Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 216760, 15 pages
doi:10.1155/2010/216760
Research Article
Global Existence, Uniqueness, and
Asymptotic Behavior of Solution for p-Laplacian
Type Wave Equation
Caisheng Chen, Huaping Yao, and Ling Shao
Department of Mathematics, Hohai University, Nanjing, Jiangsu, 210098, China
Correspondence should be addressed to Caisheng Chen,
Received 10 May 2010; Accepted 13 July 2010
Academic Editor: Michel C. Chipot
Copyright q 2010 Caisheng Chen 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 study the global existence and uniqueness of a solution to an initial boundary value problem
for the nonlinear wave equation with the p-Laplacian operator u
tt
−div|∇u|
p−2
∇u−Δu
t
gx, u
fx. Further, the asymptotic behavior of solution is established. The nonlinear term g likes
gx, uax|u|
α−1
u − bx|u|
β−1
u with appropriate functions ax and bx,whereα>β≥ 1.
1. Introduction
u
0
x
,u
t
x, 0
u
1
x
, in Ω; u
x, t
0, on ∂Ω ×
0, ∞
, 1.2
where 2 ≤ p<nand Ω is a boundary domain in R
n
with smooth boundary ∂Ω.The
assumptions on f, g, u
0
Moreover, if f 0andugu ≥ Gu, then there exist positive constants c and γ such that
E
t
≤ c exp
−γt
,t≥ 0, if n 2, 1.4
E
t
≤ c
1 t
−n/n−2
,t≥ 0, if n ≥ 3,
1.5
where
E
t
1
2
with Gx, u
u
0
fx, sds.
Gao and Ma in 2 also considered the global existence of solution f or 1.1-1.2.In
Theorem 3.1of2, the similar results to 1.4-1.5 for asymptotic behavior of solution were
obtained if the nonlinear function gx, ugu satisfies
g
u
≤ a
|
u
|
σ−1
b, ug
u
≥ ρG
u
≥ 0, in Ω × R
α−1
u − bx|u|
β−1
u with α>β≥ 1anda, b ≥ 0. Obviously, the sign condition
ugu ≥ 0 fails to hold for this type of function.
For these purposes, we must establish the global existence of solution for 1.1-1.2.
Several methods have been used to study the existence of solutions to nonlinear wave
equation. Notable among them is the variational approach through the use of Faedo-Galerkin
approximation combined with the method of compactness and monotonicity, see 7.To
prove the uniqueness, we need to derive the various estimates for assumed solution ut.
For the decay property, like 1.5, we use the method recently introduced by Martinez 8 to
study the decay rate of solution to the wave equation u
tt
− Δu gu
t
0inΩ × R
, where Ω
is a bounded domain of R
n
.
This paper is organized as follows. In Section 2, some assumptions and the main
results are stated. In Section 3, we use Faedo-Galerkin approximation together with a
combination of the compactness and the monotonicity methods to prove the global existence
of solution to problem 1.1-1.2. Further, we establish the uniqueness of solution by some a
priori estimate t o assumed solutions. The proof of asymptotic behavior of solution is given in
Section 4.
Journal of Inequalities and Applications 3
2. Assumptions and Main Results
We first give some notations and definitions. Let Ω be a bounded domain in R
u
t represent for u
t
t and so on.
If T>0isgivenandX is a Banach space, we denote by L
p
0,T; X the space of
functions which are L
p
over 0,T and which take their values in X. In this space, we consider
the norm
u
L
p
0,T;X
T
0
u
t
p
p/p − 1,p>1.
A
2
Let gx, u ∈ C
1
Ω × R and satisfy
ug
x, u
h
1
x
|
u
|
≥ k
0
G
x, u
h
1
x
g
u
x, u
≤ k
1
|
u
|
α−1
h
3
x
, in Ω × R 2.3
with some k
0
,k
1
> 0 and the nonnegative functions h
1
x ∈ L
p
1,p
0
, u
t
∈ L
2
0,T; W
1,2
0
, u
tt
∈ L
2
0,T; W
−1,p
,
and u satisfies 1.2 with u
0
,u
1
∈ W
1,p
0
and the integral identity
Ω
u
tt
. Then the problem 1.1-1.2 admits
a solution ut satisfying
u ∈ C
0, ∞
; ,W
1,2
0
∩ L
∞
0, ∞
; ,W
1,p
0
,
u
t
∈ L
2
0, ∞
p
p
t
0
∇u
t
s
2
2
ds ≤ C
1
A B
, ∀t ≥ 0,
2.6
where
A
u
0
p
p
p
p
,i 1, 2,H
3
h
3
λ
1
λ
1
,λ
1
n/2.
