×[0,T*]×ℝ×ℝ+2),μ1(x,t,z1,z2,z3)≥μ*>0, for all (x,t,z1,z2,z3)∈[0,1]×[0,T*]×ℝ×ℝ+2;
(H₈) f∈CN+1([0,1]×[0,T*]×ℝ3×ℝ+2), f1∈CN([0,1]×[0,T*])×ℝ3×ℝ+2).

We also use the notations

f[u](x,t)=f(x,t,u,ux,ut,‖u(t)‖2,‖ux(t)‖2),μi[u](x,t)=μi (x,t,u,‖u(t)‖2,‖ux(t)‖2), i=1,2.

Let u₀ be a unique weak solution of problem (P₀) (as in Theorem 3.4) corresponding to ɛ = 0, i.e.,

(P0){u0″-λΔu0′-∂∂x[μ1[u0](x,t)u0x]+∫0tg(t-s)∂∂x[μ2[u0](x,s)u0x(x,s)] ds=f[u0], 0<x<1, 0<t<T,u0x(0,t)-h0u0(0,t)=u0(1,t)=0,u0(x,0)=u˜0(x), u0′(x,0)=u˜1(x),u0∈W1(M,T).

Considering the sequence of weak solutions u_r, 1 ≤ r ≤ N, of the following problems:

(P˜r){ur″-λΔur′-∂∂x[μ1 [u0](x,t)urx]+∫0tg (t-s)∂∂x[μ2[u0](x,s)urx (x,s)] ds=Fr,0<x<1,0<t<T,urx(0,t)-h0ur(0,t)=ur(1,t)=0,ur(x,0)=ur′(x,0)=0,ur∈W1(M,T),

where F_r, 1 ≤ r ≤ N, are defined by the recurrence formulas

(4.3)Fr={f[u0],r=0,πr[N,f]+πr-1[N-1,f1] +∑i=1r∂∂x(ρi[μ1]∇ur-i-∫0tg (t-s) ρi[μ2]∇ur-i(s)ds),1≤r≤N,

with π_r[N, f] = π_r[N, f; u₀, u₁, …, u_r], ρ_r[μ_i] = ρ_r[μ_i, N, u₀, ū, σ⁽¹⁾, σ⁽²⁾], 0 ≤ r ≤ N, defined by the formulas:

Formula π_r[N, f] = π_r[N, f; u₀, ū, σ⁽¹⁾, σ⁽²⁾]: (4.4)πr[N,f]={f[u0],r=0,∑1≤|m|≤r1m!Dmf[u0]Φ˜r[m,N,u¯,σ(1),σ(2)],1≤r≤N,
in which
(4.5)Φ˜r[m,N,u¯,σ(1),σ(2)]=∑(k1,…,k5)∈A¯r(m,N)PN[m1][u¯]k1PN[m2][∇u¯]k2PN[m3][u¯′]k3P2N[m4][σ(1)]k4P2N[m5][σ(2)]k5
and A¯r(m,N)={(k1,…,k5)∈ℤ+5:k1+…+k5=r,mi≤ki≤miN,i=1,…,3;mj≤kj≤2mjN,j=4,5},m=(m1,…,m5)∈ℤ+5,∣m∣=m1+…+m5,m!=m1!…m5!,Dmf=D3m1D4m2D5m3D6m4D7m5f, and σ(1)=(σ1(1),…,σ2N(1)),σ(2)=(σ1(2),…,σ2N(2))
, are defined by
(4.6)σi(1)={2〈u0,u1〉,i=12〈u0,ui〉+∑j=1i〈uj,ui-j〉,2≤i≤N,∑j=1i〈uj,ui-j〉,N+1≤i≤2N,σi(2)={2〈∇u0,∇u1〉,i=12〈∇u0,∇ui〉+∑j=1i〈∇uj,∇ui-j〉,2≤i≤N,∑j=1i〈∇uj,∇ui-j〉,N+1≤i≤2N;
Formula ρ_r[μ_i] = ρ_r[μ_i, N, u₀, ū, σ⁽¹⁾, σ⁽²⁾]:
(4.7)ρr[μi]={μi[u0]r=0,∑1≤∣γ∣≤r1γ!Dγμi[u0]Φr[N,u¯,σ(1),σ(2),γ],1≤r≤N,
where (4.8)Φr[N,u¯,σ(1),σ(2),γ]=∑(k1,k2,k3)∈A˜r(γ,N)PN[γ1][u¯]k1P2N[γ2][σ(1)]k2P2N[γ3][σ(2)]k3,
with γ=(γ1,γ2,γ3)∈ℤ+3, 1 ≤ |γ| = γ₁ + … + γ₃ ≤ r, γ! = γ₁!…γ₃!, Dγμi=D3γ1D4γ2D5γ3μi,A˜r(γ,N)={(k1,k2,k3)∈ℤ+3:|γ|=r,γ1≤k1≤γ1N,γ2≤k2≤2γ2N,γ3≤k3≤2γ3N}. Then, we have the following lemma.

