# Pullback Exponential Attractor for Second Order Nonautonomous Lattice System

Authors: Shengfan Zhou; Hong Chen; Zhaojuan Wang
Journal: Discrete Dynamics in Nature and Society
 (2014)

---

## Abstract

We first present some sufficient conditions for the existence of a pullback exponential attractor for continuous process on the product space of the weighted spaces of infinite sequences. Then we prove the existence and continuity of a pullback exponential attractor for second order lattice system with time-dependent coupled coefficients in the weighted space of infinite sequences. Moreover, we obtain the upper bound of fractal dimension and attracting rate for the attractor.


---

## Body

## 1. Introduction

Lattice dynamical systems (LDSs), which include coupled systems of infinite ordinary differential equations and coupled map lattices, have drawn more and more attention because these systems appear in various fields [1, 2]. In recent years, global attractors, uniform attractors, pullback attractors (or kernel sections), and random attractor for autonomous, nonautonomous, and stochastic LDSs have been studied; see [3–12]. However, these attractors sometimes attract orbits at a relatively slow speed, so that it might take an unexpected long time to be reached. Besides, it is usually difficult to estimate the attracting rate in terms of physical parameters of the model. Appropriate alternatives are exponential attractors and pullback exponential attractors which contain the global attractors and pullback attractors and attract all bounded sets exponentially [13–20]. For LDSs, [13, 16, 21] studied the existence of exponential attractors for first order, second order, and partly dissipative autonomous LDSs, respectively. Zhou and Han [20] presented some sufficient conditions for the existence of pullback exponential attractors for LDSs in l
ρ
2 and provided applications to first order and partly dissipative nonautonomous LDSs. As we know, there are no results on pullback exponential attractors for second order nonautonomous LDSs which have been studied extensively [4–7, 9–13, 21–23]. In this paper, we study the following second order nonautonomous LDSs:
(1)
d
2
u
i
d
t
2
+
α
d
u
i
d
t
=
-
λ
i
u
i
+
∑
j
=
-
q
q
‍
η
i
,
j
(
t
)
u
i
+
j
+
f
i
(
u
i
,
t
)
+
g
i
(
t
)
,
i
∈
ℤ
,
where for i
∈
ℤ, u
i
∈
ℝ, g
i
∈
C
(
ℝ
,
ℝ
); f
i
∈
C
1
(
ℝ
×
ℝ
,
ℝ
); η
i
,
j
(
t
), j
=
-
q
,
…
,
q
(
q
∈
ℕ
) are locally integrable in t, and λ
i, α are positive constants. First we present some sufficient conditions for the existence of a pullback exponential attractor for a continuous process on the product space ∏
j
=
1
m
‍
l
ρ
j
p of infinite sequences, which are direct results of [20]. Then we prove the existence of a pullback exponential attractor for system (1) in weighted space l
ρ
2
×
l
ρ
2; moreover, we obtain the upper bound of fractal dimension and attracting rate for the attractor.