Further, if 1 ≤ α ≤ n p/n − p and 2 ≤ p ≤ 4, the solution satisfying 2.5-2.6 is
unique.
Theorem 2.3. Let u be a solution of 1.1-1.2 with f 0. In addition, let 2 <p<nand
g
x, u
u ≥ pG
x, u
≥ 0, in Ω × R. 2.8
Then there exists C
dx ≤ C
0
1 t
−p/p−2
, ∀t ≥ 0
. 2.9
The following theorem shows that the asymptotic estimate 2.9 can be also derived if
assumption 2.8 fails to hold.
Theorem 2.4. Let u be a solution of 1.1-1.2 with f 0. In addition, let 2 <p<nand
g
x, u
λ
|
u
|
α−1
u −
|
u
|
β−1
u, in Ω × R
2.10
with p<β1 < 2p, β < α < np/n−p. Then there exists C
0
C
α1
α1
≤ C
0
1 t
−p/p−2
, ∀t ≥ 0.
2.11
Journal of Inequalities and Applications 5
3. Proof of Theorem 2.2
In this section, we assume that all assumptions in Theorem 2.2 are satisfied. We first prove
the global existence of a solution to problem 1.1-1.2 with the Faedo-Galerkin method as
in 1, 2, 7, 9.
Let r be an integer for which the embedding H
r
0
Ω W
r,2
0
Ω → W
1,p
0
Ω is
continuous. Let w
j
j 1, 2, be eigenfunctions of the spectral problem
w
2
, ,w
m
, }
yields a basis for both H
r
0
Ω and L
2
Ω. For each integer m,letV
m
span{w
1
,w
2
, ,w
m
}.
We look for an approximate solution to problem 1.1-1.2 in the form
u
m
t
m
j1
T
jm
,w
j
g,w
j
f, w
j
,j 1, 2, m 3.3
with the p-Laplacian operator Δ
p
u div|∇u|
p−2
∇u and the initial conditions
u
m
0
u
0m
,u
m
m
. We claim that for any T>0, such a solution can be extended to the whole interval 0,T
by using the first a priori estimate below. We denote by C
k
the constant which is independent
of m and the initial data u
0
and u
1
.
Multiplying 3.3 by T
jm
t and summing the resulting equations over j,wegetafter
integration by parts
E
m
t
∇u
m
t
t
p
p
Ω
G
x, u
m
dx −
Ω
f
x
u
m
dx. 3.7
6 Journal of Inequalities and Applications
By 2.2 and Young inequality, we have
Ω
G
p
,
Ω
f
x
u
m
dx ≥−ε
∇u
m
p
p
− C
ε
f
p
p
.
∇u
m
t
p
p
− C
1
H
1
F
,
3.9
or
u
m
t
2
2
t
2
2
∇u
m
t
p
p
t
0
∇u
m
s
2
2
ds ≤ C
≤ k
1
u
m
α1
α1
Ω
|
h
2
||
u
m
|
dx
≤ C
2
∇u
m
∇u
m
p
p
H
2
.
3.12
Then it follows 3.5 and 3.6 that
E
m
t
≤ E
m
0
1
2
u
1m
≤ C
2
u
1
2
2
∇u
0
p
p
∇u
0
α
p
H
1
H
2
F
≤ C
∇u
m
s
2
2
ds ≤ C
2
A B
, ∀t ≥ 0. 3.14
Journal of Inequalities and Applications 7
With this estimate we can extend the approximate solution u
m
t to the interval 0,T
and we have that
{
u
m
t
}
is bounded in L
∞
0,T; W
1,2
0
. 3.17
Now we recall that operator −Δ
p
u − div|∇u|
p−2
∇u is bounded, monotone, and
hemicontinuous from W
1,p
0
to W
−1,p
with p ≥ 2. Then we have
−Δ
p
u
m
t
is bounded L
∞
∞
0,T; W
1,p
0
, 3.20
u
m
u
weakly star in L
∞
0,T; L
2
, 3.21
u
m
u
weakly in L
2
0,T; L
2
, 3.24
and u
m
→ u a.e. in Ω × 0,T.
