KYUNGPOOK Math. J. 2019; 59(4): 735-769  
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
* Corresponding Author.
Received: September 11, 2018; Accepted: March 4, 2019; Published online: December 23, 2019.
© Kyungpook Mathematical Journal. All rights reserved.

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Abstract

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

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

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

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

and initial conditions

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

ρhutt=(P0+Eh2L0Lux2(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

ρutt-(1+EALT00Lu2dx)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

utt-M(u2)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 = uɛ of order N + 1 in a small parameter ɛ for the equation

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, μ2CN+1([0,1]×[0,T*]××+2), μ1(x, t, z1, z2, z3) ≥ μ* > 0, for all (x,t,z1,z2,z3)[0,1]×[0,T*]××+2,fCN+1([0,1]×[0,T*]×3×+2),f1CN([0,1]×[0,T*]×3×+2). This result is a relative generalization of [14, 15, 16, 17, 24, 25].

2. Preliminaries

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

uLp(0,T;X)={(0Tu(t)Xpdt)1/p,if   1p<,esssup0<t<Tu(t)X,if   p=.

With fCk([0,1]×[0,T*]×3×+2), f = f(x, t, y1, y2, y3, y4, y5), (x, t) ∈ [0, 1] × [0, T*], (y1,y2,y3,y4,y5)3×+2, we put D1f=fx,D2f=ft,Di+2f=fyi, i = 1, …, 5 and Dαf=D1α1D7α7f,α=(α1,,α7)+7, |α| = α1 + … + α7k, D(0,…,0)f = f.

Similarly, with μCk([0,1]×[0,T*]××+2), μ = μi (x, t, z1, z2, z3), (x, t) ∈ [0, 1] × [0, T*], (z1,z2,z3)×+2, we put D1μ=μx,D2μ=μy,Dj+2μ=μyj, j = 1, 2, 3, and Dβμ=D1β1D5β5μ,β=(β1,,β5)+5, |β| = β1 + … + β5k, D(0,…,0)μ = μ.

On H1H1 (Ω), we shall use the following norm

vH1=(v2+vx2)12.

We set

V={vH1(0,1):v(1)=0},

and

a(u,v)=ux,vx+h0u(0)v(0),for all u,vV,

Then, V is a closed subspace of H1 and three norms v ↦ ||v||H1, v ↦ ||vx|| and vva=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 VL2 = (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 H1C0 (Ω̄) is compact and

vC0(Ω¯)2vH1   forallvH1.

Lemma 2.2

Let h0 ≥ 0. Then the embedding VC0 (Ω̄) is compact and

{vC0(Ω¯)vxvaforallvV,12vH1vxva1+h0vH1forallvV.

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.,

(i)|a(u,v)|(1+h0)ux   vx,forallu,vV,(ii)a(v,v)vx2,forallvV.

Lemma 2.4

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

(i)0<λ1λ2λjλj+1,limj+λj=+,(ii)a(wj,v)=λjwj,vforallvV,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, wj satisfies the following boundary value problem

{-Δwj=λjwj,   inΩ,wjx(0)-h0wj(0)=wj(1)=0,wjVC(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

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

uV˜T={vL(0,T;VH2):vL(0,T;VH2),vL2(0,T;V)L(0,T;L2)},

and u satisfies the following variational equation

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 vV, and a.e., t ∈ (0, T), together with the initial conditions

u(0)=u˜0,u(0)=u˜1,

where, for each ūT and i = 1, 2, {ai[ū](t; ·, ·)}0≤tT is the family of symmetric bilinear forms on V × V defined by

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

for all u, vV, 0 ≤ tT, with h0 ≥ 0 is given constant, and

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, ũ1VH2;

(H2) gH1 (0, T*);

(H3) μ1C2([0,1]×[0,T*]××+2),μ1(x,t,y1,y2,y3)μ*>0,(x,t,y1,y2,y3)[0,1]×[0,T*]××+2;

(H4) μ2C1([0,1]×[0,T*]××+2);

(H5) fC1([0,1]×[0,T*]×3×+2).

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

KM(f)=fC1(AM)=fC0(AM)+i=17DifC0(AM),K˜M(μi)=μiC2(A˜M)=|β|2DβμiC0(A˜M),i=1,2,

with

{fC0(AM)=sup(x,t,y1,,y5)AMf(x,t,y1,,y5),μiC0(A˜M)=sup(x,t,z1,,z3)A˜Mμi(x,t,z1,,z3),i=1,2,AM=(x,t,y1,,y5):{0x1,0tT*,y1,y2,y32M,0y4,y5M2},A˜M={(x,t,z1,,z3):0x1,0tT*,z1M,0z2,z3M2}.

