# A High Order Iterative Scheme for a Nonlinear Kirchhoff Wave Equation in the Unit Membrane

Authors: Le Thi Phuong Ngoc; Nguyen Thanh Long
Journal: International Journal of Differential Equations
 (2011)

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## Abstract

A high-order iterative scheme is established in order to get a convergent sequence at a rate of orderN (N≥1) to a local unique weak solution of a nonlinear Kirchhoff wave equation in the unit membrane. This extends a recent result in (EJDE, 2005, No. 138) where a recurrent sequence converges at a rate of order 2.


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## Body

## 1. Introduction

In this paper we consider the initial and boundary value problem(1.1)utt-B(‖ur‖02)(urr+1rur)=f(r,t,u),0<r<1,0<t<T,|lim⁡r→0+rur(r,t)|<∞,ur(1,t)+hu(1,t)=0,u(r,0)=ũ0(r),ut(r,0)=ũ1(r),
where B,f,ũ0,ũ1 are given functions satisfying conditions specified later, ∥ur∥02=∫01r|ur(r,t)|2dr, and h>0 is a given constant.Equation (1.1)1 herein is the bidimensional nonlinear wave equation describing nonlinear vibrations of the unit membrane Ω1={(x,y):x2+y2<1}. In the vibration process, the area of the unit membrane and the tension at various points change in time. The condition on the boundary ∂Ω1 describes elastic constraints, where the constant h1 has a mechanical signification. The boundary condition |lim⁡r→0+⁡rur(r,t)|<∞ is satisfied automatically if u is a classical solution of the problem (1.1), for example, with u∈C1([0,1]×(0,T))∩C2((0,1)×(0,T)). This condition is also used in connection with Sobolev spaces with weight r (see [1–3]).Equation (1.1)1 is related to the Kirchhoff equation(1.2)ρhutt=(P0+Eh2L∫0L|∂u∂y(y,t)|2dy)uxx
presented by Kirchhoff in 1876 (see [4]). 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 (1.2) 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.The Kirchhoff wave equation of the form (1.1)1 received much attention. Many interesting results about the existence, stability, regularity in time variable, asymptotic behavior, and asymptotic expansion of solutions can be found, for example, in [2, 3, 5–14] and references therein.In [2], in a special case, sufficient conditions were established for a quadratic convergence to the solution of (1.1) with f(r,t,u)=f(r,u) and B(∥ur∥02)=b0+∥ur∥02,b0>0. Based on the ideas about recurrence relations for a third-order method for solving the nonlinear operator equation F(u)=0 in [15], we extend the above result by the construction of a high-order iterative scheme for (1.1)1,  where f and B are more generalized.In this paper, we associate with (1.1)1 a recurrent sequence {um} defined by(1.3)∂2um∂t2-B(‖umr‖02)(∂2um∂r2+1r∂um∂r)=∑i=0N-11i!∂if∂ui(r,t,um-1)(um-um-1)i,0<r<1,0<t<T, with um satisfying (1.1)2-3. The first term u0 is chosen as u0≡0. IfB∈C1(ℝ+) and f∈CN([0,1]×ℝ+×ℝ), we prove that the sequence {um} converges at a rate of order N to a unique weak solution of the problem (1.1). This result is a relative generalization of [2, 3, 8, 9, 14, 16].


## 2. Preliminary Results, Notations, Function Spaces

PutΩ=(0,1). We omit the definitions of the usual function spaces Cm(Ω¯),Lp(Ω),Hm(Ω), and Wm,p(Ω). For any function v∈C0(Ω¯) we define ∥v∥0 as ∥v∥0=(∫01rv2(r)dr)1/2 and define the space V0 as completion of the space C0(Ω¯) with respect to the norm ∥·∥0. Similarly, for any function v∈C1(Ω¯) we define ∥v∥1 as ∥v∥1=(∥v∥02+∥vr∥02)1/2 and define the space V1 as completion of the space C1(Ω¯) with respect to the norm ∥·∥1. Note that the norms ∥·∥0 and ∥·∥1 can be defined, respectively, from the inner products (2.1)〈u,v〉=∫01ru(r)v(r)dr,〈u,v〉+〈ur,vr〉.
Identifying V0 with its dual V0′ we obtain the dense and continuous embedding V1↪V0≡V0′↪V1′. The inner product notation will be reutilized to denote the duality pairing between V1 and V1′.We then have the following lemmas, the proofs of which can be found in [1].Lemma 2.1.
There exist two constantsK1>0 and K2>0 such that, for all v∈C1(Ω¯), we have(i)
∥vr∥02+v2(1)≥∥v∥02,(ii)
|v(1)|≤K1∥v∥1,(iii)
r|v(r)|≤K2∥v∥1,forallr∈Ω¯.Lemma 2.2.
The embeddingV1↪V0 is compact.Remark 2.3.
In Lemma2.1, the two constants K1 and K2 can be given explicitly as K1=1+2 and K2=1+5. We also note that lim⁡r→0+⁡rv(r)=0 for all v∈V1 (see [17, page 128/Lemma 5.40]). On the other hand, by H1(ɛ,1)↪C0([ɛ,1]),0<ɛ<1 and ɛ∥v∥H1(ɛ,1)≤∥v∥1 for all v∈V1, it follows that v|[ɛ,1]∈C0([ɛ,1]). From both relations we deduce that rv∈C0(Ω¯) for all v∈V1.
Now, let the bilinear forma(·,·) be defined by(2.2)a(u,v)=hu(1)v(1)+∫01rur(r)vr(r)dr,u,v∈V1,
where h is a positive constant. Then, there exists a unique bounded linear operator A:V1→V1′ such that a(u,v)=〈Au,v〉 for all u,v∈V1. We then have the following lemma.Lemma 2.4.
The symmetric bilinear forma(·,·) defined by (2.2) is continuous on V1×V1 and coercive on V1, that is,(i)
|a(u,v)|≤C1∥u∥1∥v∥1,(ii)
a(v,v)≥C0∥v∥12,for all u,v∈V1, where C0=(1/2)min⁡{1,h} and C1=1+(1+2)h.The proof of Lemma2.4 is straightforward and we omit it.Lemma 2.5.