## 2. Preliminaries

In this section, we present some sufficient conditions for the existence of a pullback exponential attractor for a continuous process on the product space ofm weighted space of infinite sequences.Letρ
j
:
ℤ
→
ℝ
+ be positive-valued function such that 0
<
ρ
j
(
i
)
=
ρ
j
,
i
≤
M
j
0, where M
j
0 are positive constants, j
=
1
,
…
,
m, and let
(2)
l
ρ
j
p
=
{
u
=
(
u
i
)
i
∈
ℤ
:
∑
i
∈
ℤ
‍
ρ
j
,
i
|
u
i
|
p
<
∞
,
u
i
∈
ℝ
}
,
p
>
1
,
which is Banach space with norm ∥
u
∥
ρ
j
p
=
∑
i
∈
ℤ
‍
ρ
j
,
i
|
u
i
|
p for u
=
(
u
i
)
i
∈
ℤ, v
=
(
v
i
)
i
∈
ℤ
∈
l
ρ
j
p. Let l
ρ
,
m
p
=
∏
j
=
1
m
‍
l
ρ
j
p be a product space of m spaces l
ρ
j
p, j
=
1
,
…
,
m. Write
(3)
l
ρ
j
,
N
p
=
{
u
=
(
u
i
)
i
∈
ℤ
∈
l
ρ
j
p
:
u
i
=
0
for
|
i
|
>
N
}
,
N
∈
ℕ
.
Then l
ρ
j
,
N
p is (
2
N
+
1
)-dimensional subspaces of l
ρ
j
p. Let l
ρ
,
m
,
N
p
=
∏
j
=
1
m
‍
l
ρ
j
,
N
p and define a bounded projection P
N
:
l
ρ
,
m
p
→
l
ρ
,
m
,
N
p as follows: for φ
=
(
φ
i
)
i
∈
ℤ
=
(
u
i
(
1
)
,
…
,
u
i
(
m
)
)
i
∈
ℤ, u
(
j
)
=
(
u
i
(
j
)
)
i
∈
ℤ
∈
l
ρ
j
p,
(4)
(
P
N
φ
)
i
=
{
φ
i
,
|
i
|
≤
N
,
0
,
|
i
|
>
N
.Consider a two-parameter continuous process onE: W
(
t
,
τ
)
:
E
→
E, t
≥
τ
∈
ℝ, satisfying the following: W
(
τ
,
τ
)
=
I, for all τ
∈
ℝ; W
(
t
,
r
)
W
(
r
,
τ
)
=
W
(
t
,
τ
), -
∞
<
τ
≤
r
≤
t
<
∞; (
t
,
τ
,
φ
τ
)
→
W
(
t
,
τ
)
φ
τ is continuous for -
∞
<
τ
≤
t
<
∞.Definition 1.
A family{
ℳ
(
t
)
}
t
∈
ℝ of subsets of l
ρ
,
m
p is called a pullback exponential attractor for the continuous process {
W
(
t
,
τ
)
}
t
≥
τ, if (i)
eachℳ
(
t
)
(
t
∈
ℝ
) is a compact set of l
ρ
,
m
p and its fractal dimension is uniformly bounded in t; that is, sup
⁡
t
∈
ℝ
dim
⁡
f
ℳ
(
t
)
<
∞;
(ii)
it is positively invariant; that is,W
(
t
,
τ
)
ℳ
(
τ
)
⊂
ℳ
(
t
) for all -
∞
<
τ
≤
t
<
∞;
(iii)
there exist an exponentα
>
0 and two positive-valued functions Q, 𝒯
:
ℝ
+
→
ℝ
+ such that for any bounded set B
⊂
l
ρ
,
m
p
(5)
d
h
(
W
(
t
,
τ
)
B
,
ℳ
(
t
)
)
≤
Q
(
∥
B
∥
l
ρ
,
m
p
)
e
-
α
(
t
-
τ
)
,
τ
∈
ℝ
,
τ
+
𝒯
(
∥
B
∥
l
ρ
,
m
p
)
≤
t
<
∞
,where “d
h
(
·
,
·
)” is the Hausdorff semidistance between two subsets of l
ρ
,
m
p.
As a direct consequence of Theorem  2 of [20], we have the following theorem.Theorem 2.
Let{
W
(
t
,
τ
)
}
t
≥
τ be a continuous process in l
ρ
,
m
p. Assume that there exists a uniformly bounded absorbing set B
0
⊂
l
ρ
,
m
p for {
W
(
t
,
τ
)
}
t
≥
τ: for any bounded set B
⊂
l
ρ
,
m
p, there exists a constant T
B
≥
0 such that W
(
t
,
τ
)
B
⊆
B
0 for all t
≥
τ
+
T
B, τ
∈
ℝ. For any t
∈
ℝ, set X
(
t
)
=
∪
s
≥
T
B
0
W
(
t
,
t
-
s
)
B
0
¯
⊆
B
0. If(
A
1
)
there existT
*
>
T
B
0 and L
T
*
=
L
(
T
*
)
>
0 such that, for every τ
∈
ℝ, any t
∈
[
0
,
T
*
], and φ
τ
,
ψ
τ
∈
X
(
τ
),
(6)
∥
W
(
t
+
τ
,
τ
)
φ
τ
-
W
(
t
+
τ
,
τ
)
ψ
τ
∥
l
ρ
,
m
p
≤
L
T
*
∥
φ
τ
-
ψ
τ
∥
l
ρ
,
m
p
;
(
A
2
)
there exist a positive constantγ
∈
[
0
,
1
/
2
) and a (
2
N
+
1
)
m-dimensional bounded projection P
N
:
l
ρ
,
m
p
→
l
ρ
,
m
,
N
p such that, for every τ
∈
ℝ and φ
τ
,
ψ
τ
∈
X
(
τ
),
(7)
∥
(
I
-
P
N
)
(
W
(
t
+
τ
,
τ
)
φ
τ
-
W
(
t
+
τ
,
τ
)
ψ
τ
)
∥
l
ρ
,
m
p
≤
γ
∥
φ
τ
-
ψ
τ
∥
l
ρ
,
m
p
.Then {
W
(
t
,
τ
)
}
t
≥
τ possesses a pullback exponential attractor {
ℳ
(
t
)
}
t
∈
R satisfying
(8)
ℳ
(
t
)
⊆
X
(
t
)
⊆
B
0
,
dim
⁡
f
ℳ
(
t
)
≤
ln
⁡
N
θ
-
ln
⁡
a
θ
,
and, for any bounded set B
⊂
l
ρ
,
m
p,
(9)
d
h
(
W
(
t
,
τ
)
B
,
ℳ
(
t
)
)
≤
L
T
*

Re
α
T
B
a
θ
2
e
-
α
(
t
-
τ
)
,
aaaaaaaaaaaaaaa
-
∞
<
τ
+
T
B
<
t
≤
+
∞
,
where N
θ is the minimal number of closed balls of E with radius θ covering the closed unit ball B
N
(
0,1
) of l
ρ
,
m
,
N
p centered at 0, and a
θ
=
2
(
γ
+
θ
L
T
*
) with 0
<
θ
<
(
1
-
2
γ
)
/
2
L
T
*, α
=
-
ln
⁡
θ
/
T
*, and R
=
sup
⁡
φ
∈
B
0
∥
φ
∥
l
ρ
,
m
p.


## 3. Pullback Exponential Attractor for Second Order Nonautonomous Lattice System

In this section, we study the existence of a pullback exponential attractor for the continuous processes associated with the second order nonautonomous lattice system (1). Note that system (1) with initial conditions can be written as a vector form
(10)
u
¨
+
α
u
˙
=
-
λ
u
+
A
(
t
)
u
+
f
(
u
,
t
)
+
g
(
t
)
,
t
>
τ
,
τ
∈
R
,
u
(
τ
)