Lemma 4.2

Let π_r[N, f] = π_r[N, f; u₀, ū, σ⁽¹⁾, σ⁽²⁾], ρ_r[μ_i] = ρ_r[μ_i, N, u₀, ū, σ⁽¹⁾, σ⁽²⁾], 0 ≤ r ≤ N, be the functions defined by formulas (4.4) and (4.7). Leth=∑r=0Nurɛr. Then we have

(4.9)f(h)=∑r=0Nπr[N,f]ɛr+|ɛ|N+1R¯N(1)[f,ɛ],

(4.10)μi[h]=∑r=0Nρr[μi]ɛr+|ɛ|N+1R˜N(1)[μi,ɛ],

with‖R˜N(1)[μi,ɛ]‖L∞(0,T;L2)+‖R¯N(1)[f,ɛ]‖L∞(0,T;L2)≤C, where C is a constant depending only on N, T, f, μ₁, μ₂, u_r, 0 ≤ r ≤ N.

Proof

(i) In the case of N = 1, the proof of (4.9) is easy, hence we omit the details. We only prove the case of N ≥ 2. Let h=u0+∑i=1Nuiɛi≡u0+h1. We rewrite as below

(4.11)f[h]=f(x,t,h(x,t),∇h(x,t),h′(x,t),‖h(t)‖2,‖∇h(t)‖2)=f(x,t,u0+h1,∇u0+∇h1,u0′+h1′,‖u0+h1‖2,‖∇u0+∇h1‖2)=f(x,t,u0+h1,∇u0+∇h1,u0′+h1′,‖u0‖2+ξ4,‖∇u0‖2+ξ5),

where ξ₄ = ||u₀ + h₁||² − ||u₀||², ξ₅ = ||∇u₀ + ∇h₁||² − ||∇u₀||².

By using Taylor’s expansion of the function f[u₀ + h₁] around the point [u0]=(x,t,u0,∇u0,u0′,‖u0‖2,‖∇u0‖2) up to order N + 1, we obtain

(4.12)f[u0+h1]=f[u0]+D3f[u0]h1+D4f[u0]∇h1+D5f[u0]h1′+D6f[u0]ξ4+D7f[u0]ξ5+∑2≤|m|≤Nm=(m1,…,m5)∈ℤ+51m!Dmf[u0]h1m1(∇h1)m2(h1′)m3ξ4m4ξ4m5+RN(1)[f,h1,ξ4,ξ5],

where

(4.13)RN(1)[f,h1,ξ4,ξ5]=∑|m|=N+1m∈ℤ+5N+1m!(∫01(1-θ)NDmf(θ)dθ)h1m1(∇h1)m2(h1′)m3ξ4m4ξ5m5=|ɛ|N+1R¯N(1)[f,h1,ξ4,ξ5],Dmf(θ)=Dmf(x,t,u0+θh1,∇u0+θ∇h1,u0′+θh1′,‖u0‖2+θξ4,‖∇u0‖2+θξ5).

By the formula (4.1), it follows that

(4.14)h1m1=(∑i=1Nuiɛi)m1=∑k=m1m1NPN[m1][u¯]kɛk,(∇h1)m2=(∑i=1N∇uiɛi)m2=∑k=m2m2NPN[m2][∇u¯]kɛk,(h1′)m3=(∑i=1Nui′ɛi)m3=∑k=m3m3NPN[m3][u¯′]kɛk,

where ū = (u₁, …, u_N), ∇ū = (∇u₁, …, ∇u_N), u¯′=(u1′,…,uN′).

On the other hand,

(4.15)ξ4=‖u0+h1‖2-‖u0‖2=2〈u0,h1〉+‖h1‖2≡∑i=12Nσi(1)ɛi,

with σi(1), 1 ≤ i ≤ 2N are defined by (4.6)₁.

By the formula (4.1), it follows from (4.15) that

(4.16)ξ4m4=(∑i=12Nσi(1)ɛi)m4=∑k=m42m4NP2N[m4][σ(1)]kɛ4,

where σ(1)=(σ1(1),…,σ2N(1)).