8 Journal of Inequalities and Applications
Since the embedding W
1,2
0
→ L
2
is compact, we get, from 3.18 and 3.19,
u
m
−→ u
strongly in L
2
0,T; L
2
. 3.25
Using the growth condition 2.3 and 3.25,weseethat
T
0
Ω
g
x, u
m
g
x, u
weakly in L
α1/α
0,T; L
α1/α
. 3.28
With these convergences, we can pass to the limit in the approximate equation and
then
d
dt
u
t
,v
To prove the uniqueness, we assume that ut and vt are two solutions which satisfy
2.5-2.6 and u0v0,u
t
0v
t
0. Setting Utu
t
t,Vtv
t
t,andWt
Ut − V t.Weseefrom1.1 and 1.2 that
W
t
− ΔW − div
|
∇u
|
p−2
∇u −
|
∇v
|
p−2
∇v
g
x, v
|
∇v
|
p−2
∇v
∇Wdx
Ω
g
x, v
− g
x, u
Wdx,
W
t
2
2
2
Ω
g
x, v
− g
x, u
Wdxds
3.31
Journal of Inequalities and Applications 9
Now setting U
u 1 − v, 0 ≤ ≤ 1, then
t
0
Ω
|
∇u
|
|
∇U
|
p−2
∇U
d
|
∇W
|
dx dτ
≤
p − 1
t
0
Ω
1
0
|
∇u
τ
|
|
∇v
τ
|
,
|
∇
u
τ
− v
τ
|
≤
τ
0
t
0
Ω
τ
0
|
∇u
τ
|
p−2
|
∇v
τ
|
p−2
|
∇W
s
||
p−2
p
∇Ws
2
∇Wτ
2
ds dτ
≤ C
1
A B
p−2/p
t
0
τ
0
∇Ws
2
∇Wτ
∇Ws
2
2
ds
3.34
with C
2
C
1
A B
p−2/p
.
For the term of the right side to 3.31, we have
G
1
t
0
Ω
g
x, v
− g
|
W
|
dx dτ
≤
t
0
Ω
1
0
g
u
x, U
u
τ
λ
1
d
u
s
s − v
s
s
λ
2
Wτ
λ
2
d ds dτ
3.35
with λ
1
n/2, λ
2
2n/n − 2.
10 Journal of Inequalities and Applications
By the assumption A
2
and 1 ≤ α ≤ n p/n − p,weseethat
|
α−1
|
h
3
|
n/2
dx
≤ C
3
Ω
|
u
τ
|
nα−1/2
|
v
τ
|
nα−1/2
.
3.36
By the estimate 2.6, we have
∇ut
p
,
vt
p
≤ C
2
A B
1/p
, ∀t ≥ 0.
3.37
Therefore, there exists C
4
> 0, depending u
0
,v
0
,f,h
i
such that
u
s
s − v
s
s
λ
2
≤ C
0
∇
u
s
s
− v
s
s
2
C
0
∇Ws
τ
0
W
s
λ
2
Wτ
λ
2
dsdτ ≤ C
4
t
0
∇W
s
2
ds
2
2
ds ≤
C
2
C
4
t
t
0
∇Ws
2
2
. 3.41
The integral inequality 3.41 shows that there exists T
1
> 0, such that
W
t
0, 0 ≤ t ≤ T
1
. 3.42
Consequently, ut − vtu0 − v00, 0 ≤ t ≤ T
2
Ω
t
s
∇u
τ
τdτ
2
dx ≤
Ω
t
s
|
∇u
τ
. We complete the proof of Theorem 2.2.
4. Proof of Theorem 2.3
Let us first state a well-known lemma that will be needed later.