For each T ∈ (0, T*], we denote

VT={vL(0,T;VH2):vL(0,T;VH2),vL2(0,T;V)},

it is a Banach space with respect to the norm

vVT=max{vL(0,T;VH2);vL(0,T;VH2);vL2(0,T;V)}.

For every M >0, we put

W(M,T)={vVT:vVTM},W1(M,T)={vW(M,T):vL(0,T;L2)}.

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

um-1W1(M,T),

based on the associate problem (3.2), we find umW1(M, T) (m ≥ 1) satisfying the linear variational problem

{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,vV,um(0)=u˜0,um(0)=u˜1,

where

{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,vV;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 {um} ⊂ 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 {wj} for V given by Lemma 2.4. Put

um(k)(t)=j=1kcmj(k)(t)wj,

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

{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,1jk,um(k)(0)=u˜0k,u˙m(k)(0)=u˜1k,

in which

{u˜0k=j=1kαj(k)wju˜0strongly in VH2,u˜1k=j=1kβj(k)wju˜1strongly in VH2.

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

{c¨mj(k)(t)+λλjc˙mj(k)(t)+i=1ka1ij(m)(t)cmi(k)(t)=i=1k0tg(t-s)a2ij(m)(s)cmi(k)(s)ds+fmj(t),cmj(k)(0)=αj(k),c˙mj(k)(0)=βj(k),1jk,

where

aγij(m)(t)=aγ(m)(t;wi,wj),fmj(t)=Fm(t),wj,γ=1,2,1i,jk.

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 ≤ jk on interval [0,Tm(k)][0,T].

Step 2. A priori estimate

First, we put

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

where

{pm(k)(t)=u˙m(k)(t)2+a1(m)(t;um(k)(t),um(k)(t))+2λ0tu˙m(k)(s)a2ds,qm(k)(t)=u˙m(k)(t)a2+μ1(m)(t)Δum(k)(t)2+2λ0tΔu˙m(k)(s)2ds,rm(k)(t)=λΔu˙m(k)(t)2+20tu¨m(k)(s)a2ds.

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

Sm(k)(t)=Sm(k)(0)+2J1(m,k)(0),Δu˜1k+20tFm(s),u˙m(k)(s)ds+20ta(Fm(s),u˙m(k)(s)+u¨m(k)(s))ds+0ta˙1(m)(s;um(k)(s),um(k)(s))ds+0tds01μ˙1(m)(x,s)|Δum(k)(x,s)|2ds-20tμ1x(m)(s)umx(k)(s),Δu˙m(k)(s)ds+20tJ˙1(m,k)(s),Δu˙m(k)(s)ds-2J1(m,k)(t),Δu˙m(k)(t)+20tdτ0τg(τ-s)a2(m)(s;um(k)(s),u˙m(k)(τ))ds+20tdτ0τg(τ-s)J2(m,k)(s),Δu˙m(k)(τ)ds+20tg(t-s)J2(m,k)(s),Δu˙m(k)(t)ds-2g(0)0tJ2(m,k)(τ),Δu˙m(k)(τ)dτ-20tdτ0τg(τ-s)J2(m,k)(s),Δu˙m(k)(τ)dssm(k)(0)+2J1(m,k)(0),Δu˜1k+j=112Ij,

where

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

Put

a˙i(m)(t;u,v)=μ˙i(m)(t)ux,vx+h0μ˙i(m)(0,t)u(0)v(0),forallu,vV.

Then we have

(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)uava,forallu,vV,(iii)|ai(m)(t;u,v)|K˜M(μi)uava,forallu,vV,t[0,T*],(iv)a1(m)(t;v,v)μ*va2,forallvV,

where γM = 1 + M + 4M2, i = 1, 2.

Lemma 3.3

We have

(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 Ij on the right-hand side of (3.21) as follows.