=
(
u
i
,
τ
)
i
∈
Z

=
u
τ
,
u
˙
(
τ
)
=

(
u
˙
i
(
τ
)
)
i
∈
ℤ
=

(
u
1
i
,
τ
)
i
∈
ℤ

=
u
1
τ
,
where u
=
(
u
i
)
i
∈
ℤ, (
A
(
t
)
u
)
i
=
∑
j
=
-
q
q
η
i
,
j
(
t
)
u
i
+
j, λ
u
=
(
λ
i
u
i
)
i
∈
ℤ, f
(
u
,
t
)
=
(
f
i
(
u
i
,
t
)
)
i
∈
ℤ, g
(
t
)
=
(
g
i
(
t
)
)
i
∈
ℤ.Throughout this section, letρ be a positive weight function from ℤ to (
0
,
M
0
)
⊂
ℝ
+ satisfying(P0)
0
<
ρ
(
i
)
≤
M
0, ρ
i
=
ρ
(
i
)
≤
c
0
ρ
(
i
±
1
), i
∈
ℤ, for some positive constants M
0 and c
0.Consider the weighted Hilbert space of infinite sequences(11)
l
ρ
2
=
{
u
=
(
u
i
)
i
∈
Z
:
∑
i
∈
Z
‍
ρ
i
u
i
2
<
∞
,
u
i
∈
R
}
endowed with inner product (
u
,
v
)
ρ
=
∑
i
∈
Z
‍
ρ
i
u
i
v
i and norm ∥
u
∥
ρ
2
=
(
u
,
v
)
ρ
=
∑
i
∈
Z
‍
ρ
i
u
i
2 for u
=
(
u
i
)
i
∈
Z, v
=
(
v
i
)
i
∈
Z
∈
l
ρ
2. For any u, v
∈
l
ρ
2, define an inner product on l
ρ
2 by (
u
,
v
)
λ
,
ρ
=
∑
i
∈
Z
‍
ρ
i
λ
i
u
i
v
i, then the norm ∥
·
∥
λ
,
ρ include by (
·
,
·
)
λ
,
ρ is equivalent to the norm ∥
·
∥
ρ include by (
·
,
·
)
ρ. Let
(12)
l
λ
,
ρ
2
=
(
l
ρ
2
,
(
·
,
·
)
λ
,
ρ
,
∥
·
∥
λ
,
ρ
)
,
E
=
l
λ
,
ρ
2
×
l
ρ
2
.We make the following assumptions onλ
i, η
i
,
j, f
i, and g
i, for i
∈
ℤ, j
=
-
q
,
…
,
q.(H1)
There exist two positive constantsλ
_ and λ
¯ such that
(13)
0
<
λ
_
≤
λ
i
≤
λ
¯
<
+
∞
,
∀
i
∈
ℤ
.
(H2)
Letη
(
t
)
=
{
sup
⁡
|
η
i
,
j
(
t
)
|
:
i
∈
ℤ
,
j
=
-
q
⋯
q
}
<
∞
(
q
∈
ℕ
) satisfy the following:
(H2a)
η
(
t
)
∈
L
loc
⁡
1
(
ℝ
) in t;
(H2b)
there exists a continuous positive valued functionQ
:
ℝ
+
→
ℝ
+ such that
(14)
∫
t
-
l
t
‍
η
(
s
)
d
s
≤
Q
(
l
)
,
0
≤
lim
⁡
l
→
+
∞
1
l
∫
t
-
l
t
‍
η
(
s
)
d
s
≤
η
¯
<
∞
,
aaaaaaaaaa
∀
t
∈
ℝ
,
l
∈
ℝ
+
,
where η
¯ is a positive constant;
(H2c)
there existI
0
∈
ℕ and η
0
>
0 such that
(15)
sup
⁡
{
|
η
i
,
j
(
t
)
|
:
|
i
|
>
I
0
,
j
=
-
q
⋯
q
}
≤
η
0
,
∀
t
∈
ℝ
.
(H3)
For alli
∈
ℤ, let f
i satisfy the following:
(H3a)
f
i
(
x
,
t
) is differentiable in x and continuous in t;
(H3b)
for anyi
∈
ℤ, f
i
(
0
,
t
)
=
0, sup
⁡
x
,
t
∈
ℝ
|
f
i
(
x
,
t
)
|
≤
β
i
(
t
);
(H3c)
there exist functionsΓ
∈
C
(
ℝ
+
×
ℝ
,
ℝ
+
) and Γ
~
∈
C
(
ℝ
+
×
ℝ
+
,
ℝ
+
) such that
(16)
sup
⁡
ρ
i
|
x
|
≤
r
|
f
i
′
(
x
,
t
)
|
≤
Γ
(
r
,
t
)
,
i
∈
ℤ
,
t
∈
ℝ
,
r
∈
ℝ
+
,
Γ
(
0
,
t
)
=
0
,
∫
τ
τ
+
l
‍
Γ
2
(
r
,
t
)
d
t
≤
Γ
~
(
r
,
l
)
,
∀
τ
∈
ℝ
,
l
∈
ℝ
+
.
(H4)
g
(
t
)
=
(
g
i
(
t
)
)
i
∈
ℤ
∈
G, β
(
t
)
=
(
β
i
(
t
)
)
i
∈
ℤ
∈
G, where
(17)
G
=
{
∑
|
i
|
>
I
(
η
)
‍
g
∈
C
b
(
ℝ
,
l
ρ
2
)
:
∀
η
>
0
,
∃
I
(
η
)
∈
ℕ
ia
such
that
sup
⁡
t
∈
ℝ
∑
|
i
|
>
I
(
η
)
‍
ρ
i
g
i
2
(
t
)
<