There exists an orthonormal Hilbert basis{wj} of the space V0 consisting of eigenfunctions  wj corresponding to eigenvalues λj such that(i)
0<λ1≤λ2≤⋯≤λj↑+∞asj→∞,(ii)
a(wj,v)=λj〈wj,v〉forallv∈V1andj∈ℕ.Note that it follows from (ii) that {wj/λj} is automatically an orthonormal set in V1 with respect to a(·,·) as inner product. The eigensolutions wj are indeed eigensolutions for the boundary value problem
(2.3)Awj≡-1rddr(rdwjdr)=λjwj,inΩ,|lim⁡r→0+rdwjdr(r)|<+∞,dwjdr(1)+hwj(1)=0.The proof of Lemma2.5 can be found in ([18, page 87, Theorem 7.7]) with V=V1,H=V0 and a(·,·) as defined by (2.2).For any functionv∈C2(Ω¯) we define ∥v∥2 as (2.4)‖v‖2=(‖v‖02+‖vr‖02+‖Av‖02)1/2
and define the space V2 as completion of C2(Ω¯) with respect to the norm ∥·∥2. Note that V2 is also a Hilbert space with respect to the scalar product (2.5)〈u,v〉+〈ur,vr〉+〈Au,Av〉
and that V2 can be defined also as V2={v∈V1:Av∈V0}.We then have the following two lemmas the proof of which can be found in [1].Lemma 2.6.
The embeddingV2↪V1 is compact.Lemma 2.7.
For allv∈V2 we have
(2.6)(i)‖vr‖L∞(Ω)≤12‖Av‖0,(ii)‖vrr‖0≤32‖Av‖0,(iii)‖v‖L∞(Ω)2≤(2‖v‖0+12‖Av‖0)‖v‖0.For a Banach spaceX, we denote by ∥·∥X its norm, by X′ its dual space, and by Lp(0,T;X),1≤p≤∞ the Banach space of all real measurable functions u:(0,T)→X such that (2.7)‖u‖Lp(0,T;X)=(∫0T‖u(t)‖Xpdt)1/p<∞,for1≤p<∞,‖u‖L∞(0,T;X)=esssup⁡0<t<T‖u(t)‖Xforp=∞.Let(2.8)u(t),u′(t)=ut(t)=u̇(t),u′′(t)=utt(t)=ü(t),ur(t)=∇u(t),urr(t)
denote (2.9)u(r,t),∂u∂t(r,t),∂2u∂t2(r,t),∂u∂r(r,t),∂2u∂r2(r,t),
respectively.Withf∈Ck(Ω¯×ℝ+×ℝ),f=f(r,t,u), we put D1f=∂f/∂r,D2f=∂f/∂t,D3f=∂f/∂u, and Dγf=D1γ1D2γ2D3γ3f,γ=(γ1,γ2,γ3)∈ℤ+3,|γ|=γ1+γ2+γ3=k.


## 3. The Hight Order Iterative Schemes

FixT*>0, we make the following assumptions: (H1)
ũ0∈V2andũ1∈V1;(H2)
B∈C1(ℝ+)andthereexistconstantsb*>0,α>1,d0,d1>0suchthat
(i)
b*≤B(η)≤d0(1+ηα),forallη≥0,(ii)
|B′(η)|≤d1(1+ηα-1),forallη≥0;(H3)
f∈CN(Ω¯×[0,T*]×ℝ)andsatisfiesthefollowingcondition:forallM>0,
(3.1)K̂*i(1)(M,f)=sup⁡(r,t,u)∈A¯M|(r)-i∂if∂ui(r,t,u)|<+∞,i=0,1,…,N-1,K̂*i(2)(M,f)=sup⁡(r,t,u)∈A¯M|(r)-i∂if∂r∂ui-1(r,t,u)|<+∞,i=1,…,N-1,whereA¯M={(r,t,u)∈[0,1]×[0,T*]×ℝ:|u|≤M2+1/2}. We put (3.2)K̂*i(M,f)={K̂*0(1)(M,f),i=0,max⁡{K̂*i(1)(M,f),K̂*i(2)(M,f)},i=1,…,N-1.WithB and f satisfying assumptions (H2) and (H3), respectively, for each M>0 given, we introduce the following constants:(3.3)K̃M(B)=sup⁡0≤η≤M2(B(η)+|B′(η)|),K¯0(M,f)=sup⁡(r,t,u)∈A¯M|f(r,t,u)|,K¯N(M,f)=∑|γ|≤NK¯0(M,Dγf).For eachT∈(0,T*] and M>0 we get (3.4)W(M,T)={v∈L∞(0,T;V2):v′∈L∞(0,T;V1),v′′∈L2(0,T;V0),with‖v‖L∞(0,T;V2),‖v′‖L∞(0,T;V1),‖v′′‖L2(0,T;V0)≤M},W1(M,T)={v∈W(M,T):v′′∈L∞(0,T;V0)}.We will choose as first initial termu0≡0, suppose that(3.5)um-1∈W1(M,T),
and associate with the problem (1.1) the following variational problem.Findum∈W1(M,T)(m≥1) so that(3.6)〈um′′(t),v〉+bm(t)a(um(t),v)=〈Fm(t),v〉,∀v∈V1,um(0)=ũ0,um′(0)=ũ1,
where(3.7)bm(t)=B(‖∇um(t)‖02),Fm(r,t)=∑i=0N-11i!D3if(r,t,um-1)(um-um-1)i.Then, we have the following theorem.Theorem 3.1.
Let assumptions(H1)–(H3) hold. Then there exist a constant M>0  depending on T*,ũ0,ũ1,B,h and a constant T>0 depending on T*,ũ0,ũ1,B,h,f such that, for u0≡0, there exists a recurrent sequence {um}⊂W1(M,T) defined by (3.6), (3.7).Proof.
The proof consists of several steps.
Step 1.
The Faedo-Galerkin approximation (introduced by Lions [19]). Consider as in Lemma 2.5 the basis {wj} for V1 and put
(3.8)um(k)(t)=∑j=1kcmj(k)(t)wj,
where the coefficients cmj(k) satisfy the system of the following nonlinear differential equations:
(3.9)〈üm(k)(t),wj〉+bm(k)(t)a(um(k)(t),wj)=〈Fm(k)(t),wj〉,1≤j≤k,um(k)(0)=ũ0k,u̇m(k)(0)=ũ1k,
where
(3.10)ũ0k=∑j=1kαj(k)wj⟶ũ0stronglyinV2,ũ1k=∑j=1kβj(k)wj⟶ũ1stronglyinV1.(3.11)bm(k)(t)=B(‖∇um(k)(t)‖02),Fm(k)(r,t)=∑i=0N-11i!D3if(r,t,um-1)(um(k)-um-1)i=∑j=0NΨj(r,t,um-1)(um(k))j,Ψj(r,t,um-1)=∑i=jN-11j!(i-j)!(-1)i-jD3if(r,t,um-1)um-1i-j,0≤j≤N-1.
Let us suppose thatum-1 satisfies (3.5). Then we have the following lemma.
Lemma 3.2.
Let assumptions(H1)–(H3) hold. For fixed M>0 and T>0,  then, the system (3.8)–(3.11) has a unique solution um(k)(t) on an interval [0,Tm(k)]⊂[0,T].
