Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 805158, 15 pages
doi:10.1155/2012/805158
Research Article
Global Existence of Strong Solutions to
a Class of Fully Nonlinear Wave Equations with
Strongly Damped Terms
Zhigang Pan,1 Hong Luo,2 and Tian Ma1
1
2
Yangtze Center of Mathematics, Sichuan University, Chengdu, Sichuan 610041, China
College of Mathematics and Software Science, Sichuan Normal University, Chengdu,
Sichuan 610066, China
Correspondence should be addressed to Zhigang Pan, pzg555@yeah.net
Received 20 February 2012; Revised 8 May 2012; Accepted 9 May 2012
Academic Editor: Kuppalapalle Vajravelu
Copyright q 2012 Zhigang Pan et al. 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.
We consider the global existence of strong solution u, corresponding to a class of fully nonlinear
wave equations with strongly damped terms utt − kΔut fx, Δu gx, u, Du, D2 u in a bounded
and smooth domain Ω in Rn , where fx, Δu is a given monotone in Δu nonlinearity satisfying
some dissipativity and growth restrictions and gx, u, Du, D2 u is in a sense subordinated to
fx, Δu. By using spatial sequence techniques, the Galerkin approximation method, and some
0, ∞, W 2,p Ω ∩
monotonicity arguments, we obtained the global existence of a solution u ∈ L∞
loc
1,p
W0 Ω.
1. Introduction
We are concerned with the following mixed problem for a class of fully nonlinear wave
equations with strongly damped terms in a bounded and C∞ domain Ω ⊂ Rn :
utt − kΔut fx, Δu g x, u, Du, D2 u ,
u0, x ϕ,
ut, x 0,
ut 0, x ψ,
in 0, ∞ × Ω,
in Ω,
on 0, ∞ × ∂Ω,
1.1
2
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where
ut ∂u
,
∂t
utt ∂2 u
,
∂t2
Δ
n
∂2
,
2
i1 ∂xi
α1 · · · αn 2,
D
∂
∂
,...,
,
∂x1
∂xn
x x1 , . . . , xn ,
D2 ∂x1α1
∂2
,
· · · ∂xnαn
1.2
k > 0.
Equations of type 1.1 are a class essential nonlinear wave equations describing the speed
of strain waves in a viscoelastic configuration e.g., a bar if the space dimension N 1 and
a plate if N 2 made up of the material of the rate type 1, 2. They can also be seen as
field equations governing the longitudinal motion of a viscoelastic bar obeying the nonlinear
Voigt model 3. Concerning damped cases, there is much to the global existence of solutions
for the problem:
utt ut − uxx fu,
u0, x u0 x,
in 0, ∞ × Ω,
ut 0, x u1 x,
in Ω;
1.3
they discussed the global existence of weak solutions and regularity in R1 and Rn 4–8. On
the other hand, Ikehata and Inoue 9 considered the global existence of weak solutions for
two-dimensional problem in an exterior domain Ω ⊂ R2 with a compact smooth boundary
∂Ω for a semilinear strongly damped wave equation with a power-type nonlinearity |u|q and
q > 6:
utt t, x − Δut, x − Δut t, x |ut, x|q
u0, x u0 x,
ut, x 0,
in 0, ∞ × Ω,
ut 0, x u1 x in Ω,
1.4
on 0, ∞ × ∂Ω.
Cholewa and Dlotko 10 discussed the global solvability and asymptotic behavior of
solutions to semilinear Cauchy problem for strongly damped wave equation in the whole of
Rn . They assume the nonlinear term f grows like |u|q and q < n 2/n − 2 if n ≥ 3. Similar
problems attracted attention of the researchers for many years 11–13. Especially, Yang 14
studied the global existence of weak solutions to the more general equation including 1.4,
but he did not discuss the regularity of weak solution for the quasilinear wave equation. We
are interested in discussing the global existence and regularity of weak solutions for strongly
damped wave equation with the dissipative terms g containing Du and the nonlinear terms
f containing Δu. Here fx, Δu is a given monotone in Δu nonlinearity satisfying some
dissipativity and growth restrictions and gx, u, Du, D2 u is in a sense subordinated to
fx, Δu.
In 15, we have investigated the existence of global solutions to a class of nonlinear
damped wave operator equations. In this paper, our first aim is to study the global existence
of strong solutions to the more general equation including 1.4, which is the motivation that
we establish our abstract strongly damped wave equation model with. The second aim is to
deal with the global existence of strong solutions to a class of fully nonlinear wave equations
with strongly damped terms under some weakly growing conditions.
