Hindawi Publishing Corporation
International Journal of Differential Equations
Volume 2011, Article ID 679528, 31 pages
doi:10.1155/2011/679528
Research Article
A High Order Iterative Scheme for a Nonlinear
Kirchhoff Wave Equation in the Unit Membrane
Le Thi Phuong Ngoc1 and Nguyen Thanh Long2
1
2
Nhatrang Educational College, 01 Nguyen Chanh Street, Nhatrang City, Vietnam
Department of Mathematics and Computer Science, University of Natural Science,
Vietnam National University Ho Chi Minh City, 227 Nguyen Van Cu Street, District 5,
Ho Chi Minh City, Vietnam
Correspondence should be addressed to Nguyen Thanh Long, longnt2@gmail.com
Received 5 May 2011; Accepted 16 October 2011
Academic Editor: Bashir Ahmad
Copyright q 2011 L. T. P. Ngoc and N. T. Long. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
A high-order iterative scheme is established in order to get a convergent sequence at a rate of
order N 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.
1. Introduction
In this paper we consider the initial and boundary value problem
utt − B
ur 20
1
urr ur fr, t, u, 0 < r < 1, 0 < t < T,
r
√
lim rur r, t < ∞,
1.1
r → 0
ur 1, t hu1, t 0,
ur, 0 u
0 r,
ut r, 0 u
1 r,
1 are given functions satisfying conditions specified later, ur 20 where B, f, u
0 , u
dr, and h > 0 is a given constant.
1
0
r|ur r, t|2
2
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Equation 1.11 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 |limr → 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 1.11 is related to the Kirchhoff equation
ρhutt Eh
P0 2L
L ∂u 2
y, t dy uxx
0 ∂y
1.2
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 1.11 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 fr, t, u fr, u and Bur 20 b0 ur 20 , b0 > 0. Based
on the ideas about recurrence relations for a third-order method for solving the nonlinear
operator equation Fu 0 in 15, we extend the above result by the construction of a highorder iterative scheme for 1.11 , where f and B are more generalized.
In this paper, we associate with 1.11 a recurrent sequence {um } defined by
∂2 um
2
−
B
u
mr
0
∂t2
∂2 um 1 ∂um
r ∂r
∂r 2
N−1
i0
1 ∂i f
r, t, um−1 um − um−1 i ,
i! ∂ui
1.3
0 < r < 1, 0 < t < T , with um satisfying 1.12-3 . The first term u0 is chosen as u0 ≡ 0. If B ∈
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
Put Ω 0, 1. We omit the definitions of the usual function spaces Cm Ω, Lp Ω, H m Ω, and
1
1/2
W m,p Ω. For any function v ∈ C0 Ω we define v0 as v0 0 rv2 rdr 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 v1 as v1 v20 vr 20 1/2
and define the space V1 as
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3
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
u, v 1
rurvrdr,
u, v ur , vr .
2.1
0
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 constants K1 > 0 and K2 > 0 such that, for all v ∈ C1 Ω, we have
i vr 20 v2 1 ≥ v20 ,
ii |v1| ≤ K1 v1 ,
√
iii r|vr| ≤ K2 v1 , for all r ∈ Ω.
Lemma 2.2. The embedding V1 → V0 is compact.
Remark
2.3. In Lemma
can be given explicitly as K1 2.1,√the two constants K1 and K2 √
√
1 2 and K2 1 5. We also note that limr → 0 rvr 0 for all v ∈ V1 see
17, page 128/Lemma 5.40. On the other hand, by H 1 ε, 1 → C0 ε, 1, 0 < ε < 1 and
√
εvH 1 ε,1 ≤ v1 for all v ∈ V1 , it follows that v|ε,1 ∈ C0 ε, 1. From both relations we
√
deduce that rv ∈ C0 Ω for all v ∈ V1 .
Now, let the bilinear form a·, · be defined by
au, v hu1v1 1
u, v ∈ V1 ,
rur rvr rdr,
2.2
0
where h is a positive constant. Then, there exists a unique bounded linear operator A : V1 →
V1 such that au, v Au, v for all u, v ∈ V1 . We then have the following lemma.
Lemma 2.4. The symmetric bilinear form a·, · defined by 2.2 is continuous on V1 ×V1 and coercive
on V1 , that is,
i |au, v| ≤ C1 u1 v1 ,
ii av, v ≥ C0 v21 ,
for all u, v ∈ V1 , where C0 1/2 min{1, h} and C1 1 1 √
2h.
The proof of Lemma 2.4 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 ↑ ∞ as j → ∞,
ii awj , v λj wj , v for all v ∈ V1 and j ∈ N.
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International Journal of Differential Equations
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
dwj
−1 d
r
λj wj , in Ω,
Awj ≡
r dr
dr
dwj
√ dwj r < ∞,
1 hwj 1 0.
lim r
r → 0
dr
dr
2.3
The proof of Lemma 2.5 can be found in 18, page 87, Theorem 7.7 with V V1 , H V0 and a·, · as defined by 2.2.
For any function v ∈ C2 Ω we define v2 as
1/2
v2 v20 vr 20 Av20
2.4
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
u, v ur , vr Au, Av
2.5
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 embedding V2 → V1 is compact.
Lemma 2.7. For all v ∈ V2 we have
1
vr L∞ Ω ≤ √ Av0 ,
2
3
ii vrr 0 ≤
Av0 ,
2
1
iii v2L∞ Ω ≤ 2v0 √ Av0 v0 .
2
i
2.6
For a Banach space X, we denote by · X its norm, by X its dual space, and by
L 0, T ; X, 1 ≤ p ≤ ∞ the Banach space of all real measurable functions u : 0, T → X
such that
p
uLp 0,T ;X T
0
1/p
p
utX dt
< ∞,
uL∞ 0,T ;X ess suputX
0<t<T
for 1 ≤ p < ∞,
for p ∞.
2.7
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5
Let
ut,
u t ut t u̇t,
u t utt t üt,
ur t ∇ut,
urr t
2.8
denote
∂u
r, t,
∂t
ur, t,
∂2 u
r, t,
∂t2
∂2 u
r, t,
∂r 2
∂u
r, t,
∂r
2.9
respectively.