η
}and |
|
|
g
|
|
|
ρ
=
sup
⁡
t
∈
ℝ
∥
g
i
(
t
)
∥
ρ, |
|
|
β
|
|
|
ρ
=
sup
⁡
t
∈
ℝ
∥
β
i
(
t
)
∥
ρ, C
b
(
ℝ
,
l
ρ
2
) denotes the space of all continuous bounded functions from ℝ into l
ρ
2.Letting(18)
v
=
u
˙
+
ε
0
u
,
ε
0
=
λ
_
α
3
λ
_
+
α
2
,
φ
=
(
u
v
)
,
then the system (10) is equivalent to the following evolution equation:
(19)
φ
˙
+
C
(
φ
)
=
F
(
φ
,
t
)
,
t
>
τ
,
τ
∈
ℝ
,
φ
(
τ
)
=
(
u
τ
v
τ
)
=
(
u
τ
u
1
τ
+
ε
0
u
τ
)
,
where
(20)
F
(
φ
,
t
)
=
(
0
A
(
t
)
u
+
g
(
t
)
+
f
(
u
,
t
)
)
,
C
(
φ
)
=
(
ε
0
u
-
v
λ
u
-
ε
0
(
α
-
ε
0
)
u
+
(
α
-
ε
0
)
v
)
.Definition 3.
The functionφ
:
[
τ
,
τ
+
T
)
→
E
(
T
>
0
) is called a mild solution of the following lattice differential equations:
(21)
d
φ
(
t
)
d
t
=
H
(
φ
,
t
)
,
φ
=
(
φ
i
)
i
∈
ℤ
,
H
=
(
H
i
)
i
∈
ℤ
,
i
i
i
t
≥
τ
∈
ℝ
,
if φ
∈
C
(
[
τ
,
τ
+
T
)
,
E
) and
(22)
φ
i
(
t
,
τ
)
=
φ
i
(
τ
)
+
∫
τ
t
‍
H
i
(
φ
(
s
)
,
s
)
d
s
,
i
∈
ℤ
,
t
∈
[
τ
,
τ
+
T
)
.Theorem 4.
Assume that (P0) and (H1)–(H4) hold. Then for any fixedτ
∈
ℝ and any initial data φ
(
τ
)
=
(
u
τ
,
v
τ
)
∈
E, problem (19) admits a unique mild solution φ
(
·
,
τ
;
φ
τ
)
∈
C
(
[
τ
,
+
∞
)
,
E
) with φ
(
τ
,
τ
;
φ
τ
)
=
φ
τ and φ
(
·
,
τ
;
φ
τ
) being continuous in φ
τ
∈
E, and the mapping
(23)
U
(
t
,
τ
)
:
φ
(
τ
)
=
(
u
τ
,
v
τ
)
∈
E
⟶
φ
(
t
,
τ
)
=
(
u
(
t
,
τ
)
,
v
(
t
,
τ
)
)
∈
E
,
t
≥
τ
,
τ
∈
ℝ
,
generates a continuous process {
U
(
t
,
τ
)
}
t
≥
τ on E.Proof.
Let(24)
F
~
(
u
,
t
)
=
A
(
t
)
u
+
f
(
u
,
t
)
+
g
(
t
)
,
and then F
~
(
u
,
t
) is continuous in u and locally integrable in t from l
ρ
2
×
ℝ into l
ρ
2. For any bounded set B
⊂
E with sup
⁡
φ
∈
B
∥
φ
∥
≤
r, let
(25)
h
B
(
t
)
=
(
(
2
q
+
1
)
η
(
t
)
+
Γ
(
r
λ
_
,
t
)
)
r
+
|
|
|
g
|
|
|
≥
0
,
t
∈
ℝ
.
Then for φ
=
(
u
,
v
), φ
(
j
)
=
(
u
(
j
)
,
u
(
j
)
)
∈
B, j
=
1
,
2, by (P0), (H1)–(H4), we have
(26)
∥
F
(
φ
,
t
)
∥
E
≤
h
B
(
t
)
,
∥
F
(
φ
(
1
)
,
t
)
-
F
(
φ
(
2
)
,
t
)
∥
E
≤
h
B
(
t
)
·
∥
φ
(
1
)
-
φ
(
2
)
∥
E
,
t
∈
ℝ
,
∫
τ
τ
+
1
‍
h
B
(
t
)
d
t
<
∞
,
∀
τ
∈
ℝ
.
By the approximating method [24], it follows that problem (19) possesses a unique local mild solution φ
(
·
,
τ
;
φ
τ
)
∈
C
(
[
τ
,
T
max
⁡
)
,
E
). Similar to the proof of Theorem 5 below concerning the existence of a uniformly bounded absorbing set, we can prove that T
max
⁡
=
+
∞.In the following part of this section, we assume that (P0) and (H1)–(H4) and(27)
ε
0
>
2
q
~
(
1
λ
_
+
1
)
η
¯
,
q
~
=
q
+
∑
k
=
0
q
‍
c
0
k
hold.Theorem 5.
Assume that (P0) and (H1)–(H4) hold. There exists a uniform bounded closed absorbing ballB
