Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2007, Article ID 56350, 13 pages
doi:10.1155/2007/56350
Research Article
Existence and Asymptotic Stability of Solutions for Hyperbolic
Differential Inclusions with a Source Term
Jong Yeoul Park and Sun Hye Park
Received 10 October 2006; Revised 26 December 2006; Accepted 16 January 2007
Recommended by Michel Chipot
We study the existence of global weak solutions for a hyperbolic differential inclusion
with a source term, and then investigate the asymptotic stability of the solutions by using
Nakao lemma.
Copyright © 2007 J. Y. Park and S. H. Park. 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.
1. Introduction
In this paper, we are concerned with the global existence and the asymptotic stability of
weak solutions for a hyperbolic differential inclusion with nonlinear damping and source
terms:
ytt − Δyt − div |∇ y | p−2 ∇ y + Ξ = λ| y |m−2 y
Ξ(x,t) ∈ ϕ yt (x,t)
in Ω × (0, ∞),
a.e. (x,t) ∈ Ω × (0, ∞),
y = 0 on ∂Ω × (0, ∞),
y(x,0) = y0 (x),
(1.1)
yt (x,0) = y1 (x) in x ∈ Ω,
where Ω is a bounded domain in RN with sufficiently smooth boundary ∂Ω, p ≥ 2, λ > 0,
and ϕ is a discontinuous and nonlinear set-valued mapping by filling in jumps of a locally
bounded function b.
Recently, a class of differential inclusion problems is studied by many authors [2, 6, 7,
11, 14–16, 19]. Most of them considered the existence of weak solutions for differential
inclusions of various forms. Miettinen [6] Miettinen and Panagiotopoulos [7] proved the
existence of weak solutions for some parabolic differential inclusions. J. Y. Park et al. [14]
showed the existence of a global weak solution to the hyperbolic differential inclusion
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Journal of Inequalities and Applications
(1.1) with λ = 0 by making use of the Faedo-Galerkin approximation, and then considered asymptotic stability of the solution by using Nakao lemma [8]. The background of
these variational problems are in physics, especially in solid mechanics, where nonconvex, nonmonotone, and multivalued constitutive laws lead to differential inclusions. We
refer to [11, 12] to see the applications of differential inclusions.
On the other hand, it is interesting to mention the existence and nonexistence of global
solutions for nonlinear wave equations with nonlinear damping and source terms [4, 5,
10, 13, 18] in the past twenty years. Thus, in this paper, we will deal with the existence and
the asymptotic behavior of a global weak solution for the hyperbolic differential inclusion
(1.1) involving p-Laplacian, a nonlinear, discontinuous, and multivalued damping term
and a nonlinear source term. The difficulties come from the interaction between the pLaplacian and source terms. As far as we are concerned, there is a little literature dealing
with asymptotic behavior of solutions for differential inclusions with source terms.
The plan of this paper is as follows. In Section 2, the main results besides notations and
assumptions are stated. In Section 3, the existence of global weak solutions to problem
(1.1) is proved by using the potential-well method and the Faedo-Galerkin method. In
Section 4, the asymptotic stability of the solutions is investigated by using Nakao lemma.
2. Statement of main results
We first introduce the following abbreviations: QT = Ω × (0,T), ΣT = ∂Ω × (0,T),
· p = · L p (Ω) , · k,p = · W k,p (Ω) . For simplicity, we denote · 2 by · . For every
1,q
q ∈ (1, ∞), we denote the dual of W0 by W −1,q with q = q/(q − 1). The notation (·, ·)
for the L2 -inner product will also be used for the notation of duality pairing between dual
spaces.
Throughout this paper, we assume that p and m are positive real numbers satisfying
2≤ p<m<
Np
+1
2(N − p)
(2 ≤ p < m < ∞ if p ≥ N).
(2.1)
Define the potential well
p
1,p
ᐃ = y ∈ W0 (Ω) | I(y) = ∇ y p − λ y m
m > 0 ∪ {0 }.