Similarly, we have

(4.17)ξ5m5=(∑i=12Nσi(2)ɛi)m5=∑k=m52m5NP2N[m5][σ(2)]kɛk,

where σ(2)=(σ1(2),…,σ2N(2)), are defined by (4.6)₂.

Therefore, it follows from (4.14), (4.16), (4.17) that

(4.18)h1m1(∇h1)m2(h1′)m3ξ4m4ξ5m5=∑r=|m|NΦ˜r[m,N,u¯,σ(1),σ(2)]ɛr+|ɛ|N+1R¯N(2)[m,N,u¯,σ(1),σ(2)],

where

(4.19)Φ˜r[m,N,u¯,σ(1),σ(2)]=∑(k1,…,k5)∈A¯r(m,N)PN[m1][u¯]k1PN[m2][∇u¯]k2PN[m3][u¯′]k3P2N[m4][σ(1)]k4P2N[m5][σ(2)]k5,

with Ā_r(m, N) as in (4.5) and

|ɛ|N+1R¯N(2)[m,N,u¯,σ(1),σ(2)]=∑r=N+1(|m|+m4+m5)NΦ˜r[m,N,u¯,σ(1),σ(2)]ɛr.

Hence, we deduce from (4.12), (4.18) that

(4.20)f[u0+h1]=f[u0]+∑1≤|m|≤Nm=(m1,…,m5)∈ℤ+51m!Dmf[u0]∑r=|m|NΦ˜r[m,N,u¯,σ(1),σ(2)]ɛr+|ɛ|N+1R¯N(1)[f,ɛ]=f[u0]+∑r=1N(∑1≤|m|≤r1m!Dmf[u0]Φ˜r[m,N,u¯,σ(1),σ(2)])ɛr+|ɛ|N+1R¯N(1)[f,ɛ]=∑r=0Nπr[N,f]ɛr+|ɛ|N+1R¯N(1)[f,ɛ],

where

(4.21)R¯N(1)[f,ɛ]=∑2≤|m|≤Nm=(m1,…,m5)∈ℤ+51m!Dmf[u0]R¯N(2)[m,N,u¯,σ(1),σ(2)]+R¯N(1)[f,h1,ξ4,ξ5],

with π_r[N, f], 0 ≤ r ≤ N are defined by (4.4), and ‖R¯N(1)[f,ɛ]‖L∞(0,T;L2)≤C, where C is a constant depending only on N, T, f, u_r, r = 0, 1, …, N.

Hence, the formula (4.9) is proved.

(ii) In the case of μ_i[h] = μ_i (x, t, h(x, t), ||h(t)||², ||∇h(t)||²). Applying the formulas (4.4) – (4.6) and (4.9) with f = f(x, t, z₁, z₂, z₃), D_jf = 0, j = 4, 5, D3f=D3μi=∂μi∂z1,D6f=D4μi=∂μi∂z2,D7f=D5μi=∂μi∂z3 and π_r[N, f] = ρ_r[μ_i], 0 ≤ r ≤ N, we obtain formulas (4.6) – (4.8) and the formula (4.10) is proved.

This completes the proof of the lemma 4.2.

Remark 4.1

Lemma 4.2 is a generalization of a formula contained in [15] (formula (4.38), p. 262) and it is useful to obtain the following Lemma 4.3. These Lemmas are the key to obtain the asymptotic expansion of the weak solution u = u_ɛ of order N + 1 in a small parameter ɛ.

Let u = u_ɛ ∈ W₁(M, T) be a unique weak solution of the problem (P_ɛ). Then v=u-∑r=0Nurɛr≡u-h=u-u0-h1 satisfies the problem

(4.22){v″-λΔv′-∂∂x[μ1[v+h]vx]+∫0tg(t-s)∂∂x[μ2[v+h]vx(x,s)]ds=f[v+h]-f[h]+ɛ(f1[v+h]-f1[h])+∂∂x[(μ1[v+h]-μ1[h])hx]-∫0tg(t-s)∂∂x[(μ2[v+h]-μ2[h])hx(x,s)]ds+Eɛ(x,t),0<x<1,0<t<T,vx(0,t)-h0v(0,t)=v(1,t)=0,v(x,0)=v′(x,0)=0,

where

(4.23)Eɛ(x,t)=f[h]-f[u0]+ɛf1[h]+∂∂x[(μ1[h]-μ1[u0])hx]-∫0tg(t-s)∂∂x[(μ2[h]-μ2[u0])hx(x,s)]ds-∑r=1NFrɛr.