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3
This paper is organized as follows:
i in Section 2, we recall some preliminary tools and definitions;
ii in Section 3, we put forward our abstract strongly damped wave equation model
and proof the global existence of strong solution of it;
iii in Section 4, we provide the proof of the main results about the mixed problem
1.1.
2. Preliminaries
We introduce two spatial sequences:
X ⊂ H3 ⊂ X2 ⊂ X1 ⊂ H,
X2 ⊂ H2 ⊂ H1 ⊂ H,
2.1
where H, H1 , H2 , and H3 are Hilbert spaces, X is a linear space, and X1 , X2 are Banach spaces.
All embeddings of 2.1 are dense. Let
L : X −→ X1
be one-for-one dense linear operator,
Lu, vH u, vH1 ,
∀u, v ∈ X.
2.2
Furthermore, L has eigenvectors {ek } satisfying
Lek λk ek ,
k 1, 2, . . .,
2.3
and {ek } constitutes common orthogonal basis of H and H3 .
We consider the following abstract wave equation model:
d2 u
d
k Lu Gu,
2
dt
dt
u0 ϕ,
k > 0,
2.4
ut 0 ψ,
where G : X2 × R −→ X1∗ is a map, R 0, ∞, and L : X2 −→ X1 is a bounded linear
operator, satisfying
Lu, LvH u, vH2 ,
∀u, v ∈ X2 .
2.5
Definition 2.1. We say that u ∈ W 1,∞ 0, T , H1 ∩ L∞ 0, T , X2 is a global weak solution of
the 2.4 provided for ϕ, ψ ∈ X2 × H1
ut , vH kLu, vH for each v ∈ X1 and 0 ≤ t ≤ T < ∞.
t
0
Gu, vdτ ψ, v H k Lϕ, v H ,
2.6
4
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Definition 2.2. Let un , u0 ∈ Lp 0, T , X2 . We say that un u0 in Lp 0, T , X2 is uniformly
weakly convergent if {un } ⊂ L∞ 0, T , H is bounded, and
un u0 , in Lp 0, T , X2 ,
T
lim
|un − u0 , vH |2 dt 0, ∀v ∈ H.
n→∞
2.7
0
Lemma 2.3 see 16. Let {un } ∈ Lp 0, T , W m,p Ω m ≥ 1 be bounded sequences and {un }
uniformly weakly convergent to u0 ∈ Lp 0, T , W m,p Ω. Then, for each |α| ≤ m − 1, it follows that
Dα un −→ Dα u0 ,
in L2 0, T × Ω.
2.8
Lemma 2.4 see 17. Let Ω ⊂ Rn be an open set and f : Ω × RN −→ R1 satisfy Caratheodory
condition and
N
fx, ξ
≤ C |ξi |pi /p bx.
2.9
i1
If {uik } ⊂ Lpi Ω 1 ≤ i ≤ N is bounded and uik convergent to ui in Ω0 for all bounded Ω0 ⊂ Ω,
then for each v ∈ Lp Ω, the following equality holds
lim
k→∞
Ω
fx, u1k , . . . , uNk v dx Ω
fx, u1 , . . . , uN v dx.
2.10
3. Model Results
Let G A B : X2 × R −→ X1∗ . Assume
A1 there is a C1 functional F : X2 −→ R1 such that
Au, Lv −DFu, v,
∀u, v ∈ X;
3.1
A2 functional F : X2 −→ R1 is coercive, that is,
Fu −→ ∞, ⇐⇒ uX2 −→ ∞;
3.2
A3 B satisfies
|Bu, Lv| ≤ C1 Fu for g ∈ L1loc 0, ∞.
k
v2H1 C2 ,
2
∀u, v ∈ X,
3.3
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5
Theorem 3.1. Set G : X2 × R −→ X1∗ , for each ϕ, ψ ∈ X2 × H1 , then the following assertions hold.
1 If G A satisfies (A1) and (A2), then 2.4 has a globally weak solution
u ∈ W 1,∞ 0, ∞, H1 ∩ W 1,2 0, ∞, H2 ∩ L∞ 0, ∞, X2 .