Estimation of I1

The Cauchy-Schwartz inequality leads to

I1=20tFm(s),u˙m(k)(s)ds20tFm(s)u˙m(k)(s)dsTKM2(f)+0tSm(k)(s)ds.
Estimation of the terms I2 and I3

We note that, by (3.13)2 we have

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

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

2abβa2+1βb2,a,b,β>0,

and

Sm(k)(t)2λ0tu˙m(k)(s)a2ds+20tu¨m(k)(s)a2dsmin{1,λ}0t(u˙m(k)(s)a+u¨m(k)(s)a)2ds,

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

I2=20ta(Fm(s),u˙m(k)(s)+u¨m(k)(s))ds20tFm(s)a(u˙m(k)(s)a+u¨m(k)(s)a)ds1βmin{1,λ}0tFm(s)a2ds+βmin{1,λ}0t(u˙m(k)(s)a+u¨m(k)(s)a)2dsT(4γM2+h0)βmin{1,λ}KM2(f)+βSm(k)(t),β>0;I3=0ta˙1(m)(s;um(k)(s),um(k)(s))dsγMK˜M(μ1)0tum(k)(s)a2dsγMK˜M(μ1)μ*0tSm(k)(s)ds.
Estimation of I4

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

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

|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

I4=0tds01μ˙1(m)(x,s)|Δum(k)(x,s)|2dsγMK˜M(μ1)0tΔum(k)(s)2dsγMK˜M(μ1)μ*0tSm(k)(s)ds.
Estimation of I5

Similarly, by the following formula

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

|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))+λΔu˙m(k)(t)2μ*umx(k)(t)2+λΔu˙m(k)(t)2min{μ*,λ}(umx(k)(t)2+Δu˙m(k)(t)2),

we shall estimate the following term I5 as follows

I5=-20tμ1x(m)(s)umx(k)(s),Δu˙m(k)(s)ds4γMK˜M(μ1)0tumx(k)(s)Δu˙m(k)(s)ds2γMK˜M(μ1)min{μ*,λ}0tSm(k)(s)ds.
Estimation of I6

By Lemma 3.3, (ii) and the inequality Sm(k)(t)λΔu˙m(k)(t)2, we have

I6=20tJ˙1(m,k)(s),Δu˙m(k)(s)ds20tJ˙1(m,k)(s)Δu˙m(k)(s)ds2λd1(M)K˜M(μ1)0tSm(k)(s)ds.
Estimation of I7

By Lemma 3.3, (ii), we have

J1(m,k)(t)2=(J1(m,k)(0)+0tJ˙1(m,k)(s)ds)22J1(m,k)(0)2+2T0tJ˙1(m,k)(s)2ds2J1(m,k)(0)2+2Td12(M)K˜M2(μ1)0tSm(k)(s)ds.

We deduce from the inequalities (3.38) and Sm(k)(t)λΔu˙m(k)(t)2 that

I7=-2J1(m,k)(t),Δu˙m(k)(t)2λJ1(m,k)(t)Sm(k)(t)1βλJ1(m,k)(t)2+βSm(k)(t)2βλJ1(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)a2μ*um(k)(t)a2+u˙m(k)(t)a22μ*um(k)(t)au˙m(k)(t)a,

we have

I8=20tdτ0τg(τ-s)a2(m)(s;um(k)(s),u˙m(k)(τ))ds2K˜M(μ2)0tdτ0τg(τ-s)Sm(k)(s)μ*Sm(k)(τ)ds2μ*K˜M(μ2)0tdτ0τg(τ-s)Sm(k)(s)Sm(k)(τ)ds2μ*K˜M(μ2)TgL2(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)a2μ*um(k)(t)a2+u˙m(k)(t)a22μ*um(k)(t)au˙m(k)(t)a,Sm(k)(t)λΔu˙m(k)(t)2,

it is not difficult to estimate the following terms

I9=20tdτ0τg(τ-s)J2(m,k)(s),Δu˙m(k)(τ)ds20tdτ0τg(τ-s)J2(m,k)(s)Δu˙m(k)(τ)ds2λd0(M)K˜M(μ2)0tdτ0τg(τ-s)Sm(k)(s)Sm(k)(τ)ds2λd0(M)K˜M(μ2)TgL2(0,T*)0tSm(k)(s)ds;I10=20tg(t-s)J2(m,k)(s),Δu˙m(k)(t)ds2λd0(M)K˜M(μ2)0tg(t-s)Sm(k)(s)Sm(k)(t)ds1βλd02(M)K˜M2(μ2)gL2(0,T*)20tSm(k)(s)ds+βSm(k)(t),β>0;I11=-2g(0)0tJ2(m,k)(τ),Δu˙m(k)(τ)dτ2g(0)0tJ2(m,k)(τ)Δu˙m(k)(τ)dτ2λg(0)d0(M)K˜M(μ2)0tSm(k)(τ)dτ;I12=-20tdτ0τg(τ-s)J2(m,k)(s),Δu˙m(k)(τ)ds20tdτ0τg(τ-s)J2(m,k)(s)Δu˙m(k)(τ)ds2λd0(M)K˜M(μ2)0tdτ0τg(τ-s)Sm(k)(s)Sm(k)(τ)ds2λd0(M)K˜M(μ2)TgL2(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