Lemma 4.3

Under the assumptions (H₁), (H₂), (H₇) and (H₈), there exists a constant C̄_*such that

(4.24)‖Eɛ‖L∞(0,T;L2)≤C¯*|ɛ|N+1,

where C̄_*is a constant depending only on N, T, f, f₁, μ₁, μ₂, u_r, 0 ≤ r ≤ N.

Proof

In the case of N = 1, the proof of Lemma 4.3 is easy. The details are omitted. We only consider N ≥ 2.

By using formulas (4.9), (4.10) for the functions f₁[h], μ₁[h] and μ₂[h], we obtain

(4.25)f1[h]=∑r=0N-1πr[N-1,f1]ɛr+|ɛ|NR¯N-1(1)[f1,ɛ],μi[h]=∑r=0Nρr[μi]ɛr+|ɛ|N+1R˜N(1)[μi,ɛ], i=1,2.

By (4.25)₁, we rewrite ɛf₁[h] as follows

(4.26)ɛf1[h]=∑r=1Nπr-1[N-1,f1]ɛr+ɛ|ɛ|NR¯N-1(1)[f1,ɛ].

First, we deduce from (4.9) and (4.26), that

(4.27)f[h]-f[u0]+ɛf1[h]=∑r=1N(πr[N,f]+πr-1[N-1],f1)ɛr+|ɛ|N+1R¯N(1)[f,f1,ɛ],

where R¯N(1)[f,f1,ɛ]=R¯N(1)[f,ɛ]+ɛ|ɛ|R¯N-1(1)[f1,ɛ] is bounded in L^∞(0, T; L²) by a constant depending only on N, T, f, f₁, u_r, 0 ≤ r ≤ N.

On the other hand, we deduce from (4.10) and (4.25)₂ that

(4.28)∂∂x[(μ1[h]-μ1[u0])hx]=∂∂x[(∑r=1Nρr[μ1]ɛr+|ɛ|N+1R˜n(1)[μ1,ɛ])hx]=∑r=1N∑i=1r∂∂x(ρi[μ1]∇ur-i)ɛr+|ɛ|N+1R˜N2[μ1,ɛ],

where

(4.29)R˜N(2)[μ1,ɛ]=∂∂x[R˜N(1)[μ1,ɛ]hx+1|ɛ|N+1∑r=N+12N∑i=1rρi[μ1]∇ur-iɛr].

Similarly

(4.30)∂∂x[(μ2[h]-μ2[u0])hx]=∑r=1N∑i=1r∂∂x(ρi[μ2]∇ur-i)ɛr+|ɛ|N+1R˜N(2)[μ2,ɛ],

where

(4.31)R˜N(2)[μ2,ɛ]=∂∂x[R˜N(1)[μ2,ɛ]hx+1|ɛ|N+1∑r=N+12N∑i=1rρi[μ2]∇ur-iɛr].

Combining (4.3), (4.4), (4.7), (4.23), (4.27), (4.28) and (4.30), we then obtain

(4.32)Eɛ(x,t)=|ɛ|N+1(R¯N(1)[f,f1,ɛ]+R˜N(2)[μ1,ɛ]-∫0tg(t-s)R˜N(2)[μ2,ɛ]ds)

By the functions u_r ∈ W₁(M, T), 0 ≤ r ≤ N, we obtain from (4.27), (4.29), (4.31) and (4.32) that

(4.33)‖Eɛ‖L∞(0,T;L2)≤C¯*|ɛ|N+1,

where C̄_* is a constant depending only on N, T, f, f₁, μ₁, μ₂, u_r, 0 ≤ r ≤ N.

This completes thee proof of lemma 4.3.

Now, we estimate v=u-∑r=0Nurɛr.

By multiplying the two sides of (4.22) by v′, we verify without difficulty that

(4.34)S(t)=2∫0t〈Eɛ(s),v′(s)〉ds+∫0ta˙1[v+h](s,v(s),v(s))ds-2∫0t[a1[v+h](s,h(s),v′(s))-a1[h](s,h(s),v′(s))]ds+2∫0tdτ∫0τg(τ-s)a2[v+h](s,v(s),v′(τ))ds+2∫0t〈f[v+h]-f[h]+ɛ(f1[v+h]-f1[h]),v′(s)〉ds+2∫0tdt∫0τg(τ-s) [a2[v+h](s,h(s),v′(τ))-a2[h](s,h(s),v′(τ))]ds≡∑i=16Si,

where

(4.35)S(t)=‖v′(t)‖2+a1[v+h](t,v(t),v(t))+2λ∫0t‖v′(s)‖a2ds ≥‖v′(t)‖2+μ*‖v(t)‖a2+2λ∫0t‖v′(s)‖a2ds.