0
=
B
0
(
0
,
r
0
)
⊂
E of {
U
(
t
,
τ
)
}
t
≥
τ centered at 0 with radius r
0 (independent of τ, t) such that, for any bounded subset B
⊂
E, there exists T
B
≥
0 yielding U
(
t
+
τ
,
τ
)
B
⊆
B
0 for all t
≥
T
B.Proof.
Letφ
(
t
)
=
(
u
i
(
t
)
v
i
(
t
)
)
i
∈
ℤ be a mild solution of (19), t
≥
τ. Since the set of continuous functions is dense in L
loc
⁡
1
(
ℝ
), by (H2a), there exist sequences of continuous functions η
i
,
j
m
(
t
), m
∈
ℕ, such that
(28)
lim
⁡
m
→
+
∞
∫
τ
t
‍
|
η
i
,
j
m
(
s
)
-
η
i
,
j
(
s
)
|
d
s
=
0
,
|
η
i
,
j
m
(
t
)
|
≤
|
η
i
,
j
(
t
)
|
≤
|
η
(
t
)
|
,
∀
τ
,
t
∈
ℝ
.
Consider the following lattice differential equations:(29)
φ
˙
m
+
C
(
φ
m
)
=
F
m
(
φ
m
,
t
)
,
t
>
τ
,
φ
m
(
τ
)
=
φ
τ
=
(
u
τ
v
τ
)
=
(
u
τ
u
1
τ
+
ε
0
u
τ
)
,
τ
∈
ℝ
,
where
(30)
φ
m
=
(
u
m
v
m
)
,
F
m
(
φ
m
,
t
)
=
(
0
A
m
(
t
)
u
m
+
f
(
u
m
,
t
)
+
g
(
t
)
)
,
C
(
φ
m
)
=
(
ε
0
u
m
-
v
m
λ
u
m
-
ε
0
(
α
-
ε
0
)
u
m
+
(
α
-
ε
0
)
v
m
)
,
(
A
m
(
t
)
u
m
)
i
=
∑
j
=
-
q
q
‍
η
i
,
j
m
(
t
)
u
i
+
j
m
.
Combining the continuity of F
m
(
φ
m
,
t
) in t with the proof of Theorem 4, (36) has a unique strong solution φ
m
(
·
,
τ
,
φ
τ
)
=
(
u
m
(
·
,
τ
,
φ
τ
), v
m
(
·
,
τ
,
φ
τ
)
)
∈
C
(
[
τ
,
+
∞
)
,
E
)
∩
C
1
(
(
τ
,
+
∞
)
,
E
) satisfying
(31)
φ
i
m
(
t
)
=
φ
τ
+
∫
τ
t
‍
(
F
i
(
φ
i
m
(
s
)
,
s
)
-
C
(
φ
i
m
(
s
)
)
i
)
d
s
,
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
t
∈
[
τ
,
τ
+
T
]
.
Taking the inner product (
·
,
·
)
E of (29) with φ
m
(
t
), we obtain
(32)
1
2
d
∥
φ
m
∥
E
2
d
t
+
(
C
(
φ
m
)
,
φ
m
)
E
=
(
F
m
(
φ
m
,
t
)
,
φ
m
)
E
.
By some computation, we have that
(33)
(
C
(
φ
m
)
,
φ
m
)
E
≥
ε
0
2
∥
φ
m
∥
E
2
+
α
2
∥
v
m
∥
ρ
2
,
(34)
(
F
m
(
φ
m
)
,
φ
m
)
E
≤
1
α
(
|
|
|
β
|
|
|
ρ
2
+
|
|
|
g
|
|
|
ρ
2
)
+
α
2
∥
v
m
∥
ρ
2
+
1
2
q
~
(
1
λ
_
+
1
)
η
(
t
)
·
∥
φ
m
∥
E
2
.
Then we have that, for t
≥
τ,
(35)
d
∥
φ
m
(
t
)
∥
E
2
d
t
≤
(
-
ε
0
+
q
~
(
1
λ
_
+
1
)
η
(
t
)
)
·
∥
φ
m
∥
E
2
+
2
α
(
|
|
|
β
|
|
|
ρ
2
+
|
|
|
g
|
|
|
ρ
2
)
.
Applying Gronwall’s inequality to (35) on [
τ
,
t
] (t
≥
τ), we obtain
(36)
∥
φ
m
(
t
)
∥
2
≤
e
-
ε
0
(
t
-
τ
)
+
∫
τ
t
‍
q
~
(
(
1
/
λ
_
)
+
1
)
η
(
s
)
d
s
∥
φ
τ
m
∥
E
2
+
2
α
(
|
|
|
β
|
|
|
ρ
2
+
|
|
|
g
|
|
|
ρ
2
)
×
∫
τ
t
‍
e
-
ε
0
(
t
-
s
)
+
∫
s
t
‍
q
~
(
(
1
/
λ
_
)
+
1
)
η
(
r
)
d
r
d
s
≐
χ
2
(
t
,
τ
)
.
It then follows that, for T
>
0,
(37)
∥
F
i
m
(
φ
i
m
,
t
)
-
(
C
(
φ
m
)
)
i
∥
ℝ
2
≤
κ
(
T
,
τ
)
,
a
a
a
a
a
a
a
a
a
a
a
a
a
t
∈
[
τ
,