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

where

S¯m,k=2Sm(k)(0)+4J1(m,k)(0),Δu˜1k+24λJ1(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)+TgL2(0,T*)+TgL2(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)gL2(0,T*))K˜M(μ2)gL2(0,T*).
Estimation of m,k

Notice that the formula J1(m,k)(0)=x[μ1(x,0,u˜0(x),u˜02,u˜0x2)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

S¯m,k=2Sm(k)(0)+4J1(m,k)(0),Δu˜1k+24λJ1(m,k)(0)2=2a1(m)(0;u˜0k,u˜0k)+2μ1(m)(0)Δu˜0k2+2u˜1k2+2u˜1ka2+2λΔu˜1k2+4x(μ1(m)(·,0)u˜0kx),Δu˜1k+24λx(μ1(m)(·,0)u˜0k)212M2for all m,k.

Hence, from (3.43)2,3, we can choose T ∈ (0, T*], such that

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

and

kT=(1+12λ+1μ*)TD¯1(M)exp(TD¯2(M))<1,

where

D¯1(M)=8(1+2M)2KM2(f)+6(1+4M)2M2λ(K˜M2(μ1)+K˜M2(μ2)gL1(0,T*)2),D¯2(M)=1+γMK˜M(μ1)μ*+3K˜M2(μ2)T*gL2(0,T*)2λμ*.

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

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

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

um(k)W(M,T),   for all mand k.
Step 3. Limiting process

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

{um(k)uminL(0,T;VH2)weak*,u˙m(k)uminL(0,T;VH2)weak*,u¨m(k)uminL2(0,T;V)weak,umW(M,T).

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

The proof of Theorem 3.1 is complete.

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

vW1(T)=vL(0,T;V)+vL(0,T;L2)+vL(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

Let (H1) – (H5) hold. Then

(i) Prob. (1.1) – (1.3) has a unique weak solution uW1(M, T), where M >0 and T > 0 are chosen constants as in Theorem 3.1.

Furthermore,

(ii) The recurrent sequence {um} 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

um-uW1(T)CTkTm,   forallm,

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

Proof

(a) Existence of the solution

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

{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,vV,wm(0)=wm(0)=0,

where

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, φV, with i = 1, 2.

Note that

|a¯i(m)(t;u,φ)|μi(m+1)(t)-μi(m)(t)C0(Ω¯)uaφa,   u,   φV,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

Zm(t)=0ta˙1(m+1)(s;wm(s),wm(s))ds+20tFm+1(s)-Fm(s),wm(s)ds+20tdτ0τg(τ-s)a2(m+1)(s;wm(s),wm(τ))ds-20ta¯1(m)(s;um(s),wm(s))ds+20tdτ0τg(τ-s)a¯2(m)(s;um(s),wm(τ))dsJ1+J2+J3+J4+J5,

where

Zm(t)=wm(t)2+a1(m+1)(t;wm(t),wm(t))+2λ0twm(s)a2dswm(t)2+μ*wm(t)a2+2λ0twm(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

J10t|a˙1(m+1)(s;wm(s),wm(s))|dsγMK˜M(μ1)μ*0tZm(s)ds.
Second integral J2

By the following inequality

Fm+1(t)-Fm(t)2(1+2M)KM(f)[wm-1(t)+wm-1(t)]2(1+2M)KM(f)wm-1W1(T),

we obtain

|J2|20tFm+1(s)-Fm(s)wm(s)ds4(1+2M)KM(f)wm-1W1(T)0twm(s)ds4T(1+2M)2KM2(f)wm-1W1(T)2+0tZm(s)ds.
Third integral J3