With f ∈ Ck Ω × R × R, f fr, t, u, we put D1 f ∂f/∂r, D2 f ∂f/∂t, D3 f γ
γ
γ
∂f/∂u, and Dγ f D11 D22 D33 f, γ γ1 , γ2 , γ3 ∈ Z3 , |γ| γ1 γ2 γ3 k.
3. The Hight Order Iterative Schemes
Fix T ∗ > 0, we make the following assumptions:
H1 u
0 ∈ V2 and u
1 ∈ V1 ;
H2 B ∈ C1 R and there exist constants b∗ > 0, α > 1, d0 , d1 > 0 such that
i b∗ ≤ Bη ≤ d0 1 ηα , for all η ≥ 0,
ii |B η| ≤ d1 1 ηα−1 , for all η ≥ 0;
H3 f ∈ CN Ω × 0, T ∗ × R and satisfies the following condition : for all M > 0,
1 M, f
K
∗i
2 M, f
K
∗i
√ ∂i f
−i
sup r
r, t, u < ∞,
i
∂u
i 0, 1, . . . , N − 1,
r,t,u∈AM
√ ∂i f
−i
t,
u
sup r
r,
< ∞,
∂r∂ui−1
3.1
i 1, . . . , N − 1,
r,t,u∈AM
√
where AM {r, t, u ∈ 0, 1 × 0, T ∗ × R : |u| ≤ M 2 1/ 2}. We put
∗i
K
⎧ 1 ⎨K
∗0 M, f ,
M, f ⎩max K
2 M, f ,
1 M, f , K
∗i
∗i
i 0,
i 1, . . . , N − 1.
3.2
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International Journal of Differential Equations
With B and f satisfying assumptions H2 and H3 , respectively, for each M > 0
given, we introduce the following constants:
M B sup
K
B η B η ,
0≤η≤M2
sup fr, t, u,
K 0 M, f 3.3
r,t,u∈AM
K N M, f K 0 M, Dγ f .
|γ |≤N
For each T ∈ 0, T ∗ and M > 0 we get
WM, T v ∈ L∞ 0, T ; V2 : v ∈ L∞ 0, T ; V1 , v ∈ L2 0, T ; V0 ,
with vL∞ 0,T ;V2 , v L∞ 0,T ;V1 , v L2 0,T ;V0 ≤ M ,
3.4
W1 M, T v ∈ WM, T : v ∈ L∞ 0, T ; V0 .
We will choose as first initial term u0 ≡ 0, suppose that
um−1 ∈ W1 M, T ,
3.5
and associate with the problem 1.1 the following variational problem.
Find um ∈ W1 M, T m ≥ 1 so that
um t, v bm taum t, v Fm t, v,
0 ,
um 0 u
∀v ∈ V1 ,
um 0 u
1 ,
3.6
where
bm t B ∇um t20 ,
Fm r, t N−1
i0
1 i
D fr, t, um−1 um − um−1 i .
i! 3
3.7
Then, we have the following theorem.
Theorem 3.1. Let assumptions H1 –H3 hold. Then there exist a constant M > 0 depending on
0 , u
1 , B, h and a constant T > 0 depending on T ∗ , u
0 , u
1 , B, h, f such that, for u0 ≡ 0, there exists
T ∗, u
a recurrent sequence {um } ⊂ W1 M, T defined by 3.6, 3.7.
Proof. The proof consists of several steps.
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7
Step 1. The Faedo-Galerkin approximation introduced by Lions 19. Consider as in
Lemma 2.5 the basis {wj } for V1 and put
k
um t k
k
cmj twj ,
3.8
j1
k
where the coefficients cmj satisfy the system of the following nonlinear differential equations:
k
k
k
k
üm t, wj bm ta um t, wj Fm t, wj ,
k
um 0 u
0k ,
1 ≤ j ≤ k,
k
u̇m 0 u
1k ,
3.9
where
u
0k k
j1
u
1k
k
αj wj −→ u
0 strongly in V2 ,
3.10
k
k
βj wj −→ u
1 strongly in V1 .
j1
k 2
k
bm t B ∇um t ,
0
k
Fm r, t N−1
i0
N
i 1 i
k
k j
D3 fr, t, um−1 um − um−1 Ψj r, t, um−1 um ,
i!
j0
Ψj r, t, um−1 N−1
ij
1
i−j
−1i−j D3i fr, t, um−1 um−1 ,
j! i − j !
3.11
0 ≤ j ≤ N − 1.
Let us suppose that um−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–
k
k
3.11 has a unique solution um t on an interval 0, Tm ⊂ 0, T .
Proof of Lemma 3.2. The system of 3.8–3.11 is rewritten in the form
k
k
k
k
c̈mj t −λj bm tcmj t Fm t, wj ,
k
k
cmj 0 αj ,
k
1 ≤ j ≤ k,
k
3.12
ċmj 0 βj ,
and it is equivalent to the system of integral equations
k
cmj t
k
αj
k
βj t
− λj
t
τ
dτ
0
0
k
k
bm scmj sds
t
dτ
0
τ
0
k
Fm s, wj ds,
3.13
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International Journal of Differential Equations
for 1 ≤ j ≤ k. Omitting the index m, it is written as follows:
c Fc,
3.14
where Fc F1 c, . . . , Fk c, c c1 , . . . , ck ,
Fj ct qj t − λj
t
τ
dτ
0
N−1
t
i1
qj t k
αj
bsc
j sds
0
τ
Ψi ·, s, um−1 ui s, wj ds,
dτ
0
k
βj t
1 ≤ j ≤ k,
0
t
τ
dτ
0
Ψ0 ·, s, um−1 , wj ds,
3.15
1 ≤ j ≤ k,
0
bct
bt
B ∇ut20 ,
ut k
cj twj .
j1
k
k
For every Tm ∈ 0, T and ρ > 0 that will be chosen later, we put X C0 0, Tm ;
Rk , S {c ∈ Y : cX ≤ ρ}, where cX sup0≤t≤Tmk |ct|1 , |ct|1 kj1 |cj t|, for each
c c1 , . . . , ck ∈ X. Clearly S is a closed nonempty subset in X, and we have the operator
k
F : X → X. In what follows, we will choose ρ > 0 and Tm > 0 such that
i S is mapped into itself by F,
ii F : S → S is contractive.