Put M_* = (N + 2)M, it is not difficult to prove that the following inequalities hold

(4.36)S1=2∫0t〈Eɛ(s),v′(s)〉ds≤TC¯*2∣ɛ∣2N+2+∫0tS(s)ds;S2=∫0ta˙1[v+h](s,v(s),v(s))ds≤γM*K˜M*(μ1)∫0t‖v(s)‖a2ds ≤1μ*γM*K˜M*(μ1)∫0tS(s)ds;S3=-2∫0t[a1[v+h](s,h(s),v′(s))-a1[h](s,h(s),v′(s))]ds ≤2(1+4M*)M*K˜M*(μ1)∫0t‖v(s)‖a‖v′(s)‖ads ≤16S(t)+3λμ*(1+4M*)2M*2K˜M*2(μ1)∫0tS(s)ds;S4=2∫0tdτ∫0τg(τ-s)a2[v+h](s,v(s),v′(τ))ds ≤2K˜M*(μ2)∫0tdτ∫0τ∣g(τ-s)∣ ‖v(s)‖a‖v′(τ)‖ads ≤16S(t)+3λμ*TK˜M*2(μ2)‖g‖L2(0,T*)2∫0tS(s)ds;S5=2∫0t〈f[v+h]-f[h]+ɛ(f1[v+h]-f1[h]),v′(s)〉ds ≤8(1+1μ*)(1+2M*) (KM*(f)+KM*(f1))∫0tS(s)ds;S6=2∫0tdτ∫0τg(τ-s) [a2[v+h](s,h(s),v′(τ))-a2[h](s,h(s),v′(τ))]ds ≤2∫0tdτ∫0τ∣g(τ-s)∣ ‖μ2[v+h](s)-μ2[h](s)‖C0([0,1])‖h(s)‖a ‖v′(τ)‖ads ≤2(1+4M*)M*K˜M*(μ2)∫0tdτ∫0τ∣g(τ-s)∣ ‖v(s)‖a ‖v′(τ)‖ads ≤16S(t)+3λμ*T(1+4M*)2M*2K˜M*2(μ2)‖g‖L2(0,T*)2∫0tS(s)ds.

Combining (4.34) and (4.36), we then obtain

(4.37)S(t)≤2TC¯*2|ɛ|2N+2+2D˜T(M)∫0tS(s)ds,

where

D˜T(M)=1+1μ*γM*K˜M*(μ1)+8(1+1μ*) (1+2M*) (KM*(f)+KM*(f1))+3λμ*[(1+4M*)2M*2K˜M*2(μ1)+(1+(1+4M*)2M*2)T‖g‖L2(0,T*)2K˜M*2(μ2)].

By Gronwall’s lemma, we obtain from (4.37) that

S(t)≤2TC¯*2|ɛ|2N+2exp(2TD˜T(M)).

Hence

‖v‖W1(T)≤(1+1μ*+12λ)C¯*2Texp(TD˜T(M))|ɛ|N+1,

or

(4.38)‖uɛ-∑r=0Nurɛr‖W1(T)≤CT|ɛ|N+1.

Thus, we have the following theorem 4.4.

Theorem 4.4

Let (H₁), (H₂), (H₇) and (H₈) hold. Then there exist constants M >0 and T > 0 such that, for every ɛ, with |ɛ| < 1, Prob. (P_ɛ) has a unique weak solution u_ɛ ∈ W₁(M, T) satisfying an asymptotic estimation up to order N + 1 as in (4.38), where the functions u_r, r = 0, 1, …, N are weak solutions of Prob. (P₀), (P̃_r), r = 0, 1, …, N, respectively.

Remark 4.2

Typical examples about asymptotic expansion of solutions in a small parameter can be found in many papers, such as [14, 15, 16]. In the case of many small parameters, there is only partial results, for example, we refer to [17, 24, 25] for the asymptotic expansion of solutions in two or three small parameters.

Acknowledgements

Go to

Abstract
1. Introduction
2. Preliminaries
3. The Existence and Uniqueness Theorem
Lemma 3.2
Lemma 3.3
Theorem 3.4
4. Asymptotic Expansion of the Solution with respect to a Small Parameter
Acknowledgements
References

The authors wish to express their sincere thanks to the referees for the valuable comments and important remarks.

References

Go to

Abstract
1. Introduction
2. Preliminaries
3. The Existence and Uniqueness Theorem
Lemma 3.2
Lemma 3.3
Theorem 3.4
4. Asymptotic Expansion of the Solution with respect to a Small Parameter
Acknowledgements
References

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