Proof of Lemma3.2.
The system of (3.8)–(3.11) is rewritten in the form
(3.12)c̈mj(k)(t)=-λjbm(k)(t)cmj(k)(t)+〈Fm(k)(t),wj〉,1≤j≤k,cmj(k)(0)=αj(k),ċmj(k)(0)=βj(k),
and it is equivalent to the system of integral equations
(3.13)cmj(k)(t)=αj(k)+βj(k)t-λj∫0tdτ∫0τbm(k)(s)cmj(k)(s)ds+∫0tdτ∫0τ〈Fm(k)(s),wj〉ds,
for 1≤j≤k. Omitting the index m, it is written as follows:
(3.14)c=F[c],
where ℱ[c]=(ℱ1[c],…,ℱk[c]),c=(c1,…,ck),
(3.15)Fj[c](t)=qj(t)-λj∫0tdτ∫0τb̃(s)cj(s)ds+∑i=1N-1∫0tdτ∫0τ〈Ψi(⋅,s,um-1)ui(s),wj〉ds,1≤j≤k,qj(t)=αj(k)+βj(k)t+∫0tdτ∫0τ〈Ψ0(⋅,s,um-1),wj〉ds,1≤j≤k,b̃(t)=b̃[c](t)=B(‖∇u(t)‖02),u(t)=∑j=1kcj(t)wj.
For everyTm(k)∈(0,T] and ρ>0 that will be chosen later, we put X=C0([0,Tm(k)];ℝk),S={c∈Y:∥c∥X≤ρ}, where ∥c∥X=sup⁡0≤t≤Tm(k)|c(t)|1,|c(t)|1=∑j=1k|cj(t)|, for each c=(c1,…,ck)∈X. Clearly S is a closed nonempty subset in X, and we have the operator ℱ:X→X. In what follows, we will choose ρ>0 and Tm(k)>0 such that(i)
Sismappedintoitselfbyℱ,(ii)
ℱ:S→Siscontractive.
Proof (i).
First we note that, for allc=(c1,…,ck)∈S,
(3.16)u(t)=∑j=1kcj(t)wj,‖u(t)‖0=∑j=1kcj2(t)≤|c(t)|1,‖∇u(t)‖02≤a(u(t),u(t))=∑i,j=1kci(t)cj(t)a(wi,wj)=∑j=1kλjcj2(t)≤λk‖u(t)‖02,‖u(t)‖0≤|c(t)|1≤‖c‖X≤ρ,‖∇u(t)‖0≤λk|c(t)|1≤λkρ,‖u(t)‖1≤1C0a(u(t),u(t))≤1C0λk‖u(t)‖0≤λkC0|c(t)|1≤λkC0ρ,
so
(3.17)b̃(t)=B(‖∇u(t)‖02)≤d0(1+‖∇u(t)‖02α)≤d0(1+λk2αρ2α).
On the other hand, by
(3.18)|um-1|≤M2+12≡θ,|Ψ0(r,t,um-1)|=|∑i=0N-11i!(-1)iD3if(r,t,um-1)um-1i|≤∑i=0N-1|1i!D3if(r,t,um-1)um-1i|≤K¯N(M,f)∑i=0N-11i!(M2+12)i=K¯N(M,f)∑i=0N-1θii!,
we have
(3.19)|〈Ψ0(t,um-1),wj〉|≤‖Ψ0(t,um-1)‖0‖wj‖0=‖Ψ0(t,um-1)‖0≤12K¯N(M,f)∑i=0N-1θii!.
By Lemma 2.1, (iii), and the assumption (H3), we deduce from (3.16) that
(3.20)|〈Ψi(s,um-1)ui(s),wj〉|=|∑l=iN-11i!(l-i)!(-1)l-i〈D3lf(r,t,um-1)um-1l-iui(s),wj〉|=|∑l=iN-11i!(l-i)!(-1)l-i〈(r)-lD3lf(r,s,um-1)(r)l-ium-1l-i(r)iui(s),wj〉|≤∑l=iN-11i!(l-i)!K̂*l(M,f)θl-iK2i‖u(s)‖1i〈(r)l-i,|wj|〉≤∑l=iN-11i!(l-i)!K̂*l(M,f)θl-iK2i(λkC0ρ)i12+l-i≤∑l=iN-11i!(l-i)!12+l-iK̂*l(M,f)θl-iK2i(λkC0ρ)i,1≤i≤N-1.
It follows that(3.21)|Fj[c](t)|≤|qj(t)|+λkd0(1+λk2αρ2α)∫0tdτ∫0τ|cj(s)|ds+12t2∑i=1N-1∑l=iN-11i!(l-i)!12+l-iK̂*l(M,f)θl-iK2i(λkC0ρ)i.
Thus
(3.22)|F[c](t)|1≤|q(t)|1+λkd0(1+λk2αρ2α)∫0tdτ∫0τ|c(s)|1ds+12kt2∑i=1N-1∑l=iN-11i!(l-i)!12+l-iK̂*l(M,f)θl-iK2i(λkC0ρ)i≤‖q‖T+12D¯ρ(Tm(k))2,
where ∥q∥T=sup⁡0≤t≤T|q(t)|1 and
(3.23)D¯ρ=D¯ρ(f,ρ,k,M,m,N)=λkd0(1+λk2αρ2α)ρ+k∑i=1N-1∑l=iN-11i!(l-i)!12+l-iK̂*l(M,f)θl-iK2i(λkC0ρ)i.
Hence, we obtain(3.24)‖F[c]‖X≤‖q‖T+12D¯ρ(Tm(k))2,
choosingρ>∥q∥T andTm(k)∈(0,T], such that(3.25)12D¯ρ(Tm(k))2≤ρ-‖q‖T,12D̃ρ(Tm(k))2<1,
where
(3.26)D̃ρ=D̃ρ(ρ,k,M,T,m,N,f)≡d0λk(1+λk2αρ2α)+2d1λk2ρ2(1+λkα-1ρ2α-2)+k∑i=1N-1i(λkC0K2)iρi-1∑l=iN-11i!(l-i)!12+l-iK̂*l(M,f)θl-i.
Then
(3.27)‖F[c]‖X≤‖q‖T+12D¯ρ(Tm(k))2≤ρ,∀c∈S,
which means that ℱ maps S into itself.
Proof (ii).
We now prove that, for allc,d∈S, for all t∈[0,Tm(k)],
(3.28)|F[c](t)-F[d](t)|1≤12D̃ρt2‖c-d‖X,∀n∈N,
where D̃ρ is defined as (3.26).
Proof of (3.28) is as follows.