Proof (i). First we note that, for all c c1 , . . . , ck ∈ S,
ut k
cj twj ,
j1
! k
!
ut0 " cj2 t ≤ |ct|1 ,
j1
∇ut20 ≤ aut, ut k
k
ci tcj ta wi , wj λj cj2 t ≤ λk ut20 ,
i,j1
3.16
j1
ut0 ≤ |ct|1 ≤ cX ≤ ρ,
∇ut0 ≤ λk |ct|1 ≤ λk ρ,
1 1 λk
λk
aut, ut ≤
λk ut0 ≤
ρ,
|ct|1 ≤
ut1 ≤
C0
C0
C0
C0
so
B ∇ut2 ≤ d0 1 ∇ut2α ≤ d0 1 λ2α ρ2α .
bt
0
0
k
3.17
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9
On the other hand, by
1
|um−1 | ≤ M 2 √ ≡ θ,
2
N−1
N−1
1 i
1
i i
i
i
D fr, t, um−1 um−1 −1 D3 fr, t, um−1 um−1 ≤
|Ψ0 r, t, um−1 | i0 i! 3
i0 i!
≤ KN
1
N−1
M, f
i!
i0
1
M 2 √
2
i
3.18
θi
N−1
,
K N M, f
i!
i0
we have
θi
N−1
Ψ0 t, um−1 , wj ≤ Ψ0 t, um−1 wj Ψ0 t, um−1 ≤ √1 K N M, f
.
0
0
0
i!
2
i0
3.19
By Lemma 2.1, iii, and the assumption H3 , we deduce from 3.16 that
Ψi s, um−1 ui s, wj N−1
1
l−i
l
l−i
i
D3 fr, t, um−1 um−1 u s, wj −1
li i!l − i!
N−1
√ l−i l−i √ i i
1
l−i √ −l l
r D3 fr, s, um−1 r um−1 r u s, wj −1
li i!l − i!
≤
N−1
li
≤
N−1
li
≤
N−1
li
√ l−i 1
∗l M, f θl−i K i usi1
r , wj K
2
i!l − i!
3.20
⎛
⎞i
l−i i
1
λ
1
k ⎠
∗l M, f θ K ⎝
ρ √
K
2
i!l − i!
C0
2l−i
⎞i
⎛
l−i i
1
1
λ
k
∗l M, f θ K ⎝
ρ⎠ ,
K
√
2
i!l − i! 2 l − i
C0
1 ≤ i ≤ N − 1.
It follows that
Fj ct ≤ qj t λk d0 1 λ2α ρ2α
k
N−1
N−1
1
t2
2 i1
li
t
τ
dτ
0
cj sds
0
⎛
1
1
∗l M, f θl−i K i ⎝
K
√
2
i!l − i! 2 l − i
⎞i
λk ⎠
ρ .
C0
3.21
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International Journal of Differential Equations
Thus
2α
|Fct|1 ≤ qt1 λk d0 1 λ2α
k ρ
t
τ
dτ
0
0
|cs|1 ds
⎞i
⎛
N−1
l−i i
1 2 N−1
1
1
λ
k
∗l M, f θ K ⎝
kt
ρ⎠
K
√
2
2
i!l − i! 2 l − i
C0
i1 li
3.22
1 k 2
≤ qT Dρ Tm
,
2
where qT sup0≤t≤T |qt|1 and
Dρ Dρ f, ρ, k, M, m, N
2α
ρk
λk d0 1 λ2α
k ρ
N−1
N−1
i1 li
⎞i
⎛
l−i i
1
1
λ
k
∗l M, f θ K ⎝
ρ⎠ .
K
√
2
i!l − i! 2 l − i
C0
3.23
Hence, we obtain
1 k 2
,
FcX ≤ qT Dρ Tm
2
3.24
k
choosing ρ > qT and Tm ∈ 0, T , such that
1 k 2
D ρ Tm
≤ ρ − qT ,
2
1 k 2
< 1,
Dρ Tm
2
3.25
where
ρ ρ, k, M, T, m, N, f
ρ D
D
2α
2 2
α−1 2α−2
2d
1
λ
ρ
λ
ρ
ρ
≡ d0 λk 1 λ2α
1
k
k
k
k
N−1
⎛
i⎝
i1
⎞i
λk ⎠ i−1
K2 ρ
C0
N−1
li
3.26
1
1
∗l M, f θl−i .
K
√
i!l − i! 2 l − i
Then
1 k 2
≤ ρ,
FcX ≤ qT Dρ Tm
2
which means that F maps S into itself.
∀c ∈ S,
3.27
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11
k
Proof (ii). We now prove that, for all c, d ∈ S, for all t ∈ 0, Tm ,
|Fct − Fdt|1 ≤
1 2
Dρ t c − dX ,
2
∀n ∈ N,
3.28
ρ is defined as 3.26.
where D
Proof of 3.28 is as follows.
k
For all j 1, 2, . . . , k, for all t ∈ 0, Tm , we have
Fj ct − Fj dt ≤ λj
t
τ
cj s − dj s ds
dτ bcs
0
0
t
λj
τ dj sds
− bds
dτ bcs
0
0
N−1
t
dτ
0
i1
3.29
τ Ψi s, um−1 ui s − vi s , wj ds,
0
where
bct
B ∇ut20 ,
ut k
cj twj ,
j1
bdt
B ∇vt20 ,
vt k
3.30
dj twj ,
j1
so
|Fct − Fdt|1 ≤ λk
t
τ
dτ
0
λk
0
t
0
τ
− bds
dτ bcs
|ds|1 ds
j1 i1
dτ
0
k N−1
t
j1 i1
dτ
0
τ
t
0
0
k N−1
t
≤ λk
bcs|cs
− ds|1 ds
0
τ Ψi s, um−1 ui s − vi s , wj ds
0
bcs|cs
− ds|1 ds λk ρ
dτ
t
0
τ
− bds
dτ bcs
ds
0
τ Ψi s, um−1 ui s − vi s , wj ds ≡ J1 J2 J3 ,
0
3.31
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International Journal of Differential Equations
in which
J 1 λk
t
τ
dτ
0
0
bcs|cs
− ds|1 ds
t
τ
|cs − ds|1 ds
3.32
bcs
− bds
B ξ ∇us20 − ∇vs20 ,
3.33
≤ λk d0 1 ≡ ζ1
t
τ
dτ
0
0
2α
λ2α
k ρ
dτ
0
0
|cs − ds|1 ds.