(2.2)
1,p
Then ᐃ is a neighborhood of 0 in W0 (Ω). Indeed, Sobolev imbedding (see [1])
1,p
Lm (Ω)
W0 (Ω)
(2.3)
and Poincare’s inequality yield
m− p
m
m
m
λ y m
m ≤ λc∗ ∇ y p ≤ λc∗ ∇ y p
1,p
p
∇ y p ,
1,p
∀ y ∈ W0 (Ω),
(2.4)
where c∗ is an imbedding constant from W0 (Ω) to Lm (Ω). From this, we deduce that
I(y) > 0 (i.e., y ∈ ᐃ) as ∇ y p < (λ−1 c∗−m )1/(m− p) .
J. Y. Park and S. H. Park
3
For later purpose, we introduce the functional J defined by
1
λ
p
∇ y p − y m
m.
p
m
(2.5)
m− p
1
p
∇ y p .
I(y) +
m
mp
(2.6)
J(y) :=
Obviously, we have
J(y) =
1,p
Define the operator A : W0 (Ω) → W −1,p (Ω) by
Ay = − div |∇ y | p−2 ∇ y ,
1,p
∀ y ∈ W0 (Ω),
(2.7)
then A is bounded, monotone, hemicontinuous (see, e.g., [3]), and
p
(Ay, y) = ∇ y p ,
Ay, yt =
1 d
p
∇ y p
p dt
1,p
for y ∈ W0 (Ω).
(2.8)
Now, we formulate the following assumptions.
(H1 ) Let b : R → R be a locally bounded function satisfying
b(s)s ≥ μ1 s2 ,
b(s) ≤ μ2 |s|,
for s ∈ R,
(2.9)
where μ1 and μ2 are some positive constants.
(H2 ) y0 ∈ ᐃ, y1 ∈ L2 (Ω), and
p λ m m − p
m− p
1 2 1 0 < E(0) = y1 + ∇ y0 p − y0 m <
2
p
m
2mp λc∗m 2(m − 1)p
p/(m− p)
.
(2.10)
The multivalued function ϕ : R → 2R is obtained by filling in jumps of a function b :
R → R by means of the functions b , b , b, b : R → R as follows:
b (t) = ess inf b(s),
b (t) = ess sup b(s),
|s−t |≤
b(t) = lim+ b (t),
→0
|s−t |≤
b(t) = lim+ b (t),
→0
(2.11)
ϕ(t) = b(t),b(t) .
We will need a regularization of b defined by
n
b (t) = n
1
∞
−∞
b(t − τ)ρ(nτ)dτ,
(2.12)
where ρ ∈ C0∞ ((−1,1)), ρ ≥ 0 and −1 ρ(τ)dτ = 1. It is easy to show that bn is continuous
for all n ∈ N and b , b , b, b, bn satisfy the same condition (H1 ) with a possibly different
constant if b satisfies (H1 ). So, in the sequel, we denote the different constants by the same
symbol as the original constants.
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Journal of Inequalities and Applications
Definition 2.1. A function y(x,t) is a weak solution to problem (1.1) if for every T > 0,
1,p
y satisfies y ∈ L∞ (0,T;W0 (Ω)), yt ∈ L2 (0,T;W 1,2 (Ω)) ∩ L∞ (0,T;L2 (Ω)), ytt ∈ L2 (0,T;
W −1,p (Ω)), there exists Ξ ∈ L∞ (0,T;L2 (Ω)) and the following relations hold:
T
0
p −2
ytt (t),z + ∇ yt (t), ∇z + ∇ y(t) ∇ y(t), ∇z + Ξ(t),z dt
=
T
0
m−2
λ y(t)
y(t),z dt,
Ξ(x,t) ∈ ϕ yt (x,t)
y(0) = y0 ,
1,p
∀z ∈ W0 (Ω),
(2.13)
a.e. (x,t) ∈ QT ,
yt (0) = y1 .