3.4
2 If G A B satisfies (A1)–(A3), then 2.4 has a global weak solution
1,∞
1,2
u ∈ Wloc
0, ∞, H1 ∩ Wloc
0, ∞, H2 ∩ L∞
loc 0, ∞, X2 .
3.5
3 Furthermore, if L : X2 −→ X1 is symmetric sectorial operator, that is, Lu, v u, Lv,
and G A B satisfies
1
|Gu, v| ≤ C1 Fu v2H C2 ,
2
3.6
2,2
0, ∞, H.
then u ∈ Wloc
Proof. Let {ek } ⊂ X be a common orthogonal basis of H and H3 , satisfying 2.3. Set
n
1
αi ei | αi ∈ R ,
Xn i1
⎧
⎫
n
⎨
⎬
n X
βj tej | βj ∈ C2 0, ∞ .
⎩ j1
⎭
3.7
n X
n.
Clearly, LXn Xn , LX
By using Galerkin method, there exits un ∈ C2 0, ∞, Xn satisfying
dun
,v
dt
H
kLun , vH t
0
Gun , vdτ ψn , v H k Lϕn , v H ,
un 0 ϕn ,
3.8
un 0 ψn ,
for ∀v ∈ Xn , and
t 0
n.
for ∀v ∈ X
d 2 un
,v
dt2
H
dun
,v
k L
dt
dτ H
t
0
Gun , vdτ
3.9
6
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Firstly, we consider G A. Let v d/dtLun in 3.9. Taking into account 2.2 and
3.1, it follows that
t t
d
dun d
, Lun
Aun , Lun dτ,
dτ −
k L
dt dt
dt
0
0
H
t
1 d dun dun
dun dun
dun
0
,
,
dτ
k
DFun ,
dt dt H1
dt dt H2
dt
0 2 dt
0
d 2 un d
, Lun
dt2 dt
3.10
t 2
2
1
dun − 1 ψn 2 k dun dτ Fun − F ϕn .
H
1
2 dt H1 2
dt H2
0
We get
t 2
2
1
dun k dun dτ Fun F ϕn 1 ψn 2 .
H1
2 dt H1
dt H2
2
0
3.11
Let ϕ ∈ H3 . From 2.1 and 2.2, it is known that {en } are orthogonal basis of H1 . We
find that ϕn −→ ϕ in H3 , and ψn −→ ψ in H1 . Since H3 ⊂ X2 is imbedding, it follows that
ϕn −→ ϕ,
in X2 ,
ψn −→ ψ,
in H1 .
3.12
From 3.2, 3.11, and 3.12, we obtain that,
1,∞
1,2
0, ∞, H1 ∩ Wloc
0, ∞, H2 ∩ L∞
{un } ⊂ Wloc
loc 0, ∞, X2 is bounded.
3.13
Let
un ∗ u0 ,
1,∞
in Wloc
0, ∞, H1 ∩ L∞
loc 0, ∞, X2 ,
un u0 ,
1,2
in Wloc
0, ∞, H2 ,
3.14
1,2
which implies that un → u0 in Wloc
0, ∞, H is uniformly weakly convergent from that
H2 ⊂ H is compact imbedding.
If we have the following equality:
t
k
2
lim − |Gun − Gu0 , Lun − Lu0 |dτ un − u0 H2 0,
n→∞
2
0
then u0 is a weak solution of 2.4 in view of 3.8, 3.14.
3.15
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7
We will show 3.15 as follows. It follows that from 2.5,
t 0
d
d
Lun − Lu0 , Lun − Lu0
dt
dt
1
dτ 2
H
t
0
d
un − u0 , un − u0 H2 dτ
dt
2
1
1
un t − u0 t2H2 − ϕn − ϕH2 .
2
2
3.16
Taking into account 2.2, 2.5, and 3.9, we get that
−
t
k
un − u0 2H2
2
0
t
2
d
k
d
Lun − Lu0 , Lun − Lu0
dτ ϕn − ϕH2
Gu0 − Gun , Lun − Lu0 k
dt
dt
2
0
H
t
dun
, u0
Gu0 , Lun − Lu0 Gun , Lu0 − Gun , Lun − k
dt
0
H2
2
k
du0
d
dτ ϕn − ϕH2
, un − u0
Lun , Lun
−k
k
dt
dt
2
H2
H
t
du0
dun
, u0
, un − u0
−k
Gu0 , Lun − Lu0 Gun , Lu0 − k
dt
dt
0
H2
H2
2
k
d 2 un
d
d
dτ ϕn − ϕH2
Lu
Lu
−
k
,
Lu
k
,
Lu
n
n
n
n
2
dt
dt
2
dt
H
H
t
Gun − Gu0 , Lun − Lu0 dτ Gu0 , Lun − Lu0 Gun , Lu0 0
d
dun dun
dun
dτ
, u0
u0 , un − u0
,
−k
−k
dt
dt
dt dt H1
H2
H2
2
dun
k
−
, un
ψn , ϕn H1 ϕn − ϕH2 .