For allj=1,2,…,k, for all t∈[0,Tm(k)], we have(3.29)|Fj[c](t)-Fj[d](t)|≤λj∫0tdτ∫0τ|b̃[c](s)(cj(s)-dj(s))|ds+λj∫0tdτ∫0τ|(b̃[c](s)-b̃[d](s))dj(s)|ds+∑i=1N-1∫0tdτ∫0τ|〈Ψi(s,um-1)(ui(s)-vi(s)),wj〉|ds,
where
(3.30)b̃[c](t)=B(‖∇u(t)‖02),b̃[d](t)=B(‖∇v(t)‖02),u(t)=∑j=1kcj(t)wj,v(t)=∑j=1kdj(t)wj,
so
(3.31)|F[c](t)-F[d](t)|1≤λk∫0tdτ∫0τb̃[c](s)|c(s)-d(s)|1ds+λk∫0tdτ∫0τ|b̃[c](s)-b̃[d](s)||d(s)|1ds+∑j=1k∑i=1N-1∫0tdτ∫0τ|〈Ψi(s,um-1)(ui(s)-vi(s)),wj〉|ds≤λk∫0tdτ∫0τb̃[c](s)|c(s)-d(s)|1ds+λkρ∫0tdτ∫0τ|b̃[c](s)-b̃[d](s)|ds+∑j=1k∑i=1N-1∫0tdτ∫0τ|〈Ψi(s,um-1)(ui(s)-vi(s)),wj〉|ds≡J1+J2+J3,
in which
(3.32)J1=λk∫0tdτ∫0τb̃[c](s)|c(s)-d(s)|1ds≤λkd0(1+λk2αρ2α)∫0tdτ∫0τ|c(s)-d(s)|1ds≡ζ1∫0tdτ∫0τ|c(s)-d(s)|1ds.
In order to consider J2, we also note that
(3.33)b̃[c](s)-b̃[d](s)=B′(ξ)(‖∇u(s)‖02-‖∇v(s)‖02),
where
(3.34)ξ=θ‖∇u(s)‖02+(1-θ)‖∇v(s)‖02,0≤ξ≤λkρ2,0<θ<1,
and B′(ξ) satisfy the following inequality:
(3.35)|B′(ξ)|≤d1(1+ξα-1)≤d1(1+λkα-1ρ2α-2).
It implies that(3.36)|b̃[c](s)-b̃[d](s)|=|B′(ξ)(‖∇u(s)‖02-‖∇v(s)‖02)|≤d1(1+λkα-1ρ2α-2)2λkρ|c(s)-d(s)|1=2λkρd1(1+λkα-1ρ2α-2)|c(s)-d(s)|1,
and then
(3.37)J2=λkρ∫0tdτ∫0τ|b̃[c](s)-b̃[d](s)|ds≤2λk2ρ2d1(1+λkα-1ρ2α-2)∫0tdτ∫0τ|c(s)-d(s)|1ds≡ζ2∫0tdτ∫0τ|c(s)-d(s)|1ds.
It remains to estimateJ3. By(3.38)(ru(s))i-(rv(s))i=∑j=0i-1(ru(s))j(rv(s))i-j-1r(u(s)-v(s)),
we obtain
(3.39)|(ru(s))i-(rv(s))i|≤K2i∑j=0i-1‖u(s)‖1j‖v(s)‖1i-j-1‖u(s)-v(s)‖1≤K2i∑j=0i-1‖u(s)‖1j‖v(s)‖1i-j-1‖u(s)-v(s)‖1≤K2i∑j=0i-1(λkC0ρ)j(λkC0ρ)i-j-1λkC0|c(s)-d(s)|1=K2i∑j=0i-1(λkC0)iρi-1|c(s)-d(s)|1=i(λkC0K2)iρi-1|c(s)-d(s)|1.
On the other hand,
(3.40)Ψi(r,t,um-1)=∑l=iN-11i!(l-i)!(-1)l-iD3lf(r,t,um-1)um-1l-i,(r)-i|Ψi(r,t,um-1)|=|∑l=iN-11i!(l-i)!(-1)l-i(r)-lD3lf(r,t,um-1)(r)l-ium-1l-i|≤|∑l=iN-11i!(l-i)!(r)-l|D3lf(r,t,um-1)|(r)l-i|um-1l-i||≤∑l=iN-11i!(l-i)!K̂*l(M,f)θl-i(r)l-i.
Hence, we deduce from (3.39), (3.40) that(3.41)J3=∑j=1k∑i=1N-1∫0tdτ∫0τ|〈(r)-iΨi(s,um-1)((ru(s))i-(rv(s))i),wj〉|ds≤∑j=1k∑i=1N-1∑l=iN-11i!(l-i)!K̂*l(M,f)θl-ii(λkC0K2)iρi-1×∫0tdτ∫0τ|〈(r)l-i,|wj|〉||c(s)-d(s)|1ds=∑j=1k∑i=1N-1∑l=iN-11i!(l-i)!K̂*l(M,f)θl-ii(λkC0K2)iρi-112+l-i×∫0tdτ∫0τ|c(s)-d(s)|1ds=k∑i=1N-1i(λkC0K2)iρi-1∑l=iN-11i!(l-i)!K̂*l(M,f)θl-i12+l-i×∫0tdτ∫0τ|c(s)-d(s)|1ds≡ζ3∫0tdτ∫0τ|c(s)-d(s)|1ds.
We deduce that(3.42)|F[c](t)-F[d](t)|1≤(ζ1+ζ2+ζ3)∫0tdτ∫0τ|c(s)-d(s)|1ds≤12D̃ρt2‖c-d‖X.
We note that
(3.43)ζ1+ζ2+ζ3=D̃ρ(ρ,k,M,T,m,N,f)=D̃ρ.
It follows from (3.28) that(3.44)‖F[c]-F[d]‖X≤12D̃ρ(Tm(k))2‖c-d‖X,∀c,d∈S.
By (3.25), it follows that ℱ:S→S is contractive. We deduce that ℱ has a unique fixed point in S; that is, the system (3.8)–(3.11) has a unique solution um(k)(t) on an interval [0,Tm(k)]. The proof of Lemma 3.2 is complete.
The following estimates allow one to take constantTm(k)=T for all m and k.
Step 2.
A priori estimates. Put(3.45)Sm(k)(t)=Xm(k)(t)+Ym(k)(t)+∫0t‖üm(k)(s)‖02ds,
where
(3.46)Xm(k)(t)=‖u̇m(k)(t)‖02+bm(k)(t)a(um(k)(t),um(k)(t)),Ym(k)(t)=a(u̇m(k)(t),u̇m(k)(t))+bm(k)(t)‖Aum(k)(t)‖02,
with A is defined by (2.2). Then it follows that
(3.47)Sm(k)(t)=Sm(k)(0)+∫0tḃm(k)(s)[a(um(k)(s),um(k)(s))+‖Aum(k)(s)‖02]ds+2∫0t〈Fm(k)(s),u̇m(k)(s)〉ds+2∫0ta(Fm(k)(s),u̇m(k)(s))ds+∫0t〈Fm(k)(s),üm(k)(s)〉ds-∫0tbm(k)(s)〈Aum(k)(s),üm(k)(s)〉ds≡Sm(k)(0)+∑j=15Ij.