Theorem 2.2. Under the assumptions (H1 ) and (H2 ), problem (1.1) has a weak solution.
Theorem 2.3. Under the same conditions of Theorem 2.2, the solutions of problem (1.1)
satisfy the following decay rates.
If p = 2, then there exist positive constants C and γ such that
E(t) ≤ C exp(−γt)
a.e. t ≥ 0,
(2.14)
and if p > 2, then there exists a constant C > 0 such that
E(t) ≤ C(1 + t)− p/(p−2)
a.e. t ≥ 0,
(2.15)
p
where E(t) = (1/2) yt (t)2 + (1/ p)∇ y(t) p − (λ/m) y(t)m
m.
In order to prove the decay rates of Theorem 2.3, we need the following lemma by
Nakao (see [8, 9] for the proof).
Lemma 2.4. Let φ : R+ → R be a bounded nonincreasing and nonnegative function for which
there exist constants α > 0 and β ≥ 0 such that
sup
t ≤s≤t+1
φ(s)
1+β
≤ α φ(t) − φ(t + 1) ,
∀t ≥ 0.
(2.16)
Then the following hold.
(1) If β = 0, there exist positive constants C and γ such that
φ(t) ≤ C exp(−γt),
∀t ≥ 0.
(2.17)
(2) If β > 0, there exists a positive constant C such that
φ(t) ≤ C(1 + t)−1/β ,
∀t ≥ 0.
(2.18)
3. Proof of Theorem 2.2
In this section, we are going to show the existence of solutions to problem (1.1) using
the Faedo-Galerkin approximation and the potential method. To this end let {w j }∞
j =1
1,p
be a basis in W0 (Ω) which are orthogonal in L2 (Ω). Let Vn = Span{w1 ,w2 ,...,wn }.
J. Y. Park and S. H. Park
5
We choose y0n and y1n in Vn such that
1,p
y0n −→ y0
Let y n (t) =
n
j =1 g jn (t)w j
y1n −→ y1
in W0 ,
in L2 (Ω).
(3.1)
be the solution to the approximate equation
m−2
yttn (t),w j + ∇ ytn (t), ∇w j + Ay n (t),w j + bn ytn (t) ,w j = λ y n (t)
y n (0) = y0n ,
y n (t),w j ,
ytn (0) = y1n .
(3.2)
By standard methods of ordinary differential equations, we can prove the existence of a
solution to (3.2) on some interval [0,tm ). Then this solution can be extended to the closed
interval [0,T] by using the a priori estimate below.
Step 1 (a priori estimate). Equation (3.1) and the condition y0 ∈ ᐃ imply that
p
m
I y0n = ∇ y0n p − λ y0n m −→ I y0 > 0.
(3.3)
Hence, without loss of generality, we assume that I(y0n ) > 0 (i.e., y0n ∈ ᐃ) for all n. Substituting w j in (3.2) by ytn (t), we obtain
2 d n
E (t) + ∇ ytn (t) + bn ytn (t) , ytn (t) = 0,
dt
(3.4)
2 1 p λ m
1
En (t) = ytn (t) + ∇ y n (t) p − y n (t)m
2
p
m
2
n 1
n
= yt (t) + J y (t) .
2
(3.5)
where
Integrating (3.4) over (0,t) and using assumption (H1 ), we have
1
y n (t)2 + J y n (t) +
t
2
t
0
∇ y n (τ)2 dτ ≤ En (0).
t
(3.6)
Since En (0) → E(0) and E(0) > 0, without loss of generality, we assume that En (0) < 2E(0)
for all n. Now, we claim that
y n (t) ∈ ᐃ,
t > 0.