dt
2
H1
3.17
From 2.1 and 3.14, we have
lim ϕn − ϕH2 0,
n→∞
t
lim
n→∞
Gu0 , Lun − Lu0 dτ 0,
0
t lim
n→∞
0
d
u0 , un − u0
dt
dτ 0.
H2
3.18
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Journal of Applied Mathematics
Then, we get
t
k
lim un − u0 2H2
2 n→∞
0
dun 2
t
dun
dτ
, u0
lim 0 Gun , Lu0 − k
dt n→∞
dt
H1
H2
dun
, un
− lim
ψ, ϕ H1 .
n→∞
dt
H1
lim −
n→∞
Gun − Gu0 , Lun − Lu0 dτ 3.19
In view of 3.9, 3.14, we obtain for all v ∈ ∪∞
n1 Xn
t du0
du0 dv
k
dτ
,v
,
−
Gun , Lvdτ dt
dt dt H1
0
0
H2
du0
,v
− ψ, v0 H1 .
dt
H1
t
lim
n→∞
3.20
1,2
p
Since ∪∞
n1 Xn is dense in W 0, T , H2 ∩ L 0, T , X2 , for all p < ∞, 3.20 holds for
1,2
p
all v ∈ W 0, T , H2 ∩ L 0, T , X2 . Thus, we have
t du0 2
du0
k
dτ
, u0
−
Gun , Lu0 dτ dt
dt H1
0
0
H2
du0
, u0
− ψ, ϕ H1 .
dt
H1
t
lim
n→∞
3.21
From 3.14 and H2 ⊂ H1 being compact imbedding, it follows that
t t dun 2
du0 2
dτ,
dτ lim
n → ∞ 0 dt H
dt H1
0
1
du0
dun
, un
, u0
, a.e. t ≥ 0.
lim
n→∞
dt
dt
H1
H1
3.22
Clearly,
t lim
n→∞
0
dun
, un
dt
dτ H1
t 0
du0
, u0
dt
dτ.
H1
Then, 3.14 follows from 3.19–3.21, which implies assertion 1.
3.23
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9
Secondly, we consider G A B. Let v d/dtLun in 3.9. In view of 2.2 and 2.8,
it follows that
t 0
d 2 un d
, Lun
dt2 dt
t
0
H
t d
dτ dτ
A Bun , Lun
dt
0
H
H
dun dun
dτ
,
k
dt dt H2
dun d
, Lun
k L
dt dt
1 d dun dun
,
2 dt dt dt H1
t dun
d
Bun , Lun
−DFun ,
dτ,
dt
dt
0
H
t t 2
2
1
d
dun − 1 ψn 2 k dun dτ Fun − F ϕn Lu
dτ,
Bu
,
n
n
dt H1
2 dt H1 2
dt
H2
0
0
H
t t
2
2
1 2
1
d
dun k dun dt Fun ψn .
Lu
Bu
,
n
n dτ F ϕn H1
2 dt H1
dt H2
dt
2
0
0
3.24
From 3.3, we have
t
t 2
dun 2
dun 2
du
1
k
n
Fun k C2 dτ
C1 Fun dt dτ ≤ C
2 dt H1
2 dt H1
H2
0
0
1 2
F ϕn ψn H1
2
t
2 dun 1
Fun dτ ft,
≤C
2 dt H1
0
3.25
where ft 1/2ψ2H1 supn Fϕn .
By using Gronwall inequality, it follows that
t
2
1
dun Fun ≤ f0eCt fτeCt−τ dτ,
2 dt H1
0
3.26
which implies that for all 0 < T < ∞,
{un } ⊂ W 1,∞ 0, T , H1 ∩ L∞ 0, T , X2 is bounded.
3.27
From 3.25 and 3.21, it follows that
{un } ⊂ W 1,2 0, T , H2 is bounded.