τ
+
T
]
,
i
∈
ℤ
,
for some κ
(
T
,
τ
)
>
0 independent of m and i, and then, for any t
,
s
∈
[
τ
,
τ
+
T
), m
∈
ℕ,
(38)
∥
φ
i
m
(
t
)
-
φ
i
m
(
s
)
∥
ℝ
2
=
∫
s
t
‍
∥
F
i
m
(
φ
i
m
,
s
)
-
(
C
(
φ
m
)
)
i
∥
ℝ
2
d
s
≤
κ
(
T
,
τ
)
·
|
t
-
s
|
implies the equicontinuity of {
φ
i
m
(
t
)
}
m
∈
ℕ
,
t
∈
[
τ
,
τ
+
T
]. By the Arzela-Ascoli Theorem, there exists a convergent subsequence {
φ
i
m
k
(
t
)
}
m
∈
ℕ of {
φ
i
m
(
t
)
}
m
∈
ℕ
,
t
∈
[
τ
,
τ
+
T
], such that
(39)
lim
⁡
k
→
∞
φ
i
m
k
(
t
)
⟶
φ
¯
i
(
t
)
=
(
u
¯
i
(
t
)
,
v
¯
i
(
t
)
)
,
i
i
t
∈
[
τ
,
τ
+
T
]
,
i
∈
ℤ
,
and φ
¯
i
(
t
) is continuous in t
∈
[
τ
,
τ
+
T
]; moreover, by (36), |
φ
¯
i
(
t
)
|
≤
χ
(
t
,
τ
) for t
∈
[
τ
,
τ
+
T
]. By the Lebesgue Convergence Theorem, we have
(40)
lim
⁡
m
→
+
∞
∫
τ
t
‍
|
η
i
,
j
m
k
(
s
)
u
i
m
k
(
s
)
-
η
i
,
j
(
s
)
u
¯
i
(
s
)
|
d
s
=
0
.
Thus, by replacing m with m
k in (31) and letting k
→
+
∞, we have
(41)
φ
¯
i
(
t
)
=
φ
τ
+
∫
τ
t
‍
(
F
i
(
φ
¯
(
s
)
,
s
)
-
(
C
(
φ
¯
(
s
)
)
i
)
d
s
,
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
t
∈
[
τ
,
τ
+
T
]
.
By the uniqueness of the mild solutions of (19), we have φ
¯
i
(
t
)
=
φ
i
(
t
) for t
∈
[
τ
,
τ
+
T
]. By replacing m with m
k and letting k
→
∞ in (36), we have that for, t
≥
τ,
(42)
∥
φ
(
t
)
∥
E
2
≤
e
-
ε
0
(
t
-
τ
)
+
q
~
(
(
1
/
λ
_
)
+
1
)
∫
τ
t
‍
η
(
s
)
d
s
·
∥
φ
τ
∥
E
2
+
1
2
r
2
(
t
,
τ
)
,
where
(43)
r
2
(
t
,
τ
)
=
4
α
(
|
|
|
β
|
|
|
ρ
2
+
|
|
|
g
|
|
|
ρ
2
)
∫
τ
t
‍
e
-
ε
0
(
t
-
s
)
+
∫
s
t
‍
q
~
(
(
1
/
λ
_
)
+
1
)
η
(
r
)
d
r
d
s
.
By (H2b), there exists T
1
>
0 such that for, s
≥
T
1 and t
∈
ℝ,
(44)
1
s
∫
t
-
s
t
‍
η
(
r
)
d
r
≤
η
¯
.
Let
(45)
k
~
=
ε
0
-
q
~
(
1
λ
_
+
1
)
η
¯
>
0
,
and then
(46)
∫
τ
t
‍
e
-
ε
0
(
t
-
s
)
+
∫
s
t
‍
q
~
(
(
1
/
λ
_
)
+
1
)
η
(
r
)
d
r
d
s
≤
∫
0
+
∞
‍
e
-
ε
0
s
+
∫
t
-
s
t
‍
q
~
(
(
1
/
λ
_
)
+
1
)
η
(
r
)
d
r
d
s
=
(
∫
0
T
1
‍
+
∫
T
1
+
∞
‍
)
e
-
ε
0
s
+
∫
t
-
s
t
‍
q
~
(
(
1
/
λ
_
)
+
1
)
η
(
r
)
d
r
d
s
≤
e
q
~
(
(
1
/
λ
_
)
+
1
)
Q
(
T
1
)
T
1
+
∫
T
1
+
∞
‍
e
-
ε
0
s
+
q
~
(
(
1
/
λ
_
)
+
1
)
η
¯
s
=
e
q
~
(
(
1
/
λ
_
)
+
1
)
Q
(
T
1
)
T
1
+
1
k
~
e
-
k
~
T
1
<
∞
.
Thus, for any bounded subset B of E and φ
τ
∈
B,
(47)
∥
φ
(
t
,
τ
,
φ
τ
)
∥
E
2
≤
e
-
ε
0
(
t
-
τ
)
+
∫
τ
t
‍
q
~
(
(
1
/
λ
_
)
+
1
)
η
(
s
)
d
s
sup
⁡
φ
τ
∈
B
∥
φ
τ
∥
E
2
+
1
2
r
0
2
,
where
(48)
r
0
2
:
=
4
α
(
|
|
|
β
|
|
|
ρ
2
+
|
|
|
g
|
|
|
ρ
2
)
(
e
q
~
(
(
1
/
λ
_
)
+
1
)
Q
(
T
1
)
T
1
+
1
k
~
e
-
k
~
T
1
)
i
<
∞
.
Again, by (H2b),
(49)
lim
⁡
t
→
+
∞
e