(3.7)
Assume that there exists a constant T > 0 such that y n (t) ∈ ᐃ for t ∈ [0,T) and y n (T) ∈
∂ᐃ, that is, I(y n (T)) = 0. From (2.6), (3.4), and (3.5), we obtain
J y n (T) =
m− p
∇ y n (T) p ≤ En (T) ≤ En (0) < 2E(0),
p
pm
(3.8)
and therefore
∇ y n (T) <
p
2pm
E(0)
m− p
1/ p
.
(3.9)
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Journal of Inequalities and Applications
Combining this with (2.4) and using (2.10), we see that
m
(m− p)/ p 2pm
∇ y n (T) p
E(0)
p
m− p
m − p n p n p
∇ y (T) p < ∇ y (T) p ,
<
2(m − 1)p
λ y n (T)m < λc∗m
(3.10)
where we used the fact that (m − p)/2(m − 1)p < 1. This gives I(y n (T)) > 0, which is a
contradiction. Therefore (3.7) is valid. From (2.6), (3.6), and (3.7),
1
y n (t)2 + m − p ∇ y n (t) p +
t
p
2
pm
t
∇ y n (s)2 ds < 2E(0).
t
0
(3.11)
By (H1 ) and (3.11), it follows that
n n 2
b y (t) ≤ μ2 y n (t)2 ≤ cE(0),
t
2
(3.12)
t
here and in the sequel we denote by c a generic positive constant independent of n and t.
It follows from (3.11) and (3.12) that
1,p
y n is bounded in L∞ 0,T;W0 (Ω) ,
ytn is bounded in L∞ 0,T;L2 (Ω) ∩ L2 0,T;W 1,2 (Ω) ,
n
n
b yt
1,p
is bounded in L
∞
(3.13)
0,T;L2 (Ω) ,
and since A : W0 (Ω) → W −1,p (Ω) is a bounded operator, it follows from (3.13) that
Ay n is bounded in L∞ 0,T;W −1,p (Ω) .
(3.14)
1,p
Finally, we will obtain an estimate for yttn . Since the imbedding W0 (Ω) Lm (Ω) is continuous, we have
n m−2 n
m−1
m−1
y (t)
y (t),z ≤ y n (t) zm ≤ c y n (t) z1,p .
m
1,p
(3.15)
From (3.2), it follows that
T
T
n
n
− Ay (t),z − ∇ y n (t), ∇z
≤
y
(t),z
dt
tt
t
0
0
m−2 n
− bn ytn (t) ,z + λ y n (t)
y (t),z dt,
∀ z ∈ Vm ,
(3.16)
and hence we obtain from (3.13)–(3.15) that
T
0
n 2
y (t)
tt
−1,p dt
≤ c.
(3.17)
J. Y. Park and S. H. Park
7
Step 2 (passage to the limit). From (3.13), (3.14), and (3.17), we can extract a subsequence from { y n }, still denoted by { y n }, such that
1,p
weakly star in L∞ 0,T;W0 (Ω) ,
y n −→ y
ytn −→ yt
weakly in L2 0,T;W 1,2 (Ω) ,
ytn −→ yt
weakly star in L∞ 0,T;L2 (Ω) ,
yttn −→ ytt
weakly in L2 0,T;W −1,p (Ω) ,
(3.18)
weakly star in L∞ 0,T;W −1,p (Ω) ,
Ay n −→ ζ
bn y n −→ Ξ weakly star in L∞ 0,T;L2 (Ω) .
1,p
Considering that the imbeddings W0 (Ω) L2 (Ω) and W 1,2 (Ω) L2 (Ω) are compact
and using the Aubin-Lions compactness lemma [3], it follows from (3.18) that
y n −→ y
ytn
−→ yt
strongly in L2 QT ,
2
(3.19)
strongly in L QT .
(3.20)
1,p
Using the first convergence result in (3.18) and the fact that the imbedding W0 (Ω) L2(m−1) (Ω) (p < m < N p/2(N − p) + 1 if N > p and p < m < ∞ if p ≥ N) is continuous,
we obtain
n m−2 n 2
y y 2
L
(QT ) =
T
0
Ω
n
y (x,t)2(m−1) dx dt ≤ c.