3.28
10
Journal of Applied Mathematics
Let
un ∗ u0 ,
in W 1,∞ 0, T , H1 ∩ L∞ 0, T , X2 ,
un u0 ,
in W 1,2 0, T , H2 ,
3.29
which implies that un → u0 in W 1,2 0, T , H is uniformly weakly convergent from that
H2 ⊂ H is compact imbedding.
The left proof is same as assertion 1.
Lastly, assume 3.6 hold. Let v d2 un /dt2 in 3.9. It follows that
dun d2 un
dτ
,
k L
dt dt2
H
H
d 2 un
dτ
A Bun ,
dt2
0
⎤
⎡
t
d2 u 2
1
n
≤ ⎣CFun 2 gt⎦dτ,
2 dt 0
H
t 0
d 2 un d 2 un
,
dt2 dt2
t
d 2 un d 2 un
d 2
k t
,
u t dxdτ
dt2 dt2 H 2 0 Ω dt n
⎤
⎡
t
d2 u 2
1
n
≤ ⎣CFun 2 gt⎦dτ,
2 dt 0
t 0
3.30
H
t 0
d 2 un d 2 un
,
dt2 dt2
k 2
≤ ψn H 2
dt H
2
k
dun 2 dt H1
⎤
⎡ d 2 u 2
1
n
⎣ CFun gτ⎦dτ.
2
0 2 dt H
t
From 3.26, the above inequality implies
t d 2 u 2
n
2 dτ ≤ C,
0 dt H
C > 0 is constant.
3.31
We see that for all 0 < T < ∞, {un } ⊂ W 2,2 0, T , H is bounded. Thus u ∈ W 2,2 0, T , H.
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11
4. Main Result
Now, we begin to consider the mixed problem 1.1. Set
F x, y y
4.1
fx, zdz.
0
We assume
p
F x, y ≥ C1 y
− C2 , p ≥ 2,
f x, y ≤ C y
p−1 1 ,
4.2
2
f x, y1 − f x, y2
y1 − y2 ≥ λ
y1 − y2 , λ > 0,
g x, z, ξ, η ≤ C |z|p/2 |ξ|p/2 η
p/2 1 ,
4.3
4.4
g x, z, ξ, η1 − g x, z, ξ, η2 ≤ K1 η1 − η2 ,
4.5
where C, C1 , C2 are constant and K1 < λK, K is the best constant satisfying
K 2 u2H 2 ≤
Ω
|Δu|2 dx.
4.6
1,p
Theorem 4.1. If the assumptions of 4.1–4.5 hold, for ϕ, ψ ∈ W 2,p Ω ∩ W0 Ω × H01 Ω,
then 1.1 is a strong solution
1,p
2,p
u ∈ L∞
Ω ∩ W0 Ω ,
loc 0, ∞, W
1
2
2
∩
L
,
ut ∈ L∞
∞,
H
∞,
H
Ω
Ω
0,
0,
0
loc
loc
utt ∈ Lp 0, T × Ω,
p 4.7
p
, ∀0 < T < ∞.
p−1
Proof. We introduce spatial sequences
$
X u ∈ C∞ Ω
Δk u
∂Ω
%
0, k 0, 1, 2, . . . ,
X1 L Ω,
p
1,p
X2 W 2,p Ω ∩ W0 Ω,
H L2 Ω,
H1 H01 Ω,
H2 H 2 Ω ∩ H01 Ω,
$
%
H3 u ∈ H 2m Ω: u|∂Ω · · · Δm−1 u|∂Ω 0 ,
4.8
12
Journal of Applied Mathematics
where the inner products of H2 and H3 are defined by
u, vH2 Ω
ΔuΔv dx,
u, vH2 Ω
Δm uΔm v dx,
4.9
where m ≥ 1 such that H3 ⊂ X2 is an embedding.
Linear operators L : X −→ X1 and L : X −→ X1 are defined by
Lu Lu −Δu.
4.10
It is known that L and L satisfy 2.2, 2.3, and 2.5. Define G A B : X2 → X1∗ by
Au, v Ω
fx, Δuv dx,
Bu, v Ω
g x, u, Du, D2 u v dx,
for v ∈ X1 .