In order to consider J2 , we also note that
where
ξ θ∇us20 1 − θ∇vs20 ,
0 ≤ ξ ≤ λk ρ 2 ,
0 < θ < 1,
3.34
and B ξ satisfy the following inequality:
B ξ ≤ d1 1 ξα−1 ≤ d1 1 λα−1 ρ2α−2 .
k
3.35
It implies that
B ξ ∇us20 − ∇vs20 bcs − bds
2α−2
≤ d1 1 λα−1
2λk ρ|cs − ds|1
k ρ
3.36
2α−2
2λk ρd1 1 λα−1
ρ
|cs − ds|1 ,
k
and then
J 2 λk ρ
t
dτ
τ
ds
bcs − bds
0
0
τ
t
2α−2
≤ 2λ2k ρ2 d1 1 λα−1
ρ
dτ
|cs − ds|1 ds
k
0
≡ ζ2
t
τ
dτ
0
0
|cs − ds|1 ds.
0
3.37
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It remains to estimate J3 . By
√
i−1
i √
i j √
i−j−1 √
√
rus −
rvs rus
rvs
rus − vs,
3.38
j0
we obtain
i−1
√
i √
i j
i−j−1
rvs ≤ K2i us1 vs1 us − vs1
rus −
j0
≤ K2i
i−1
j
i−j−1
us1 vs1 us − vs1
j0
⎞j ⎛
⎞i−j−1 ⎛
i−1
λ
λ
λk
k
k
≤ K2i ⎝
ρ⎠ ⎝
ρ⎠
|cs − ds|1
C
C
C
0
0
0
j0
3.39
⎛ ⎞i
i−1
λk ⎠ i−1
K2i ⎝
ρ |cs − ds|1
C0
j0
⎛
i⎝
⎞i
λk ⎠ i−1
K2 ρ |cs − ds|1 .
C0
On the other hand,
Ψi r, t, um−1 N−1
li
1
−1l−i D3l fr, t, um−1 ul−i
m−1 ,
i!l − i!
N−1
√ l−i l−i √ −i
1
l−i √ −l l
r |Ψi r, t, um−1 | r D3 fr, t, um−1 r um−1 −1
li i!l − i!
N−1
√ √ −l l
1
l−i l−i ≤
r D3 fr, t, um−1 r um−1 li i!l − i!
≤
N−1
li
√ 1
∗l M, f θl−i r l−i .
K
i!l − i!
Hence, we deduce from 3.39, 3.40 that
J3 k N−1
t
j1 i1
≤
dτ
0
k N−1
N−1
j1 i1 li
τ √
i √
i √ −i
r Ψi s, um−1 rus −
rvs , wj ds
0
⎛
1
∗l M, f θl−i i⎝
K
i!l − i!
⎞i
λk ⎠ i−1
K2 ρ
C0
3.40
14
International Journal of Differential Equations
×
t
dτ
0
k N−1
N−1
j1 i1 li
0
⎞i
⎛
l−i
1
λ
1
k
∗l M, f θ i⎝
K2 ⎠ ρi−1 √
K
i!l − i!
C0
2l−i
×
t
τ
dτ
0
k
N−1
τ √ l−i r , wj |cs − ds|1 ds
⎛
i⎝
i1
0
|cs − ds|1 ds
⎞i
λk ⎠ i−1 N−1
1
∗l M, f θl−i √ 1
K2 ρ
K
li
C0
i!l − i!
2l−i
t
×
≡ ζ3
t
τ
dτ
0
0
τ
dτ
0
0
|cs − ds|1 ds
|cs − ds|1 ds.
3.41
We deduce that
|Fct − Fdt|1 ≤ ζ1 ζ2 ζ3 τ
t
dτ
0
0
|cs − ds|1 ds ≤
1 2
Dρ t c − dX .
2
3.42
We note that
ρ ρ, k, M, T, m, N, f D
ρ.
ζ1 ζ 2 ζ 3 D
3.43
It follows from 3.28 that
Fc − FdX ≤
1 k 2
Dρ Tm
c − dX ,
2
∀c, d ∈ S.
3.44
By 3.25, it follows that F : S → S is contractive. We deduce that F has a unique fixed
k
k
point in S; that is, the system 3.8–3.11 has a unique solution um t on an interval 0, Tm .
The proof of Lemma 3.2 is complete.
k
The following estimates allow one to take constant Tm T for all m and k.
Step 2. A priori estimates. Put
k
Sm t
k
Xm t
k
Ym t
t k 2
üm s ds,
0
0
3.45
International Journal of Differential Equations
15
where
k 2
k
k
k
k
Xm t u̇m t bm ta um t, um t ,
0
3.46
k 2
k
k
k
k
Ym t a u̇m t, u̇m t bm tAum t ,
0
with A is defined by 2.2. Then it follows that
k
k
Sm t Sm 0 t
2
0
t
0
k
t
0
' (
k 2
k
k
k
ḃm s a um s, um s Aum s ds
0
t k
k
k
k
Fm s, u̇m s ds 2 a Fm s, u̇m s ds
k
k
Fm s, üm s ds −
≡ Sm 0 5
t
0
0
k
k
k
bm s Aum s, üm s ds
3.47
Ij .
j1
We will now require the following lemma.
Lemma 3.3. We have
i
k 2α
k
,
0 < b∗ ≤ bm t ≤ d0 1 ∇um t
0
α *
)
k 2d1 k
k
ḃm t ≤ Sm t b∗1−α Sm t ,
b∗
N−1
0 k j
k aj
Sm t ,
iii Fm t ≤
ii
0
j0
N−1
1 k j
∂ k aj
Sm t ,
iv Fm t ≤
∂r
0
0
j0
1
where aj , aj , j 0, 1, . . . , N − 1 are defined as follows:
0
aj
⎧
θi
⎪
N−1
1
⎪
0
⎪
,
K
M,
f
a
⎪
√
N
⎪
0
⎪
i!