(3.21)
This implies that
n m−2 n
y y −→ ξ
weakly in L2 QT .
(3.22)
On the other hand, we have from (3.19) that y n (x,t) → y(x,t) a.e. in QT , and thus | y n (x,
t)|m−2 y n (x,t) → | y(x,t)|m−2 y(x,t) a.e. in QT . Therefore, we conclude from (3.22) that
ξ(x,t) = | y(x,t)|m−2 y(x,t) a.e. in QT .
Letting n → ∞ in (3.2) and using the convergence results above, we have
T
0
ytt (t),z + ∇ yt (t), ∇z + ζ(t),z + Ξ(t),z dt
=
T
0
m−2
λ y(t)
y(t),z dt,
(3.23)
∀z
1,p
∈ W0 (Ω).
8
Journal of Inequalities and Applications
Step 3 ((y,Ξ) is a solution of (1.1)). Let φ ∈ C 1 [0,T] with φ(T) = 0. By replacing w j by
φ(t)w j in (3.2) and integrating by parts the result over (0,T), we have
ytn (0),φ(0)w j +
+
−
T
0
T
T
ytn (t),φt (t)w j dt =
0
T
Ay n (t),φ(t)w j dt +
0
T
0
m−2
0
∇ ytn (t),φ(t)∇w j dt
bn ytn (t) ,φ(t)w j dt
(3.24)
λ y n (t)
y n (t),φ(t)w j .
Similarly from (3.23), we get
T
yt (0),φ(0)w j +
0
T
+
0
−
T
0
yt (t),φt (t)w j dt =
T
0
T
ζ(t),φ(t)w j dt +
∇ yt (t),φ(t)∇w j dt
Ξ(t),φ(t)w j ds
0
(3.25)
m−2
λ y(t)
y(t),φ(t)w j .
Comparing between (3.24) and (3.25), we infer that
lim ytn (0) − yt (0),w j = 0,
n→∞
j = 1,2,... .
(3.26)
This implies that ytn (0) → yt (0) weakly in W −1,p (Ω). By the uniqueness of limit, yt (0) =
y1 . Analogously, taking φ ∈ C 2 [0,T] with φ(T) = φ (T) = 0, we can obtain that y(0) = y0 .
Now, we show that Ξ(x,t) ∈ ϕ(yt (x,t)) a.e. in QT . Indeed, since ytn → yt strongly in
2
L (QT ) (see (3.20)), ytn (x,t) → yt (x,t) a.e. in QT . Let η > 0. Using the theorem of Lusin
and Egoroff, we can choose a subset ω ⊂ QT such that |ω| < η, yt ∈ L2 (QT \ ω), and ytn →
yt uniformly on QT \ ω. Thus, for each > 0, there is an M > 2/ such that
n
y (x,t) − yt (x,t) < t
2
for n > M, (x,t) ∈ QT \ ω.
(3.27)
Then, if | ytn (x,t) − s| < 1/n, we have | yt (x,t) − s| < for all n > M and (x,t) ∈ QT \ ω.
Therefore, we have
b yt (x,t) ≤ bn ytn (x,t) ≤ b yt (x,t) ,
∀n > M, (x,t) ∈ QT \ ω.
(3.28)
Let φ ∈ L1 (0,T;L2 (Ω)), φ ≥ 0. Then
QT \ω
b yt (x,t) φ(x,t)dx dt ≤
≤
QT \ω
QT \ω
bn ytn (x,t) φ(x,t)dx dt
b yt (x,t) φ(x,t)dx dt.
(3.29)
J. Y. Park and S. H. Park
9
Letting n → ∞ in this inequality and using the last convergence result in (3.18), we obtain
QT \ω
b yt (x,t) φ(x,t)dx dt ≤
≤
QT \ω
Ξ(x,t)φ(x,t)dx dt
(3.30)
a.e. in QT \ ω,
(3.31)
a.e. in QT .