We will now require the following lemma.
Lemma 3.3.
We have(3.48)(i)0<b*≤bm(k)(t)≤d0(1+‖∇um(k)(t)‖02α),(ii)|ḃm(k)(t)|≤2d1b*[Sm(k)(t)+b*1-α(Sm(k)(t))α],(iii)‖Fm(k)(t)‖0≤∑j=0N-1ãj(0)(Sm(k)(t))j,(iv)‖∂∂rFm(k)(t)‖0≤∑j=0N-1ãj(1)(Sm(k)(t))j,
where ãj(0),ãj(1),j=0,1,…,N-1 are defined as follows:
(3.49)ãj(0)={ã0(0)=12K¯N(M,f)∑i=0N-1θii!,j=0,ãj(0)=K2j(b*C0)j∑i=jN-1θi-jj!(i-j)!12+i-jK̂*i(M,f),1≤j≤N-1,ãj(1)={(12+2M)K¯N(M,f)∑i=0N-1θii!,j=0,K2j(b*C0)j∑i=jN-1θi-jj!(i-j)![1i-j+3jK2-1K̂*i(M,f)+(1i-j+3+2M)K̂*i+1(M,f)],1≤j≤N-1,θ=M2+12.
Proof of Lemma3.3.
Proof (i), (ii).
Note that(3.50)Sm(k)(t)≥Xm(k)(t)≥b*a(um(k)(t),um(k)(t))≥b*‖∇um(k)(t)‖02,Sm(k)(t)≥Ym(k)(t)≥a(u̇m(k)(t),u̇m(k)(t))≥‖∇u̇m(k)(t)‖02.
We deduce that
(3.51)bm(k)(t)=B(‖∇um(k)(t)‖02)≤d0(1+‖∇um(k)(t)‖02α),|ḃm(k)(t)|=2|B′(‖∇um(k)(t)‖02)||〈∇um(k)(t),∇u̇m(k)(t)〉|≤2d1(1+‖∇um(k)(t)‖02α-2)‖∇um(k)(t)‖0‖∇u̇m(k)(t)‖0≤2d1(1+b*1-α(Sm(k)(t))α-1)1b*Sm(k)(t)=2d1b*[Sm(k)(t)+b*1-α(Sm(k)(t))α].
Proof (iii).
We have(3.52)‖Fm(k)(t)‖0≤‖Ψ0(r,t,um-1)‖0+∑j=1N-1‖Ψj(r,t,um-1)(um(k))j‖0.
By (3.18)3, we have
(3.53)‖Ψ0(r,t,um-1)‖0≤12K¯N(M,f)∑i=0N-1θii!≡ã0(0).
On the other hand, it follows from (3.49) and um-1∈W1(M,T) that
(3.54)‖Ψj(r,t,um-1)(um(k))j‖0≤∑i=jN-11j!(i-j)!‖D3if(r,t,um-1)um-1i-j(um(k))j‖0=∑i=jN-11j!(i-j)!‖(r)i-j(r)-iD3if(r,t,um-1)um-1i-j(r)j(um(k))j‖0≤∑i=jN-11j!(i-j)!‖(r)i-j‖0K̂*i(M,f)θi-jK2j‖um(k)(t)‖1j=∑i=jN-11j!(i-j)!12+i-jK̂*i(M,f)θi-jK2j‖um(k)(t)‖1j≤K2j(b*C0)j∑i=jN-1θi-jj!(i-j)!12+i-jK̂*i(M,f)(Sm(k)(t))j≡ãj(0)(Sm(k)(t))j.
It follows from (3.52)–(3.54) that
(3.55)‖Fm(k)(t)‖0≤∑j=0N-1ãj(0)(Sm(k)(t))j,
where ãj(0),0≤j≤N-1 are defined by (3.49)1.
Proof (iv).
We have(3.56)∂∂rFm(k)(r,t)=∂∂rΨ0(r,t,um-1)+∑j=1N-1jΨj(r,t,um-1)(um(k))j-1∇um(k)+∑j=1N-1(∂∂rΨj(r,t,um-1))(um(k))j.
Hence(3.57)‖∂∂rFm(k)(t)‖0≤‖∂∂rΨ0(r,t,um-1)‖0+∑j=1N-1j‖Ψj(r,t,um-1)(um(k))j-1∇um(k)‖0+∑j=1N-1‖(∂∂rΨj(r,t,um-1))(um(k))j‖0≡L1+L2+L3.
We shall estimate step by step the terms on the right-hand side of (3.57) as follows.(iv.1) EstimatingL1=∥(∂/∂r)Ψ0(r,t,um-1)∥0. We have(3.58)∂∂rΨ0(r,t,um-1)=∑i=1N-11i!(-1)iD3if(r,t,um-1)ium-1i-1∇um-1+∑i=0N-11i!(-1)iD1D3if(r,t,um-1)um-1i+∑i=0N-11i!(-1)iD3i+1f(r,t,um-1)um-1i∇um-1=a(1)+a(2)+a(3).
We will estimate step by step the terms a(1),a(2),a(3) as follows.(iv.1.1) Estimating∥a(1)∥0. We have(3.59)|a(1)|=|∑i=1N-11i!(-1)iD3if(r,t,um-1)ium-1i-1∇um-1|≤K¯N(M,f)∑i=1N-11i!iθi-1|∇um-1|=K¯N(M,f)∑i=0N-2θii!|∇um-1|≤K¯N(M,f)∑i=0N-1θii!|∇um-1|.
Hence
(3.60)‖a(1)‖0≤K¯N(M,f)∑i=0N-1θii!‖∇um-1‖0≤MK¯N(M,f)∑i=0N-1θii!.(iv.1.2) Estimating∥a(2)∥0. It follows from(3.61)|a(2)|=|∑i=0N-11i!(-1)iD1D3if(r,t,um-1)um-1i|≤K¯N(M,f)∑i=0N-1θii!
that
(3.62)‖a(2)‖0≤12K¯N(M,f)∑i=0N-1θii!.(iv.1.3) Estimating∥a(3)∥0. Similarly, with
(3.63)|a(3)|=|∑i=0N-11i!(-1)iD3i+1f(r,t,um-1)um-1i∇um-1|≤K¯N(M,f)∑i=0N-1θii!|∇um-1|,
we obtain(3.64)‖a(3)‖0≤K¯N(M,f)∑i=0N-1θii!‖∇um-1‖0≤MK¯N(M,f)∑i=0N-1θii!.