2
⎨
i0
j
N−1
θi−j
K2
⎪
1
0
⎪
∗i M, f ,
⎪
a
K
⎪
j
j
⎪
⎪
⎩
b∗ C0 ij j! i − j ! 2 i − j
j 0,
1 ≤ j ≤ N − 1,
3.48
16
International Journal of Differential Equations
⎧
θi
⎪
N−1
1
⎪
⎪
,
j 0,
⎪ √ 2M K N M, f
⎪
⎪
i!
2
⎪
i0
⎪
⎪
,
⎪
j
⎪
N−1
⎨
θi−j
K2
1
∗i M, f
jK2−1 K
aj j
⎪
⎪
b∗ C0 ij j! i − j !
⎪
⎪
⎪
⎪
⎪
⎪
1
⎪
⎪
2M K∗i1 M, f , 1 ≤ j ≤ N − 1,
⎪
⎩
i−j 3
1
θ M 2 √ .
2
3.49
Proof of Lemma 3.3. Proof (i), (ii). Note that
k 2
k
k
k
k
Sm t ≥ Xm t ≥ b∗ a um t, um t ≥ b∗ ∇um t ,
0
2
k k
k
k
k
Sm t ≥ Ym t ≥ a u̇m t, u̇m t ≥ ∇u̇m t .
3.50
0
We deduce that
k 2α
k 2
k
,
bm t B ∇um t ≤ d0 1 ∇um t
0
0
k k 2 k
k
ḃm t 2B ∇um t ∇um t, ∇u̇m t 0
k 2α−2 k k ≤ 2d1 1 ∇um t
∇um t ∇u̇m t
0
3.51
N−1
j k Ψj r, t, um−1 uk
.
Fm t ≤ Ψ0 r, t, um−1 0 m
3.52
0
0
α−1 1
k
k
≤ 2d1 1 b∗1−α Sm t
Sm t
b∗
)
*
α
2d1 k
k
Sm t b∗1−α Sm t .
b∗
Proof (iii). We have
0
j1
0
By 3.183 , we have
θi
N−1
1
0
≡ a0 .
Ψ0 r, t, um−1 0 ≤ √ K N M, f
i!
2
i0
3.53
International Journal of Differential Equations
17
On the other hand, it follows from 3.49 and um−1 ∈ W1 M, T that
Ψj r, t, um−1 uk
m
N−1
i
1
i−j
k j ≤
D fr, t, um−1 um−1 um j! i − j ! 3
j
0
0
ij
N−1
ij
≤
N−1
ij
N−1
ij
√ i−j √ −i i
1
i−j √ j
k j r
r D3 fr, t, um−1 um−1 r um j! i − j ! 0
j
√ 1
i−j j k
r
K∗i M, f θi−j K2 um t
0
1
j! i − j !
j
1
1
k
∗i M, f θi−j K j K
2 um t1
j! i − j ! 2 i − j
j
K2
≤ j
b∗ C0
≡
0
aj
N−1
ij
1
θi−j
∗i M, f
K
j! i − j ! 2 i − j
j
k
Sm t
j
k
Sm t
.
3.54
It follows from 3.52–3.54 that
N−1
0 k j
k Sm t ,
aj
Fm t ≤
0
3.55
j0
0
where aj , 0 ≤ j ≤ N − 1 are defined by 3.491 .
Proof (iv). We have
N−1
∂ k
∂
k j−1
k
Fm r, t Ψ0 r, t, um−1 jΨj r, t, um−1 um
∇um
∂r
∂r
j1
N−1
j1
∂
k j
Ψj r, t, um−1 um .
∂r
3.56
Hence
N−1
∂ k k j−1
k Fm t ≤ ∂ Ψ0 r, t, um−1 j Ψj r, t, um−1 um
∇um ∂r
∂r
0
0
0
j1
N−1
∂
k j ∂r Ψj r, t, um−1 um ≡ L1 L2 L3 .
0
j1
3.57
18
International Journal of Differential Equations
We shall estimate step by step the terms on the right-hand side of 3.57 as follows.
(iv.1) Estimating L1 ∂/∂rΨ0 r, t, um−1 0 . We have
N−1
1
∂
Ψ0 r, t, um−1 −1i D3i fr, t, um−1 iui−1
m−1 ∇um−1
∂r
i!
i1
N−1
i0
N−1
i0
1
−1i D1 D3i fr, t, um−1 uim−1
i!
3.58
1
−1i D3i1 fr, t, um−1 uim−1 ∇um−1
i!
a1 a2 a3.
We will estimate step by step the terms a1, a2, a3 as follows.
(iv.1.1) Estimating a10 . We have
N−1
1
i i
i−1
−1 D3 fr, t, um−1 ium−1 ∇um−1 |a1| i1 i!
1
N−1
≤ K N M, f
iθi−1 |∇um−1 |
i!
i1
KN
θi
N−2
M, f
|∇um−1 |
i!
i0
3.59
θi
N−1
≤ K N M, f
|∇um−1 |.
i!
i0
Hence
θi
N−1
N−1 θi
.
a10 ≤ K N M, f
∇um−1 0 ≤ MK N M, f
i0 i!
i!
i0
3.60
(iv.1.2) Estimating a20 . It follows from
N−1
θi
N−1
1
i
i
i
−1 D1 D3 fr, t, um−1 um−1 ≤ K N M, f
|a2| i0 i!
i!
i0
3.61
θi
N−1
1
.
a20 ≤ √ K N M, f
i!
2
i0
3.62
that
International Journal of Differential Equations
19
(iv.1.3) Estimating a30 . Similarly, with
N−1
θi
N−1
1
i i1
i
−1 D3 fr, t, um−1 um−1 ∇um−1 ≤ K N M, f
|a3| |∇um−1 |,
i0 i!
i!
i0
3.63
we obtain
θi
θi
N−1
N−1
.
a30 ≤ K N M, f
∇um−1 0 ≤ MK N M, f
i!
i!
i0
i0
3.64
It follows from 3.58, 3.60, 3.62, 3.64 that
∂
Ψ
L1 t,
u
r,
m−1 ≤ a10 a20 a30
∂r 0
0
θi
θi
N−1
N−1
1
2MK N M, f
≤ √ K N M, f
i!
i!