(3.32)
QT \ω
b yt (x,t) φ(x,t)dx dt.
Letting → 0+ in this inequality, we deduce that
Ξ(x,t) ∈ ϕ yt (x,t)
and letting η → 0+ , we get
Ξ(x,t) ∈ ϕ yt (x,t)
It remains to show that ζ = Ay. From the approximated problem and the convergence
results (3.18)–(3.22), we see that
T
limsup
n→∞
0
Ay n (t), y n (t) dt ≤ y1 , y0 − yt (T), y(T)
T
+
0
2
1
yt (t), yt (t) dt − ∇ y(T)
2
2
1
+ ∇ y0 −
2
T
+
0
T
(3.33)
Ξ(t), y(t) dt
0
m−2
λ y(t)
y(t), y(t) dt.
On the other hand, it follows from (3.23) that
T
0
ζ(t), y(t) dt = y1 , y0 − yt (T), y(T) +
T
0
2 1 2
1
− ∇ y(T) + ∇ y0 −
2
T
+
0
yt (t), yt (t) dt
T
2
0
Ξ(t), y(t) dt
(3.34)
m−2
λ y(t)
y(t), y(t) dt.
Combining (3.33) and (3.34), we get
T
limsup
n→∞
0
Ay n (t), y n (t) dt ≤
T
0
ζ(t), y(t) dt.
(3.35)
Since A is a monotone operator, we have
0 ≤ limsup
n→∞
≤
T
0
T
0
Ay n (t) − Az(t), y n (t) − z(t) dt
ζ(t) − Az(t), y(t) − z(t) dt,
1,p
∀z ∈ L2 0,T;W0 (Ω) .
(3.36)
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Journal of Inequalities and Applications
By Mintiy’s monotonicity argument (see, e.g., [17]),
ζ = Ay
in L2 0,T;W −1,p (Ω) .
(3.37)
Therefore the proof of Theorem 2.2 is completed.
4. Asymptotic behavior of solutions
In this section, we will prove the decay rates (2.14) and (2.15) in Theorem 2.3 by applying Lemma 2.4. To prove the decay property, we first obtain uniform estimates for the
approximated energy
2 1 p λ m
1
En (t) = ytn (t) + ∇ y n (t) p − y n (t)m
2
p
m
(4.1)
and then pass to the limit. Note that En (t) is nonnegative and uniformly bounded. Let us
fix an arbitary t > 0. From the approximated problem (3.2) and w j = ytn (t), we get
2
2
d n
E (t) + ∇ ytn (t) = − bn ytn (t) , ytn (t) ≤ −μ1 ytn (t) .
dt
(4.2)
This implies that En (t) is a nonincreasing function. Setting Fn2 (t) = En (t) − En (t + 1) and
integrating (4.2) over (t,t + 1), we have
Fn2 (t) ≥
t+1
t
∇ y n (s)2 + μ1 y n (s)2 ds ≥ λ1 + μ1
t
t
t+1
t
n 2
y (s) ds,
t
(4.3)
where λ1 > 0 is the constant λ1 v2 ≤ ∇v2 , ∀v ∈ W01,2 (Ω). By applying the mean value
theorem, there exist t1 ∈ [t,t + 1/4] and t2 ∈ [t + 3/4,t + 1] such that
n y ti ≤ 2
t
λ 1 + μ1
Fn (t),
i = 1,2.
(4.4)
Now, replacing w j by y n (t) in the approximated problem, we have
m−2 n
Ay n (t), y n (t) − λ y n (t)
y (t), y n (t)
= − yttn (t), y n (t) − ∇ ytn (t), ∇ y n (t) − bn ytn (t) , y n (t) .