It follows from (3.58), (3.60), (3.62), (3.64) that
(3.65)L1=‖∂∂rΨ0(r,t,um-1)‖0≤‖a(1)‖0+‖a(2)‖0+‖a(3)‖0≤12K¯N(M,f)∑i=0N-1θii!+2MK¯N(M,f)∑i=0N-1θii!≤(12+2M)K¯N(M,f)∑i=0N-1θii!.
(iv.2) EstimatingL2=∑j=1N-1j∥Ψj(r,t,um-1)(um(k))j-1∇um(k)∥0. By the assumption (H3), we deduce that(3.66)L2=∑j=1N-1j‖Ψj(r,t,um-1)(um(k))j-1∇um(k)‖0≤∑j=1N-1j∑i=jN-11j!(i-j)!‖D3if(r,t,um-1)um-1i-j(um(k))j-1∇um(k)‖0=∑j=1N-1j∑i=jN-11j!(i-j)!‖(r)-iD3if(r,t,um-1)(r)i-jum-1i-j(rum(k))j-1r∇um(k)‖0=∑j=1N-1j∑i=jN-11j!(i-j)!K̂*i(M,f)‖(r)i-jr∇um(k)‖0θi-jK2j-1‖um(k)(t)‖1j-1≤∑j=1N-1j∑i=jN-1θi-jj!(i-j)!K̂*i(M,f)K2j-1‖um(k)(t)‖1j≤∑j=1N-1jK2j-1(b*C0)j∑i=jN-1θi-jj!(i-j)!K̂*i(M,f)(Sm(k)(t))j.(iv.3) EstimatingL3=∑j=1N-1∥((∂/∂r)Ψj(r,t,um-1))(um(k))j∥0. We have(3.67)L3=∑j=1N-1‖(∂∂rΨj(r,t,um-1))(um(k))j‖0=∑j=1N-1‖(r)-j(∂∂rΨj(r,t,um-1))(r)j(um(k))j‖0≤∑j=1N-1‖(r)-j(∂∂rΨj(r,t,um-1))‖0K2j‖um(k)(t)‖1j≤∑j=1N-1‖(r)-j(∂∂rΨj(r,t,um-1))‖01(b*C0)jK2j(Sm(k)(t))j≡∑j=1N-1L̃3(m,j,t)K2j(b*C0)j(Sm(k)(t))j.
Now we need an estimation of the term  L̃3(m,j,t)=∥(r)-j((∂/∂r)Ψj(r,t,um-1))∥0. We have
(3.68)∂∂rΨj(r,t,um-1)=∑i=j+1N-11j!(i-j)!(-1)i-j(i-j)D3if(r,t,um-1)um-1i-j-1∇um-1+∑i=jN-11j!(i-j)!(-1)i-jD1D3if(r,t,um-1)um-1i-j+∑i=jN-11j!(i-j)!(-1)i-jD3i+1f(r,t,um-1)um-1i-j∇um-1=b(1)+b(2)+b(3).
It follows from (3.68) that
(3.69)L̃3(m,j,t)=‖(r)-j(∂∂rΨj(r,t,um-1))‖0≤‖(r)-jb(1)‖0+‖(r)-jb(2)‖0+‖(r)-jb(3)‖0.(iv.3.1) Estimating∥(r)-jb(1)∥0. We have(3.70)‖(r)-jb(1)‖0=‖(r)-j∑i=j+1N-11j!(i-j)!(-1)i-j(i-j)D3if(r,t,um-1)um-1i-j-1∇um-1‖0≤∑i=j+1N-11j!(i-j)!(i-j)‖(r)-iD3if(r,t,um-1)(r)i-j-1um-1i-j-1r∇um-1‖0≤∑i=j+1N-11j!(i-j)!(i-j)K̂*i(M,f)θi-j-1M=M∑i=jN-2θi-jj!(i-j)!K̂*i+1(M,f)≤M∑i=jN-1θi-jj!(i-j)!MK̂*i+1(M,f).(iv.3.2) Estimating∥(r)-jb(2)∥0. We have(3.71)‖(r)-jb(2)‖0=‖(r)-j∑i=jN-11j!(i-j)!(-1)i-jD1D3if(r,t,um-1)um-1i-j‖0≤∑i=jN-11j!(i-j)!‖(r)-i-1D1D3if(r,t,um-1)(r)i-j+1um-1i-j‖0≤∑i=jN-11j!(i-j)!K̂*i+1(M,f)θi-j‖(r)i-j+1‖0=∑i=jN-1θi-jj!(i-j)!1i-j+3K̂*i+1(M,f).(iv.3.3) Estimating∥(r)-jb(3)∥0. We have(3.72)‖(r)-jb(3)‖0=‖(r)-j∑i=jN-11j!(i-j)!(-1)i-jD3i+1f(r,t,um-1)um-1i-j∇um-1‖0≤∑i=jN-11j!(i-j)!‖(r)-i-1D3i+1f(r,t,um-1)(r)i-jum-1i-jr∇um-1‖0≤∑i=jN-11j!(i-j)!K̂*i+1(M,f)θi-jM=M∑i=jN-1θi-jj!(i-j)!K̂*i+1(M,f).
It follows from (3.69)–(3.72) that
(3.73)L̃3(m,j,t)=‖(r)-j(∂∂rΨj(r,t,um-1))‖0≤2M∑i=jN-1θi-jj!(i-j)!K̂*i+1(M,f)+∑i=jN-1θi-jj!(i-j)!1i-j+3K̂*i+1(M,f)=∑i=jN-1θi-jj!(i-j)!(1i-j+3+2M)K̂*i+1(M,f).
It follows from (3.67) – (3.73) that
(3.74)L3≤∑j=1N-1L̃3(m,j,t)K2j(b*C0)j(Sm(k)(t))j≤∑j=1N-1∑i=jN-1θi-jj!(i-j)!(1i-j+3+2M)K̂*i+1(M,f)K2j(b*C0)j(Sm(k)(t))j.
We deduce from (3.57), (3.65), (3.66), (3.74) that
(3.75)‖∂∂rFm(k)(t)‖0≤(12+2M)K¯N(M,f)∑i=0N-1θii!+∑j=1N-1K2j(b*C0)j∑i=jN-1θi-jj!(i-j)!×[jK2-1K̂*i(M,f)+(1i-j+3+2M)K̂*i+1(M,f)](Sm(k)(t))j≡∑j=1N-1ãj(1)(Sm(k)(t))j,
where ãj(1),0≤j≤N-1 are defined by (3.49)2.
Next, we will estimate step by step all integralsI1,…,I5.