-
ε
0
(
t
-
τ
)
+
∫
τ
t
‍
q
~
(
(
1
/
λ
_
)
+
1
)
η
(
s
)
d
s
sup
⁡
φ
τ
∈
B
∥
φ
τ
∥
E
2
=
0
.
It then follows that
(50)
B
0
=
{
φ
∈
E
:
∥
φ
τ
∥
E
≤
r
0
}
=
B
E
(
0
,
r
0
)
¯
⊂
E
is a uniformly bounded closed absorbing set for {
U
(
t
,
τ
)
}
t
≥
τ.Lemma 6.
Assume that (P0) and (H1)–(H4) hold. For anyε
>
0, there exist T
(
ε
,
B
0
)
>
T
B
0 and I
(
ε
,
B
0
)
∈
ℕ such that, for t
≥
T
(
ε
,
B
0
), the mild solution U
(
τ
+
t
,
t
)
φ
τ
=
φ
(
τ
+
t
,
τ
;
φ
τ
)
=
(
u
i
(
τ
+
t
,
τ
;
u
τ
), v
i
(
τ
+
t
,
τ
;
v
τ
)
)
i
∈
ℤ
∈
E of system (19) with φ
τ
∈
B
0
⊂
E satisfies
(51)
∑
|
i
|
>
I
(
ε
,
B
0
)
‍
∥
φ
i
(
τ
+
t
,
τ
,
φ
τ
)
∥
E
2
=
∑
|
i
|
>
I
(
ε
,
B
0
)
‍
(
λ
i
ρ
i
u
i
2
(
τ
+
t
,
τ
;
u
τ
)
+
ρ
i
v
i
2
(
τ
+
t
,
τ
;
v
τ
)
)
≤
ε
2
.Proof.
Letξ
∈
C
1
(
R
+
,
R
) be a smooth increasing function which satisfies
(52)
ξ
(
s
)
=
0
,
0
≤
s
<
1
;
0
≤
ξ
(
s
)
≤
1
,
1
≤
s
<
2
;
ξ
(
s
)
=
1
,
s
≥
2
;
|
ξ
′
(
s
)
|
≤
C
0
,
s
∈
R
+
,
C
0
>
0
.
Let φ
m
(
t
)
=
(
u
i
m
,
v
i
m
) be a solution of (29) with φ
m
(
τ
)
=
φ
τ. Let M
≥
I
0
+
q (I
0 is in (H2c)) be a suitable large integer, and set
(53)
x
i
m
=
ξ
(
|
i
|
M
)
u
i
m
,
y
i
m
=
ξ
(
|
i
|
M
)
v
i
m
,
z
m
=
(
x
m
,
y
m
)
=
(
x
i
m
,
y
i
m
)
i
∈
ℤ
.
Taking the inner product (
·
,
·
)
E of (29) with z
m, we have
(54)
1
2
d
d
t
∑
i
∈
ℤ
‍
ξ
(
|
i
|
M
)
∥
φ
i
m
∥
E
2
+
∑
i
∈
ℤ
‍
ξ
(
|
i
|
M
)
(
ε
0
2
∥
φ
i
m
∥
E
2
+
α
2
∥
v
i
m
∥
ρ
2
)
≤
(
F
m
(
φ
m
,
t
)
,
z
m
)
E
.
And we have the following estimates:
(55)
(
f
(
u
m
,
t
)
,
y
m
)
ρ
=
∑
i
∈
ℤ
‍
ξ
(
|
i
|
M
)
ρ
i
f
i
(
u
i
m
,
t
)
v
i
m
≤
1
α
∑
i
∈
ℤ
‍
ξ
(
|
i
|
M
)
ρ
i
β
i
2
(
t
)
+
α
4
∑
i
∈
ℤ
‍
ξ
(
|
i
|
M
)
ρ
i
(
v
i
m
)
2
≤
1
α
∑
|
i
|
≥
M
‍
ρ
i
β
i
2
(
t
)
+
α
4
∑
i
∈
ℤ
‍
ξ
(
|
i
|
M
)
∥
v
i
m
∥
ρ
2
,
(
g
(
t
)
,
y
m
)
ρ
≤
1
α
∑
|
i
|
≥
M
‍
ρ
i
g
i
2
(
t
)
+
α
4
∑
i
∈
ℤ
‍
ξ
(
|
i
|
M
)
∥
v
i
m
∥
ρ
2
,
(
A
m
(
t
)
u
m
,
y
m
)
ρ
≤
1
2
q
~
(
1
λ
_
+
1
)
η
(
t
)
∑
i
∈
ℤ
‍
ξ
(
|
i
|
M
)
∥
φ
i
m
∥
E
2
+
k
1
2
λ
M
_
∥
φ
m
∥
E
2
,
where
(56)
k
1
=
η
0
C
0
[
q
c
0
q
+
(
q
-
1
)
c
0
q
+
⋯
+
c
0
]
.
Therefore,
(57)
d
d
t
∑
i
∈
ℤ
‍
ξ
(
|
i
|
M
)
∥
φ
i
m
∥
E
2
≤
(
-
ε
0
+
q
~
(
1
λ
+
1
)
η
(
t
)
)
∑
i
∈
ℤ
‍
ξ
(
|
i
|
M
)
∥
φ
i
m
∥
E
2
+
2
α
∑
i
≥
M
‍
ρ
i
(
g
i
2
(
t
)
+
β
i
2
(
t
)
)
+
k
1
λ
_
M
∥
φ
m
∥
E
2
.
Applying Gronwall’s inequality to (57) on [
τ
+
T
B
0
,
τ
+
t
]
(
t
≥
T
B
0
), we have that, for φ
τ
∈
B
0,
(58)
∑
i
∈
Z
‍
ξ
(
|
i
|
M
)
∥
φ
i
m
(
τ
+
t
,
τ
;
φ
τ
)
∥
E
2
≤
e
-
ε
0
(
t
-
T
B
0
)
+
∫
τ
+
T
B
0
τ
+
t
‍
q
~
(
(
1
/
λ
_
)
+
1
)
η
(
s