2
i0
i0
≤
θi
N−1
1
.
√ 2M K N M, f
i!
2
i0
(iv.2) Estimating L2 we deduce that
L2 N−1
N−1
j
ij
N−1
N−1
j
ij
j1
N−1
N−1
j
ij
j1
≤
N−1
N−1
j
j1
≤
j1
k j−1
jΨj r, t, um−1 um k j−1
k Ψ
u
j
∇u
t,
u
r,
m−1
m
m j
j1
N−1
k
∇um 0 . By the assumption H3 ,
N−1
j1
≤
3.65
N−1
j1
ij
0
i
1
i−j
k j−1
k D
u
fr,
t,
u
∇u
u
m−1 m−1
m
m j! i − j ! 3
0
√ −i i
√ i−j i−j √ k j−1 √
1
k r
D
fr,
t,
u
r
u
ru
r∇u
m−1
m
m 3
m−1
j! i − j ! 0
j−1
√ i−j √
1
j−1 k
k ∗i M, f r∇um θi−j K2 um t
K
r
1
0
j! i − j !
j
θi−j
k
∗i M, f K j−1 K
t
u
m
2
1
j! i − j !
j
k
Sm t .
N−1
θi−j
jK2
∗i M, f
K
j
b∗ C0 ij j! i − j !
j−1
3.66
20
International Journal of Differential Equations
(iv.3) Estimating L3 N−1
j1
k j
∂/∂rΨj r, t, um−1 um 0 . We have
N−1
∂
k j L3 ∂r Ψj r, t, um−1 um 0
j1
N−1
√ −j ∂
√ j k j Ψj r, t, um−1 r um r
∂r
0
j1
N−1
j
√ −j ∂
j
k
r
K
Ψ
≤
t,
u
r,
t
u
j
m−1
m
2
1
∂r
0
j1
j
N−1
√ −j ∂
1
j
k
Ψj r, t, um−1 ≤
Sm t
j K2
r
∂r
0
j1
b∗ C0
≡
N−1
j
K2
3 m, j, t L
j
j1
b∗ C0
3.67
j
k
Sm t .
3 m, j, t √r−j ∂/∂rΨj r, t, um−1 . We have
Now we need an estimation of the term L
0
N−1
∂
1
i−j−1
Ψj r, t, um−1 −1i−j i − j D3i fr, t, um−1 um−1 ∇um−1
∂r
ij1 j! i − j !
N−1
ij
N−1
ij
1
i−j
−1i−j D1 D3i fr, t, um−1 um−1
j! i − j !
3.68
1
i−j
−1i−j D3i1 fr, t, um−1 um−1 ∇um−1
j! i − j !
b1 b2 b3.
It follows from 3.68 that
√ −j ∂
Ψj r, t, um−1 L3 m, j, t r
∂r
0
√ √ √ −j
−j
−j
≤ r b1 r b2 r b3 .
0
0
3.69
0
√ −j
(iv.3.1) Estimating r b10 . We have
√ √ −j N−1
i
1
−j
i−j−1
i−j
−1 i − j D3 fr, t, um−1 um−1 ∇um−1 r b1 r
0
ij1 j! i − j !
0
≤
√ i−j−1 i−j−1 √
1
√ −i
um−1 r∇um−1 i − j r D3i fr, t, um−1 r
0
j!
i
−
j
!
ij1
N−1
International Journal of Differential Equations
21
N−1
1
∗i M, f θi−j−1 M
i−j K
ij1 j! i − j !
≤
M
N−2
ij
N−1
θi−j
θi−j
∗i1 M, f .
∗i1 M, f ≤ M
K
MK
j! i − j !
ij j! i − j !
3.70
√ −j
(iv.3.2) Estimating r b20 . We have
√ √ −j N−1
1
−j
i−j
i−j
i
−1 D1 D3 fr, t, um−1 um−1 r b2 r
0
ij j! i − j !
0
≤
N−1
ij
≤
N−1
ij
N−1
ij
√ √ i−j1 i−j 1
−i−1
D1 D3i fr, t, um−1 r
um−1 r
0
j! i − j !
√ 1
i−j1 ∗i1 M, f θi−j K
r
0
j! i − j !
3.71
1
θi−j
∗i1 M, f .
K
j! i − j ! i − j 3
√ −j
(iv.3.3) Estimating r b30 . We have
N−1
√ √
1
−j
−j
i−j
i−j i1
r
D
fr,
t,
u
∇u
u
−1
r b3 m−1 m−1
m−1 3
0
ij j! i − j !
0
N−1
≤
ij
N−1
≤
ij
√ √ i−j i−j √
1
−i−1 i1
D3 fr, t, um−1 r
um−1 r∇um−1 r
0
j! i − j !
3.72
N−1
θi−j
1
∗i1 M, f θi−j M M
∗i1 M, f .
K
K
j! i − j !
ij j! i − j !
It follows from 3.69–3.72 that
√ −j ∂
Ψj r, t, um−1 L3 m, j, t r
∂r
0
≤ 2M
N−1
ij
N−1
ij
θi−j
N−1
1
θi−j
∗i1 M, f
∗i1 M, f K
K
j! i − j !
i−j 3
ij j! i − j !
θi−j
j! i − j !
1
∗i1 M, f .
2M K
i−j 3
3.73
22
International Journal of Differential Equations
It follows from 3.67 – 3.73 that
N−1
j
K2
3 m, j, t L3 ≤
L
j
j1
b∗ C0
≤
N−1
N−1
j1 ij
i−j
θ
j! i − j !
k
Sm t
j
j
K2
1
∗i1 M, f 2M K
j
i−j 3
b∗ C0
j
k
Sm t .
3.74
We deduce from 3.57, 3.65, 3.66, 3.74 that
θi
∂ k N−1
Fm t ≤ √1 2M K N M, f
∂r
i!
2
0
i0
N−1
j1
j
K2
b∗ C0
j
,
∗i
jK2−1 K
×
≡
N−1
j1
1
aj
N−1
ij
θi−j
j! i − j !