IntegralI1=∫0tḃm(k)(s)[a(um(k)(s),um(k)(s))+∥Aum(k)(s)∥02]ds.
Now, using the inequalities (3.48) and(3.76)sq≤1+sN0,∀s≥0,∀q,0<q≤N0=max⁡{1+α,N},
we estimate without difficulty the following integrals in the right-hand side of (3.47) as follows.
The integralI1:(3.77)I1=∫0tḃm(k)(s)[a(um(k)(s),um(k)(s))+‖Aum(k)(s)‖02]ds≤2d1b*3/2∫0t[(Sm(k)(s))2+b*1-α(Sm(k)(s))α+1]ds≤2d1b*3/2∫0t[1+(Sm(k)(s))N0+b*1-α(1+(Sm(k)(s))N0)]ds≤2d1b*3/2(1+b*1-α)∫0t[1+(Sm(k)(s))N0]ds.
The integralI2=2∫0t〈Fm(k)(s),u̇m(k)(s)〉ds:(3.78)I2=2∫0t〈Fm(k)(s),u̇m(k)(s)〉ds≤2∫0t‖Fm(k)(s)‖0‖u̇m(k)(s)‖0ds≤2∑j=0N-1ãj(0)∫0t(Sm(k)(s))j+1ds≤2∑j=0N-1ãj(0)∫0t[1+(Sm(k)(s))N0]ds.
The integralI3=2∫0ta(Fm(k)(s),u̇m(k)(s))ds:(3.79)I3=2∫0ta(Fm(k)(s),u̇m(k)(s))ds≤2∫0ta(Fm(k)(s),Fm(k)(s))a(u̇m(k)(s),u̇m(k)(s))ds≤2C1∫0t‖Fm(k)(s)‖1a(u̇m(k)(s),u̇m(k)(s))ds≤2C1∑j=0N-1ãj(1)∫0t(Sm(k)(s))j+1ds≤2C1∑j=0N-1ãj(1)∫0t[1+(Sm(k)(s))N0]ds.
The integralI4=∫0t〈Fm(k)(s),üm(k)(s)〉ds:(3.80)I4=∫0t〈Fm(k)(s),üm(k)(s)〉ds≤∫0t‖Fm(k)(s)‖0‖üm(k)(s)‖0ds≤∫0t‖Fm(k)(s)‖02ds+14∫0t‖üm(k)(s)‖02ds≤∫0t(∑j=0N-1ãj(0)(Sm(k)(s))j)2ds+14Sm(k)(t)≤N∑j=0N-1|ãj(0)|2∫0t(Sm(k)(s))jds+14Sm(k)(t)≤N∑j=0N-1|ãj(0)|2∫0t[1+(Sm(k)(s))N0]ds+14Sm(k)(t).
The integralI5=-∫0tbm(k)(s)〈Aum(k)(s),üm(k)(s)〉ds:(3.81)I5=-∫0tbm(k)(s)〈Aum(k)(s),üm(k)(s)〉ds≤∫0tbm(k)(s)‖Aum(k)(s)‖0‖üm(k)(s)‖0ds≤∫0tbm(k)(s)bm(k)(s)‖Aum(k)(s)‖02ds+14∫0t‖üm(k)(s)‖02ds≤d0∫0t[1+b*-α(Sm(k)(s))α]Sm(k)(s)ds+14Sm(k)(t)≤d0∫0t[Sm(k)(s)+b*-α(Sm(k)(s))α+1]ds+14Sm(k)(t)≤d0∫0t{(1+(Sm(k)(s))N0)+b*-α(1+(Sm(k)(s))N0)}ds+14Sm(k)(t)≤d0(1+b*-α)∫0t[1+(Sm(k)(s))N0]ds+14Sm(k)(t).
From the convergences in (3.10), we can deduce the existence of a constant M≥2 independent ofk and m such that(3.82)Sm(k)(0)≤M24.
Combining (3.47), (3.77)–(3.82), we then have(3.83)Sm(k)(t)≤M22+TD0(M)+D0(M)∫0t(Sm(k)(s))N0ds,
where
(3.84)D0(M)=2d0(1+b*-α)+4d1b*3/2(1+b*1-α)+4∑j=0N-1(ãj(0)+C1ãj(1)+12N|ãj(0)|2).
Then, we have the following Lemma.
Lemma 3.4.
There exists a constantT>0  independent of k and m such that
(3.85)Sm(k)(t)≤M2∀t∈[0,T],∀k,m.
Proof of Lemma3.4.
Put(3.86)Y(t)=M22+TD0(M)+D0(M)∫0t(Sm(k)(s))N0ds,0≤t≤T.
Clearly(3.87)Y(t)>0,0≤Sm(k)(t)≤Y(t),0≤t≤T,Y′(t)≤D0(M)YN0(t),0≤t≤T,Y(0)=M22+TD0(M).
PutZ(t)=Y-N0+1(t), after integrating of (3.87)
(3.88)Z(t)≥(M22+TD0(M))-N0+1-(N0-1)D0(M)t≥(M22+TD0(M))-N0+1-(N0-1)TD0(M),∀t∈[0,T].
Then, by(3.89)lim⁡T→0+[(M22+TD0(M))-N0+1-(N0-1)TD0(M)]=(M22)-N0+1>M-2N0+2,
we can always choose the constant T∈(0,T*] such that
(3.90)(M22+TD0(M))-N0+1-(N0-1)TD0(M)≥M-2N0+2.
Finally, it follows from (3.87), (3.88) and (3.90), that
(3.91)0≤Sm(k)(t)≤Y(t)=1Z(t)N0-1≤1(M2/2+TD0(M))-N0+1-(N0-1)TD0(M)N0-1≤M2,∀t∈[0,T].
The proof of Lemma3.4 is complete.
Remark 3.5.
The functionY(t)=[(M2/2+TD0(M))1-N0-(N0-1)D0(M)t]1/(1-N0),0≤t≤T, is the maximal solution of the following Volterra integral equation with non-decreasing kernel [20]. 									(3.92)Y(t)=M22+TD0(M)+D0(M)∫0tYN0(s)ds,0≤t≤T.
By Lemma 3.4, we can take constant Tm(k)=T for all k and m. Therefore, we have
(3.93)um(k)∈W1(M,T)∀m,k.
From (3.92), we can extract from {um(k)} a subsequence {um(ki)} such that
(3.94)um(ki)⟶uminL∞(0,T;V2)weak*,u̇m(ki)⟶um′inL∞(0,T;V1)weak*,üm(ki)⟶um′′inL2(0,T;V0)weak,(3.95)um∈W(M,T).