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

J320tdτ0τg(τ-s)|a2(m+1)(s;wm(s),wm(τ))|ds2K˜M(μ2)0tdτ0τg(τ-s)wm(s)awm(τ)ads2μ*K˜M(μ2)T*gL2(0,T*)(0twm(τ)a2dτ)1/2(0tZm(s)ds)1/22βλ0twm(τ)a2dτ+12βλμ*K˜M2(μ2)T*gL2(0,T*)20tZm(s)dsβZm(t)+12βλμ*K˜M2(μ2)T*gL2(0,T*)20tZm(s)ds,   β>0.
Fourth integral J4

By (3.55)1 and the following inequality

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

we obtain that

J4=-20ta¯1(m)(s;um(s),wm(s))ds20tμ1(m+1)(s)-μ1(m)(s)C0(Ω)um(s)awm(s)ads2(1+4M)K˜M(μ1)wm-1W1(T)0tum(s)awm(s)ads2(1+4M)MK˜M(μ1)wm-1W1(T)0twm(s)ads2βλ0twm(s)a2ds+12βλT(1+4M)2M2K˜M2(μ1)wm-1W1(T)2βZm(t)+12βλT(1+4M)2M2K˜M2(μ1)wm-1W1(T)2,   β>0.
Fifth term J5

Similarly to (3.63), we have

J5=20tdτ0τg(τ-s)a¯2(m)(s;um(s),wm(τ))ds20tdτ0τg(τ-s)μ2(m+1)(s)-μ2(m)(s)C0(Ω¯)um(s)awm(τ)ads2(1+4M)MK˜M(μ2)wm-1W1(T)0tdτ0τg(τ-s)wm(τ)ads2(1+4M)MK˜M(μ2)wm-1W1(T)gL1(0,T*)0twm(τ)adτ2βλ0twm(s)a2ds+12βλT(1+4M)2M2K˜M2(μ2)gL1(0,T*)2wm-1W1(T)2βZm(t)+12βλT(1+4M)2M2K˜M2(μ2)gL1(0,T*)2wm-1W1(T)2,   β>0.

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

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

where 1(M) and 2(M) are the constants as in (3.47).

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

wmW1(T)kTwm-1W1(T),   m,

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

um-um+pW1(T)u0-u1W1(T)(1-kT)-1kTm,m,p.

It follows that {um} is a Cauchy sequence in W1(T). Then there exists uW1(T) such that

umustrongly in W1(T).

Note that umW(M, T), then there exists a subsequence {umj} of {um} such that

{umjuinL(0,T;VH2)weak*,umjuinL(0,T;VH2)weak*,umjuinL2(0,T;V)weak,umW(M,T).

We note that

Fm-f[u]L(0,T;L2)2(1+2M)KM(f)um-1-uW1(T).

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

Fmf[u]strongly in L(0,T;L2).

We also note that

μi(m)(t)-μi[u](t)C0(Ω¯)(1+4M)K˜M(μi)um-1-uW1(T),a.e.t(0,T),i=1,2.

On the other hand, for all vV, we have

|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)ava+K˜M(μ1)um(t)-u(t)avaK˜M(μ1)[M(1+4M)um-1-uW1(T)+um-uW1(T)]va.

Hence

|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-uW1(T)+um-uW1(T)]vaφL1(0,T)0,vV,φL1(0,T).

Similarly

|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)gL1(0,T)[M(1+4M)um-1-uW1(T)+um-uW1(T)]vaφL1(0,T)0,

vV, ∀φL1(0, T).

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

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 wV and the initial conditions

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

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 uW1(M, T). The existence result follows.

(b) Uniqueness of the solution

Let u1, u2W1(M, T) be two weak solutions of Prob. (1.1) – (1.3). Then u = u1u2 satisfies the variational problem

{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,wV,u(0)=u(0)=0,

where

aj[ui](t;v,w)=μj[ui](t)vx,wx+h0μj[ui](0,t)v(0)w(0),v,wV,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

Z(t)=0ta˙1[u1](s;u(s),u(s))ds+20tf[u1](s)-f[u2](s),u(s)ds+20tdτ0τg(τ-s)a2[u1](s;u(s),u(τ))ds-20t[(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+20tdτ0τg(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

Z(t)=u(t)2+a1[u1](t;u(t),u(t))+2λ0tu(s)a2ds.

Put

Z˜M=2γMK˜M(μ1)μ*+8(1+1μ*)(1+M)KM(f)+6λμ*[K˜M2(μ2)TgL2(0,T*)2+(1+4M)2M2(K˜M2(μ1)+K˜M2(μ2)TgL2(0,T*)2)],

then it follows from (3.81) that

Z(t)Z˜M0tZ(s)ds.