M, f -
j
1
k
S
M,
f
2M
K
t
∗i1
m
i−j 3
j
k
Sm t ,
3.75
1
where aj , 0 ≤ j ≤ N − 1 are defined by 3.492 .
Next, we will estimate step by step all integrals I1 , . . . , I5 .
t k
2
k
k
k
Integral I1 0 ḃm saum s, um s Aum s0 ds.
Now, using the inequalities 3.48 and
sq ≤ 1 sN0 ,
∀s ≥ 0, ∀q, 0 < q ≤ N0 max{1 α, N},
3.76
we estimate without difficulty the following integrals in the right-hand side of 3.47 as
follows.
The integral I1 :
I1 t
0
≤2
k
ḃm s
d1
b∗3/2
' (
k 2
k
k
a um s, um s Aum s ds
t '
0
0
2
α1 (
k
k
Sm s b∗1−α Sm s
ds
International Journal of Differential Equations
≤2
≤2
t'
d1
b∗3/2
0
d1 b∗3/2
23
N0
N0 (
k
k
b∗1−α 1 Sm s
1 Sm s
ds
1 b∗1−α
t'
0
N0 (
k
1 Sm s
ds.
3.77
t k
k
The integral I2 2 0 Fm s, u̇m sds:
I2 2
t
0
t k k k
k
Fm s, u̇m s ds ≤ 2 Fm s u̇m s ds
0
0
N−1
≤2
j0
0
aj
The integral I3 2
0
t j1
N−1
N0 (
0 t '
k
k
Sm s
ds ≤ 2 aj
1 Sm s
ds.
0
t
0
j0
k
3.78
0
k
aFm s, u̇m sds:
t k
k
I3 2 a Fm s, u̇m s ds
0
t . . k
k
k
k
≤2
a Fm s, Fm s
a u̇m s, u̇m s ds
0
. t k k
k
≤ 2 C1 Fm s a u̇m s, u̇m s ds
3.79
1
0
1 t k j1
N−1
≤ 2 C1
aj
Sm s
ds
0
j0
N0 (
1 t '
N−1
k
≤ 2 C1
aj
1 Sm s
ds.
0
j0
The integral I4 I4 t
0
t
0
k
k
Fm s, üm sds:
k
k
Fm s, üm s
t k k ds ≤ Fm s üm s ds
0
0
t 1 t
k 2
k 2
Fm s ds üm s ds
0
0
4 0
0
⎛
⎞2
t N−1
0 k j
1 k
≤ ⎝ aj
Sm s ⎠ ds Sm t
4
0
j0
≤
0
24
International Journal of Differential Equations
t j
1 k
0 2
k
Sm s ds Sm t
aj ≤N
4
0
j0
N−1
≤N
t'
N0 (
1 k
0 2
k
aj 1 Sm s
ds Sm t.
4
0
j0
N−1
3.80
The integral I5 −
I5 −
t
0
≤
k
bm s
t
0
k
k
k
bm s
Aum s, üm sds:
k
k
Aum s, üm s
t
k k k
ds ≤ bm sAum s üm s ds
0
0
0
1 t
k 2
k 2
k
k
bm sbm sAum s ds üm s ds
0
0
4 0
0
t
≤ d0
t)
0
≤ d0
α *
1 k
k
k
1 b∗−α Sm s Sm sds Sm t
4
t'
0
k
Sm s
α1
k
b∗−α Sm s
(
3.81
1 k
ds Sm t
4
t /
N0 N0 0
1 k
k
k
1 Sm s
b∗−α 1 Sm s
ds Sm t
≤ d0
4
0
≤ d0 1 b∗−α
t'
0
N0 (
1 k
k
1 Sm s
ds Sm t.
4
From the convergences in 3.10, we can deduce the existence of a constant M ≥ 2 independent of k and m such that
k
Sm 0 ≤
M2
.
4
3.82
Combining 3.47, 3.77–3.82, we then have
k
Sm t ≤
M2
T D0 M D0 M
2
t 0
k
Sm s
N0
ds,
3.83
where
N−1 0 4d1 1 0 2
1
D0 M 2d0 1 b∗−α 3/2 1 b∗1−α 4 j0 aj C1 aj N aj .
2
b∗
3.84
International Journal of Differential Equations
25
Then, we have the following Lemma.
Lemma 3.4. There exists a constant T > 0 independent of k and m such that
k
3.85
∀t ∈ 0, T , ∀k, m.
Sm t ≤ M2
Proof of Lemma 3.4. Put
Y t M2
T D0 M D0 M
2
t 0
k
Sm s
N0
ds,
0 ≤ t ≤ T.
3.86
Clearly
k
Y t > 0, 0 ≤ Sm t ≤ Y t, 0 ≤ t ≤ T,
Y t ≤ D0 MY N0 t,
Y 0 0 ≤ t ≤ T,
3.87
M2
T D0 M.
2
Put Zt Y −N0 1 t, after integrating of 3.87
−N0 1
M2
T D0 M
− N0 − 1D0 Mt
2
−N0 1
M2
T D0 M
− N0 − 1T D0 M, ∀t ∈ 0, T .
2
Zt ≥
≥
3.88
Then, by
⎤ ⎡
−N0 1
−N0 1
2
M
M2
⎦
⎣
T D0 M
− N0 − 1T D0 M > M−2N0 2 ,
lim
T → 0
2
2
3.89
we can always choose the constant T ∈ 0, T ∗ such that
M2
T D0 M
2
−N0 1
− N0 − 1T D0 M ≥ M−2N0 2 .
3.90
26
International Journal of Differential Equations
Finally, it follows from 3.87, 3.88 and 3.90, that
k
0 ≤ Sm t ≤ Y t ≤
N0 −1
1
Zt
N0 −1
1
−N0 1
M2 /2 T D0 M
− N0 − 1T D0 M
≤ M2 , ∀t ∈ 0, T .
3.91
The proof of Lemma 3.4 is complete.
1/1−N0 1−N
Remark 3.5. The function Y t M2 /2 T D0 M 0 − N0 − 1D0 Mt
, 0≤t≤
T , is the maximal solution of the following Volterra integral equation with non-decreasing
kernel 20.