We can easily check from (3.9), (3.10), (3.94) that um satisfies (3.6), (3.7) in L2(0,T), weak. On the other hand, it follows from (3.6)1 and um∈W(M,T) that um′′=-bm(t)Aum+Fm∈L∞(0,T;V0), hence um∈W1(M,T) and the proof of Theorem 3.1 is complete.The following result gives a convergence at a rate of orderN of {um} to a weak solution of (1.1).First, we note thatW1(T)={v∈L∞(0,T;V1):v′∈L∞(0,T;V0)} is a Banach space with respect to the norm (see [19]): (3.96)‖v‖W1(T)=‖v‖L∞(0,T;V1)+‖v′‖L∞(0,T;V0).Then, we have the following theorem.Theorem 3.6.
Let (H1)–(H3) hold. Then, there exist constants M>0 and T>0 such that(i)
the problem (1.1) has a unique weak solution u∈W1(M,T);(ii)
the recurrent sequence{um} defined by (3.6), (3.7) converges at a rate of order N to the solution u strongly in the space W1(T) in the sense
(3.97)‖um-u‖W1(T)≤C‖um-1-u‖W1(T)N,
for all m≥1, where C is suitable constant.Furthermore, we have the estimation
(3.98)‖um-u‖W1(T)≤CTβNm,
for all m≥1, where CT and β<1 are positive constants depending only on T.Proof.
(a) Existence of the solution. First, we will prove that{um} is a Cauchy sequence in W1(T). Letvm=um+1-um. Then vm satisfies the variational problem
(3.99)〈vm′′(t),w〉+bm+1(t)a(vm(t),w)+(bm+1(t)-bm(t))〈Aum(t),w〉=〈Fm+1(t)-Fm(t),w〉,∀w∈V1,vm(0)=vm′(0)=0,
with
(3.100)f(r,t,um)-(r,t,um-1)=∑i=1N-11i!D3if(r,t,um-1)vm-1i+1N!D3Nf(r,t,λm)vm-1N,Fm+1(t)-Fm(t)=∑i=1N-11i!D3if(r,t,um-1)vmi+1N!D3Nf(r,t,λm)vm-1N,λm=um-1+θ1vm-1,(0<θ1<1),bm+1(t)-bm(t)=B(‖∇um+1(t)‖02)-B(‖∇um(t)‖02).
Takingw=vm′ in (3.99) and integrating in t we get
(3.101)γm(t)=∫0tbm+1′(s)a(vm(s),vm(s))ds-2∫0t[B(‖∇um+1(s)‖02)-B(‖∇um(s)‖02)]〈Aum(s),vm′(s)〉ds+2∑i=1N-11i!∫0t〈D3if(r,s,um-1)vmi,vm′(s)〉ds+1N!∫0t〈D3Nf(r,s,λm)vm-1N,vm′(s)〉ds=J1+⋯+J4,
where
(3.102)γm(t)=‖vm′(t)‖02+bm+1(t)a(vm(t),vm(t))≥‖vm′(t)‖02+b*C0‖vm(t)‖12≡Em(t).
We estimate without difficulty the following integralsJ1,…,J4 in the right-hand side of (3.101) as follows.
The integralJ1=∫0tbm+1′(s)a(vm(s),vm(s))ds(3.103)J1≤2C1d1(M2+M2α)∫0t‖vm(s)‖12ds≤2C1d1(M2+M2α)1b*C0∫0tEm(s)ds≡ξ1∫0tEm(s)ds.The integralJ2=-2∫0t[B(∥∇um+1(s)∥02)-B(∥∇um(s)∥02)]〈Aum(s),vm′(s)〉ds(3.104)J2≤4d1(M2+M2α)∫0t‖vm(s)‖1‖vm′(s)‖0ds≤4d1(M2+M2α)12b*C0∫0tEm(s)ds≡ξ2∫0tEm(s)ds.The integralJ3=2∑i=1N-1(1/i!)∫0t〈D3if(r,s,um-1)vmi,vm′(s)〉ds.(3.105)J3=2∑i=1N-11i!∫0t〈(r)-iD3if(r,s,um-1)(r)ivmi,vm′(s)〉ds≤2∑i=1N-11i!K̂*i(M,f)K2i∫0t‖vm(s)‖1i〈1,|vm′(s)|〉ds≤2∑i=1N-11i!K̂*i(M,f)K2i∫0t‖vm(s)‖1i‖vm′(s)‖0ds≤2∑i=1N-11i!K̂*i(M,f)K2iMi-1∫0t‖vm(s)‖1‖vm′(s)‖0ds≤2∑i=1N-11i!K̂*i(M,f)K2iMi-112b*C0∫0tEm(s)ds≡ξ3∫0tEm(s)ds.The integralJ4=(1/N!)∫0t〈D3Nf(r,s,λm)vm-1N,vm′(s)〉ds(3.106)J4=1N!∫0t〈(r)-ND3Nf(r,s,λm)(r)Nvm-1N,vm′(s)〉ds≤1N!K̂*N(M,f)K2N∫0t‖vm-1(s)‖1N〈1,|vm′(s)|〉ds≤1N!K̂*N(M,f)K2N12∫0t‖vm-1(s)‖1N‖vm′(s)‖0ds≤1N!K̂*N(M,f)K2N1222‖vm-1‖W1(T)N∫0t‖vm′(s)‖0ds≤1N!K̂*N(M,f)K2N122[T‖vm-1‖W1(T)2N+∫0t‖vm′(s)‖02ds]=ξ4[T‖vm-1‖W1(T)2N+∫0tEm(s)ds].
Combining (3.101), (3.103)–(3.106), we then obtain(3.107)Em(t)≤Tξ4‖vm-1‖W1(T)2N+(ξ1+ξ2+ξ3+ξ4)∫0tEm(s)ds≡Tξ4‖vm-1‖W1(T)2N+2ζM(1)∫0tEm(s)ds.
By using Gronwall's lemma, we obtain from (3.107), that(3.108)‖vm‖W1(T)≤μT‖vm-1‖W1(T)N,
where μT is the constant given by
(3.109)μT=(1+1b*C0)Tξ4exp⁡(TζM(1)).
Hence, we obtain from (3.108) that(3.110)‖um-um+p‖W1(T)≤(1-kT)-1(μT)-1/(N-1)(kT)Nm,
for all m and p. We take T>0 small enough, such that kT=(μT)1/(N-1)M<1. It follows that {um} is a Cauchy sequence in W1(T). Then there exists u∈W1(T) such that um→u strongly in W1(T). By the same argument used in the proof of Theorem 3.1, u∈W1(M,T) is a unique weak solution of the problem (1.1). Passing to the limit as p→+∞ for m fixed, we obtain the estimate (3.98) from (3.110) and Theorem 3.6 follows.



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*Source: 679528-2011-12-07.xml*