By Gronwall’s Lemma, we deduce Z(t) = 0, i.e., u1u2. Theorem 3.4 is proved completely.

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

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

  • (H6) f1C1([0,1]×[0,T*]×3×+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. (Pɛ) has a unique weak solution u depending on ɛ: u = uɛ. When ɛ = 0, (Pɛ) is denoted by (P0). We shall study the asymptotic expansion of the solution uɛ of Prob. (Pɛ) with respect to a small parameter ɛ.

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

{|α|=α1++αN,   α!=α1!αN!,xα=x1α1xNαN,α,β+N,αβαiβii=1,,N.

First, we shall need the following lemma.

Lemma 4.1

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

(i=1Nxiɛi)m=k=mmNPN[m][x]kɛk,

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

PN[m][x]k={xk,1kN,   m=1,αAk[m](N)m!α!xα,mkmN,   m2,

withAk[m](N)={α+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,μ2CN+1([0,1]×[0,T*]××+2),μ1(x,t,z1,z2,z3)μ*>0,for all (x,t,z1,z2,z3)[0,1]×[0,T*]××+2;

  • (H8) fCN+1([0,1]×[0,T*]×3×+2),f1CN([0,1]×[0,T*])×3×+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),u0W1(M,T).

Considering the sequence of weak solutions ur, 1 ≤ rN, 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,urW1(M,T),

where Fr, 1 ≤ rN, are defined by the recurrence formulas

Fr={f[u0],r=0,πr[N,f]+πr-1[N-1,f1]+i=1rx(ρi[μ1]ur-i-0tg(t-s)ρi[μ2]ur-i(s)ds),1rN,

with πr[N, f] = πr[N, f; u0, u1, …, ur], ρr[μi] = ρr[μi, N, u0, ū, σ(1), σ(2)], 0 ≤ rN, defined by the formulas:

  • Formula πr[N, f] = πr[N, f; u0, ū, σ(1), σ(2)]: πr[N,f]={f[u0],r=0,1|m|r1m!Dmf[u0]Φ˜r[m,N,u¯,σ(1),σ(2)],1rN,

    in which

    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)+5:k1++k5=r,mikimiN,i=1,,3;mjkj2mjN,j=4,5},m=(m1,,m5)+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

    σi(1)={2u0,u1,i=12u0,ui+j=1iuj,ui-j,2iN,j=1iuj,ui-j,N+1i2N,σi(2)={2u0,u1,i=12u0,ui+j=1iuj,ui-j,2iN,j=1iuj,ui-j,N+1i2N;

  • Formula ρr[μi] = ρr[μi, N, u0, ū, σ(1), σ(2)]:

    ρr[μi]={μi[u0]r=0,1γr1γ!Dγμi[u0]Φr[N,u¯,σ(1),σ(2),γ],1rN,

    where 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)+3, 1 ≤ |γ| = γ1 + … + γ3r, γ! = γ1!…γ3!, Dγμi=D3γ1D4γ2D5γ3μi,A˜r(γ,N)={(k1,k2,k3)+3:|γ|=r,γ1k1γ1N,γ2k22γ2N,γ3k32γ3N}. Then, we have the following lemma.

Lemma 4.2

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

f(h)=r=0Nπr[N,f]ɛr+|ɛ|N+1R¯N(1)[f,ɛ],μi[h]=r=0Nρr[μi]ɛr+|ɛ|N+1R˜N(1)[μi,ɛ],

withR˜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, ur, 0 ≤ rN.

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=1Nuiɛiu0+h1. We rewrite as below

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+h12,u0+h12)=f(x,t,u0+h1,u0+h1,u0+h1,u02+ξ4,u02+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,u02,u02) up to order N + 1, we obtain

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)+51m!Dmf[u0]h1m1(h1)m2(h1)m3ξ4m4ξ4m5+RN(1)[f,h1,ξ4,ξ5],

where

RN(1)[f,h1,ξ4,ξ5]=|m|=N+1m+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+θh1,u0+θh1,u02+θξ4,u02+θξ5).