Y t M2
T D0 M D0 M
2
t
Y N0 sds, 0 ≤ t ≤ T.
3.92
0
k
By Lemma 3.4, we can take constant Tm T for all k and m. Therefore, we have
k
um ∈ W1 M, T ∀m, k.
k
3.93
k From 3.92, we can extract from {um } a subsequence {um i } such that
k in L∞ 0, T ; V2 weak ∗ ,
k in L∞ 0, T ; V1 weak ∗ ,
um i −→ um
u̇m i −→ um
k üm i −→ um
3.94
in L2 0, T ; V0 weak,
um ∈ WM, T .
3.95
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 3.61 and um ∈ WM, T that um −bm tAum 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 order N of {um } to a weak solution
of 1.1.
First, we note that W1 T {v ∈ L∞ 0, T ; V1 : v ∈ L∞ 0, T ; V0 } is a Banach space
with respect to the norm see 19:
vW1 T vL∞ 0,T ;V1 v L∞ 0,T ;V0 .
Then, we have the following theorem.
3.96
International Journal of Differential Equations
27
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
um − uW1 T ≤ Cum−1 − uN
W1 T ,
3.97
for all m ≥ 1, where C is suitable constant.
Furthermore, we have the estimation
m
um − uW1 T ≤ CT βN ,
3.98
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 . Let vm um1 − um . Then vm satisfies the variational problem
vm t, w bm1 tavm t, w bm1 t − bm t
Aum t, w
Fm1 t − Fm t, w,
∀w ∈ V1 ,
3.99
vm 0 vm
0 0,
with
fr, t, um − r, t, um−1 N−1
i1
Fm1 t − Fm t N−1
i1
1 i
1 N
i
N
D3 fr, t, um−1 vm−1
D fr, t, λm vm−1
,
i!
N! 3
1 i
1 N
i
N
D fr, t, um−1 vm
D fr, t, λm vm−1
,
i! 3
N! 3
λm um−1 θ1 vm−1 , 0 < θ1 < 1,
bm1 t − bm t B ∇um1 t20 − B ∇um t20 .
Taking w vm
in 3.99 and integrating in t we get
γm t t
0
bm1
savm s, vm sds
t) *
B ∇um1 s20 − B ∇um s20
−2
Aum s, vm
s ds
0
3.100
28
International Journal of Differential Equations
N−1
2
i1
1
i!
t
0
i
D3i fr, s, um−1 vm
, vm
s
1 t N
N
ds D3 fr, s, λm vm−1
, vm
s ds
N! 0
J1 · · · J 4 ,
3.101
where
2
2
γm t vm
t0 bm1 tavm t, vm t ≥ vm
t0 b∗ C0 vm t21 ≡ Em t.
3.102
We estimate without difficulty the following integrals J1 , . . . , J4 in the right-hand side
of 3.101 as follows.
t savm s, vm sds
The integral J1 0 bm1
J1 ≤ 2C1 d1 M M
2
2α
t
0
vm s21 ds
t
1 t
Em sds ≡ ξ1 Em sds.
≤ 2C1 d1 M M
b∗ C0 0
0
2
3.103
2α
t
sds
The integral J2 −2 0 B∇um1 s20 − B∇um s20 Aum s, vm
J2 ≤ 4d1 M M
2
2α
t
0
≤ 4d1 M M
2
2α
s0 ds
vm s1 vm
1
2 b∗ C0
t
0
Em sds ≡ ξ2
t
3.104
Em sds.
0
t i
i
The integral J3 2 N−1
i1 1/i! 0 D3 fr, s, um−1 vm , vm sds.
N−1 1 t √ −i
√ i i r D3i fr, s, um−1 r vm
, vm s ds
J3 2 i1
i! 0
N−1 1
i t
≤ 2 i1 K∗i M, f K2 vm si1 1, vm
s ds
i!
0
t
√ N−1 1
∗i M, f K i vm si1 vm
≤ 2 i1 K
s0 ds
2
i!
0
√ N−1 1
i i−1 t
≤ 2 i1 K∗i M, f K2 M
s0 ds
vm s1 vm
i!
0
t
t
√ N−1 1
∗i M, f K i Mi−1 1
E
Em sds.
≤ 2 i1 K
≡
ξ
sds
m
3
2
i!
2 b∗ C0 0
0
3.105
International Journal of Differential Equations
29
t
N
, vm
sds
The integral J4 1/N! 0 D3N fr, s, λm vm−1
J4 ≤
≤
1
N!
t
0
√ N N
√ −N N
r
D3 fr, s, λm r vm−1
, vm
s ds
1 K∗N M, f K2N
N!
t
0
vm−1 sN
1 1, vm s ds
1
1 K∗N M, f K2N √
N!
2
t
0
vm−1 sN
1 vm s 0 ds
t
N 1
1 N
≤
K∗N M, f K2 √ 2vm−1 W1 T vm
s0 ds
N!
2 2
0
,
t
2
N 1
1 2N
≤
K∗N M, f K2 √ T vm−1 W1 T vm s0 ds
N!
2 2
0
,
t
ξ4 T vm−1 2N
W1 T 3.106
Em sds .
0
Combining 3.101, 3.103–3.106, we then obtain
Em t ≤
≡
T ξ4 vm−1 2N
W1 T T ξ4 vm−1 2N
W1 T ξ1 ξ2 ξ3 ξ4 1
2ζM
t
t
Em sds
0
3.107
Em sds.
0
By using Gronwall’s lemma, we obtain from 3.107, that
vm W1 T ≤ μT vm−1 N
W1 T ,
3.108
where μT is the constant given by
μT 1
1 b∗ C0
1
T ξ4 exp T ζM .
3.109
Hence, we obtain from 3.108 that
−1/N−1
m
um − ump ≤ 1 − kT −1 μT
kT N ,
W1 T 3.110
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
30
International Journal of Differential Equations
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.
Acknowledgments
The authors wish to express their sincere thanks to the referees for the suggestions and valuable comments. The authors are also extremely grateful for the support given by Vietnam’s
National Foundation for Science and Technology Development NAFOSTED under Project
101.01-2010.15.
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