Lemma 4.1 see 10. Let E : R
→ R
be a nonincreasing function and assume that there are
constants q ≥ 0 and γ>0, such that
∞
S
E
q1
t
dt ≤ γ
−1
E
q
0
E
S
, ∀S ≥ 0
. 4.1
t
1
2
u
t
t
2
2
1
p
∇ut
p
p
Ω
G
x, u
dx, t ≥ 0
. 4.3
Ω
u
u
tt
− Δ
p
u − Δu
t
g
x, u
dx dt 0, ∀T>S≥ 0. 4.5
Note that
T
S
E
q
t
u, u
tt
dt E
q
u
t
t
2
2
dt
−
T
S
E
q
t
u, Δ
p
u
dt
T
S
E
q
t
∇u, ∇u
t
dt.
4.6
Then we have from 4.5 that
p
T
S
E
q1
t
dt −E
q
tu, u
t
|
T
S
q
u
t
t
2
2
dt −
T
S
E
q
t
∇u, ∇u
t
dt
T
S
E
q
t
1/2
,
∇ut
p
≤ pE
1/p
t
, ∀t ≥ 0,
|
E
q
t
u, u
t
|
≤ E
q
t
u
t
2
≤ C
0
E
t
μ
1
4.8
with μ
1
q 1/2 1/p.
This gives
E
q
tu, u
t
|
T
S
≤ C
1
E
μ
q
t
∇u
t
t
2
2
dt
C
1
T
S
E
q
t
−E
t
dt ≤ C
dt ≤ C
1
T
S
E
q−1
t
E
t
ut
2
u
t
t
2
,
4.11
T
S
|
E
q
t
∇u, ∇u
t
|
dt ≤
T
S
E
q
t
∇ut
2
∇u
t
dt C
1
T
S
E
q2/p−1
t
−E
t
dt
≤
T
S
E
q1
t
S
E
q2/p
S
≤ C
1
E
S
E
μ
1
S
E
q
S
E
q2/p−1
S
0
E
S
,
4.13
for any T>S≥ 0, letting T →∞,wefindthat
∞
S
E
q1
t
dt ≤ γ
−1
E
S
E
q
0
, ∀S ≥ 0.
4.14
Ω
G
x, u
dx ≤ E
0
1 q
1 qγt
1/q
≤ C
2
E
0
1 t
−p/p−2
.
4.15
This is 2.9 and we complete the proof of Theorem 2.3.
14 Journal of Inequalities and Applications
4.2. The Proof of Theorem 2.4
By Sobolev inequality, we know that there exists λ
0
u
|
α1
−
1
β 1
|
u
|
β1
. 4.17
Obviously, there exists λ
2
> 0, such that λ>λ
2
,
λ
0
2p
|
u
|
p
G
u
≥
1
2
u
α1
α1
,
E
t
≥
1
2
u
t
t
2
2
1
2p
∇u
t
u
|
β1
−
λ
α 1 − p
α 1
|
u
|
α1
≤
β 1 − p
β 1
|
u
|
β1
β 1 − p
λ
α 1
|
u
|
q
t
Ω
pG
u
− gu
dxdt ≤
β 1 − p
T
S
E
q1
t
dt. 4.21
Then, by 4.9 and 4.11–4.14, we have
2p − β − 1
S
E
q
0
.
4.22
Journal of Inequalities and Applications 15
The applications of Lemma 4.1 and 4.19 yields that
u
t
t
2
2
∇ut
2
2
ut
α1
´
ethodes de R
´
esolution des Probl
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emes aux Limites non Lin
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Villars, Paris, France, 1969.
8 P. Martinez, “A new method to obtain decay rate estimates for dissipative systems,” ESAIM: Control,
Optimisation and Calculus of Variations, vol. 4, pp. 419–444, 1999.
9 M. Sango, “On a nonlinear hyperbolic equation with anisotropy: global existence and decay of
solution,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 7, pp. 2816–2823, 2009.
10 V. K omo rni k, Exact Controllability and Stabilization: The Multiplier Method, RAM: Research in Applied
Mathematics, John Wiley & Sons, Chichester, UK; Masson, Paris, France, 1994.
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