By the formula (4.1), it follows that

h1m1=(i=1Nuiɛi)m1=k=m1m1NPN[m1][u¯]kɛk,(h1)m2=(i=1Nuiɛi)m2=k=m2m2NPN[m2][u¯]kɛk,(h1)m3=(i=1Nuiɛi)m3=k=m3m3NPN[m3][u¯]kɛk,

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

On the other hand,

ξ4=u0+h12-u02=2u0,h1+h12i=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

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

where σ(1)=(σ1(1),,σ2N(1)).

Similarly, we have

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

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

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 Ār(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

f[u0+h1]=f[u0]+1|m|Nm=(m1,,m5)+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

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

with πr[N, f], 0 ≤ rN 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, ur, r = 0, 1, …, N.

Hence, the formula (4.9) is proved.

(ii) In the case of μi[h] = μi (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), Djf = 0, j = 4, 5, D3f=D3μi=μiz1,D6f=D4μi=μiz2,D7f=D5μi=μiz3 and πr[N, f] = ρr[μi], 0 ≤ rN, 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 = uɛ of order N + 1 in a small parameter ɛ.

Let u = uɛW1(M, T) be a unique weak solution of the problem (Pɛ). Then v=u-r=0Nurɛru-h=u-u0-h1 satisfies the problem

{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

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

EɛL(0,T;L2)C¯*|ɛ|N+1,

where C̄*is a constant depending only on N, T, f, f1, μ1, μ2, ur, 0 ≤ rN.

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

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

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

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, ur, 0 ≤ rN.

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

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

where

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

Similarly

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

where

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

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

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 urW1(M, T), 0 ≤ rN, we obtain from (4.27), (4.29), (4.31) and (4.32) that

EɛL(0,T;L2)C¯*|ɛ|N+1,

where * is a constant depending only on N, T, f, f1, μ1, μ2, ur, 0 ≤ rN.

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

S(t)=20tEɛ(s),v(s)ds+0ta˙1[v+h](s,v(s),v(s))ds-20t[a1[v+h](s,h(s),v(s))-a1[h](s,h(s),v(s))]ds+20tdτ0τg(τ-s)a2[v+h](s,v(s),v(τ))ds+20tf[v+h]-f[h]+ɛ(f1[v+h]-f1[h]),v(s)ds+20tdt0τg(τ-s)[a2[v+h](s,h(s),v(τ))-a2[h](s,h(s),v(τ))]dsi=16Si,

where

S(t)=v(t)2+a1[v+h](t,v(t),v(t))+2λ0tv(s)a2dsv(t)2+μ*v(t)a2+2λ0tv(s)a2ds.

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

S1=20tEɛ(s),v(s)dsTC¯*2ɛ2N+2+0tS(s)ds;S2=0ta˙1[v+h](s,v(s),v(s))dsγM*K˜M*(μ1)0tv(s)a2ds1μ*γM*K˜M*(μ1)0tS(s)ds;S3=-20t[a1[v+h](s,h(s),v(s))-a1[h](s,h(s),v(s))]ds2(1+4M*)M*K˜M*(μ1)0tv(s)av(s)ads16S(t)+3λμ*(1+4M*)2M*2K˜M*2(μ1)0tS(s)ds;S4=20tdτ0τg(τ-s)a2[v+h](s,v(s),v(τ))ds2K˜M*(μ2)0tdτ0τg(τ-s)v(s)av(τ)ads16S(t)+3λμ*TK˜M*2(μ2)gL2(0,T*)20tS(s)ds;S5=20tf[v+h]-f[h]+ɛ(f1[v+h]-f1[h]),v(s)ds8(1+1μ*)(1+2M*)   (KM*(f)+KM*(f1))0tS(s)ds;S6=20tdτ0τg(τ-s)[a2[v+h](s,h(s),v(τ))-a2[h](s,h(s),v(τ))]ds20tdτ0τg(τ-s)μ2[v+h](s)-μ2[h](s)C0([0,1])h(s)av(τ)ads2(1+4M*)M*K˜M*(μ2)0tdτ0τg(τ-s)v(s)av(τ)ads16S(t)+3λμ*T(1+4M*)2M*2K˜M*2(μ2)gL2(0,T*)20tS(s)ds.

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

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)TgL2(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

vW1(T)(1+1μ*+12λ)C¯*2Texp(TD˜T(M))|ɛ|N+1,

or

uɛ-r=0NurɛrW1(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. (Pɛ) has a unique weak solution uɛW1(M, T) satisfying an asymptotic estimation up to order N + 1 as in (4.38), where the functions ur, r = 0, 1, …, N are weak solutions of Prob. (P0), (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

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

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