(4.5)
J. Y. Park and S. H. Park
11
Integrating this over (t1 ,t2 ) and using (4.2) and (H1 ), we get
t2
t1
1
∇ y n (s) p ds − λ
p
p
≤
t2
t1
t2
t1
n m
y (s) ds
m
∇ y n (s) p ds − λ
p
t2
n m
y (s) ds
m
t1
= − ytn t2 , y n t2 + ytn t1 , y n t1 +
−
t2
t1
∇ ytn (s), ∇ y n (s) ds −
t2
t1
b
n
t2
t1
ytn (s)
≤ ytn t2 y n t2 + ytn t1 y n t1 +
t2
+c
t1
∇ y n (s)
t
n 2
y (s) ds
t
, y (s) ds
t2
t1
sup ∇ y n (s) p ds + μ2
t ≤s≤t+1
(4.6)
n
n 2
y (s) ds
t
t2
t1
n n y (s) y (s)ds.
t
Using Holder’s inequality, Poincaré inequality, and (4.3)–(4.6), we get
t2
t1
En (s)ds =
1
2
+
t2
t1
1
p
n 2
y (s) ds
t
t2
t1
∇ y n (s) p ds − λ
p
m
t2
t1
n m
y (s) ds ≤ cF 2 (t)
m
n
+ cFn (t) ∇ y n t2 p + ∇ y n t1 p + sup ∇ y n (s) p
1
+λ 1−
m
t2
t1
(4.7)
t ≤s≤t+1
n m
y (s) ds,
m
and hence we derive that
t2
t1
En (s)ds ≤ cFn2 (t) + cFn (t)En (t)1/ p + C1 En (t),
(4.8)
where C1 = λ(1 − (1/m))c∗m (2mp/(m − p)E(0))(m− p)/ p (mp/(m − p)) and we used the fact
p
that ∇ y n (t) p ≤ (mp/(m − p))En (t), En (t) is a nonincreasing function, and (2.4).
Young’s inequality implies that
t2
t1
1
En (s)ds ≤ cFn2 (t) + Cη Fn (t) p/(p−1) + En (t) + C1 En (t).
η
Noting that En (t + 1) ≤ 2
that
t2
t1
(4.9)
En (s)ds and En (t + 1) = En (t) − Fn2 (t), we have from (4.9)
1
1
1 n
− C1 −
E (t) ≤ c + Fn2 (t) + Cη Fn (t) p/(p−1) .
2
η
2
(4.10)
12
Journal of Inequalities and Applications
By assumption (2.10), 1/2 − C1 > 0, and hence taking η > 0 sufficiently small such that
1/2 − C1 − 1/η > 0, we obtain that
En (t) ≤ cFn2 (t) + cFn (t) p/(p−1) .
(4.11)
If p = 2 then En (t) ≤ cFn2 (t), and since En (t) is decreasing from Lemma 2.4 there exist
positive constants C and γ such that
En (t) ≤ C exp(−γt),
∀t ≥ 0.
(4.12)
If p > 2, then (4.11) and the boundedness of Fn (t) imply that
En (t) ≤ cFn (t) p/(p−1) ,
(4.13)
and then
En (t)2(p−1)/ p ≤ c2(p−1)/ p En (t) − En (t + 1) .
(4.14)
Applying Lemma 2.4 to β = (p − 2)/ p, we obtain a constant C > 0 such that
En (t) ≤ C(1 + t)− p/(p−2) ,
∀t ≥ 0.
(4.15)
Passing to the limit n → ∞ in (4.12) and (4.15), we get (2.14) and (2.15). This completes
the proof of Theorem 2.3.
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13
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Jong Yeoul Park: Department of Mathematics, Pusan National University, Pusan 609-735,
South Korea
Email address: jyepark@pusan.ac.kr
Sun Hye Park: Department of Mathematics, Pusan National University, Pusan 609-735,
South Korea
Email address: sh-park@pusan.ac.kr