4.11
We show that G A B : X2 −→ X1∗ is T -coercively weakly continuous. Let {un } ⊂
1,p
L∞ 0, T , W 2,p Ω ∩ W0 Ω satisfying 2.7 and
T
lim
n→∞
|Gun − Gu0 , Lun − Lu0 |dt
0
lim
n→∞
T 0
fx, Δun − fx, Δu0 un − u0 &
Ω
4.12
'
g x, un , Dun , D2 un − g x, u0 , Du0 , D2 u0 un − u0 dx dt 0.
We need to prove that
lim
T (
n→∞
0
Ω
'
fx, Δun g x, un , Dun , D2 un v dx dt
T (
0
Ω
'
fx, Δu0 g x, u0 , Du0 , D u0 v dx dt.
4.13
2
From 2.7 and Lemma 2.3, we obtain
un −→ u0 ,
Dun −→ Du0
in L2 0, T × Ω.
4.14
From 4.3, we get
T 0
Ω
)
&
fx, Δun − fx, Δu0 Δun − Δu0 dx dt ≥ λ
T 0
Ω
|Δun − Δu0 |2 dx dt.
4.15
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13
We have the deformation
T ( '
g x, un , Dun , D2 un − g x, u0 , Du0 , D2 u0 Δun − Δu0 dx dt
Ω
0
T ( '
g x, un , Dun , D2 u0 − g x, u0 , Du0 , D2 u0 Δun − Δu0 dx dt
Ω
0
4.16
T ( '
g x, un , Dun , D2 un − g x, un , Dun , D2 u0 Δun − Δu0 dx dt.
0
Ω
From 4.14 and Lemma 2.4, we have
T ( '
g x, un , Dun , D2 u0 − g x, u0 , Du0 , D2 u0 Δun − Δu0 dx dt 0.
lim
n→∞
Ω
0
4.17
From 4.12, 4.15–4.17, it follows that
0≥λ
T 0
Ω
|Δun − Δu0 |2 dx dt T ( '
g x, un , Dun , D2 un − g x, un , Dun , D2 u0
Ω
0
× Δun − Δu0 dx dt
T T 2
≥λ
|Δun − Δu0 |2 dx dt − K1
D un − D2 u0 |Δun − Δu0 |dx dt
0
≥
λ
2
Ω
T 0
Ω
Ω
0
|Δun − Δu0 |2 dx dt −
K12
2λ
4.18
T 2
2
D un − D2 u0 dx dt
0
Ω
2
λ2 K 2 − K12 T
2
≥
D un − D2 u0 dx dt.
2λ
0 Ω
Since λK > K1 , we have
lim
n→∞
T 2
2
D un − D2 u0 dx dt 0.
0
Ω
4.19
From 4.14, 4.19, 4.1, 4.4, and Lemma 2.3, we get 4.13.
Let F1 u Ω Fx, Δudx, where F is same as 4.1. We get
Au, Lu −DF1 u, v,
Fu −→ ∞ ⇐⇒ uX2 −→ ∞,
which implies Conditions A1, A2 of model results in Theorem 3.1.
4.20
14
Journal of Applied Mathematics
We will show 3.3 as follows. It follows that
g x, u, Du, D2 u |Δv|dx
|Bu, Lv| ≤
Ω
k
2
Ω
k
≤ v2H2
2
≤
2
2
g x, u, Du, D2 u dx
k Ω
(
'
2 p
C
D u
|∇u|p |u|p 1 dx
|Δv|2 dx 4.21
Ω
k
v2H2 CF1 u C,
2
which imply Conditions A3 of Theorem 3.1. From Theorem 3.1, 1.1 has a solution
1,p
2,p
u ∈ L∞
Ω ∩ W0 Ω ,
loc 0, ∞, W
1
2
2
∩
L
,
ut ∈ L∞
∞,
H
∞,
H
Ω
Ω
0,
0,
0
loc
loc
utt ∈ Lp 0, T × Ω,
p 4.22
p
, ∀0 < T < ∞,
p−1
satisfying
Ω
∂u
v dx − k
∂t
Ω
Δuv dx t 0
Ω
fx, Δuv dx dτ Ω
0
ψv dx − k
t Ω
Ω
g x, u, Du, D2 u v dx dτ 4.23
Δϕv dx.
Acknowledgments
The authors are grateful to the anonymous reviewers for their careful reading and useful
suggestions, which greatly improved the presentation of the paper. This paper was funded
by the National Natural Science Foundation of China no. 11071177 and the NSF of Sichuan
Science and Technology Department of China no. 2010JY0057.
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