)
d
s
·
∥
φ
m
(
τ
+
T
B
0
,
τ
;
φ
τ
)
∥
E
2
+
∫
τ
+
T
B
0
τ
+
t
‍
e
-
ε
0
(
t
+
τ
-
r
)
+
∫
r
τ
+
t
‍
q
~
(
(
1
/
λ
_
)
+
1
)
η
(
s
)
d
s
×
∑
i
≥
M
‍
2
α
ρ
i
(
g
i
2
(
r
)
+
β
i
2
(
r
)
)
d
r
+
k
1
λ
_
M
∫
τ
+
T
B
0
τ
+
t
‍
e
-
ε
0
(
t
+
τ
-
r
)
+
∫
r
τ
+
t
‍
q
~
(
(
1
/
λ
_
)
+
1
)
η
(
s
)
d
s
aaaaaaaaaawaaaaaai
×
∥
φ
m
(
r
,
τ
;
φ
τ
)
∥
E
2
d
r
.
By (44)-(45), for t
-
T
B
0
≥
T
1, we have
(59)
e
-
ε
0
(
t
-
T
B
0
)
+
∫
τ
+
T
B
0
τ
+
t
‍
q
~
(
(
1
/
λ
_
)
+
1
)
η
(
s
)
d
s
∥
φ
m
(
τ
+
T
B
0
,
τ
,
φ
τ
)
∥
E
2
≤
e
-
k
~
(
t
-
T
B
0
)
∥
φ
m
(
τ
+
T
B
0
,
τ
,
φ
τ
)
∥
E
2
⟶
0
,
as t
→
+
∞. This means that, for any ε
>
0, there exists T
2
(
ε
,
B
0
)
≥
T
B
0 such that when t
≥
T
2
(
ε
,
B
0
),
(60)
e
-
ε
0
(
t
-
T
B
0
)
+
∫
τ
+
T
B
0
τ
+
t
‍
q
~
(
(
1
/
λ
_
)
+
1
)
η
(
s
)
d
s
·
∥
φ
m
(
τ
+
T
B
0
,
τ
,
φ
τ
)
∥
E
2
≤
ε
2
3
.
By (
t
), β
(
t
)
∈
G, and (H2b), there exists I
1
(
ε
)
∈
ℕ such that, for M
>
I
1
(
ε
),
(61)
∫
τ
+
T
B
0
τ
+
t
‍
e
-
ε
0
(
t
+
τ
-
r
)
+
∫
r
τ
+
t
‍
q
~
(
(
1
/
λ
_
)
+
1
)
η
(
s
)
d
s
×
∑
i
≥
M
‍
2
α
ρ
i
(
g
i
2
(
r
)
+
β
i
2
(
r
)
)
d
r
≤
ε
2
3
.
From (47),
(62)
∫
τ
+
T
B
0
τ
+
t
‍
e
-
ε
0
(
t
+
τ
-
r
)
+
∫
r
τ
+
t
‍
q
~
(
(
1
/
λ
_
)
+
1
)
η
(
s
)
d
s
∥
φ
m
(
r
,
τ
;
φ
τ
)
∥
E
2
d
r
≤
(
t
-
T
B
0
)
e
-
ε
0
t
+
∫
0
t
‍
q
~
(
(
1
/
λ
_
)
+
1
)
η
(
s
)
d
s
sup
⁡
φ
τ
∈
B
0
∥
φ
τ
∥
E
2
+
1
2
r
0
2
∫
τ
+
T
B
0
τ
+
t
‍
e
-
ε
0
(
t
+
τ
-
r
)
+
∫
r
τ
+
t
‍
q
~
(
(
1
/
λ
_
)
+
1
)
η
(
s
)
d
s
d
r
.
By (44), we have
(63)
(
t
-
T
B
0
)
e
-
ε
0
t
+
∫
0
t
‍
q
~
(
(
1
/
λ
_
)
+
1
)
η
(
s
)
d
s
⟶
0
as
t
⟶
+
∞
,
∫
τ
+
T
B
0
τ
+
t
‍
e
-
ε
0
(
t
+
τ
-
r
)
+
∫
r
τ
+
t
‍
q
~
(
(
1
/
λ
_
)
+
1
)
η
(
s
)
d
s
∥
φ
m
(
r
,
τ
;
φ
τ
)
∥
E
2
d
r
<
∞
,
a
a
a
a
a
a
a
a
i
t
-
T
B
0
≥
T
1
.
And, for any ε
>
0, there exists I
2
(
ε
,
B
0
)
>
I
0
+
q such that, for M
>
I
2
(
ε
,
B
0
),
(64)
k
1
λ
_
M
∫
τ
+
T
B
0
τ
+
t
‍
e
-
ε
0
(
t
+
τ
-
r
)
+
∫
r
τ
+
t
‍
q
~
(
(
1
/
λ
_
)
+
1
)
η
(
s
)
d
s
∥
φ
m
(
r
,
τ
;
φ
τ
)
∥
E
2
d
r

≤

ε
2
3
.
Let T
(
ε
,
B
0
)
=
max
⁡
{
T
1
,
T
2
(
ε
,
B
0
)
} and I
(
ε
,
B
0
)
=
2
max
⁡
{
I
1
(
ε
)
,
I
2
(
ε
,
B
0
). It follows immediately that
(65)
∑
|
i
|
>
I
(
ε
,
B
0
)
(
λ
i
ρ
i
u
i
2
(
τ
+
t
,
τ
;
u
τ
)
+
ρ
i
v
i
2
(
τ
+
t
,
τ
;
v
τ
)
)
≤
ε
2
,
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
i
t
≥
T
(
ε
,
B
0
)
.In the following, we prove the existence of a pullback exponential attractor for{
U
(
t
,
τ
)
}
t
≥
τ by applying Theorem 2. For any t
∈
ℝ, set
(66)
Y
(
t
)
=
⋃
s
≥
T
B
0
U
(
t
,
t
-
s
)
B
0
¯
⊆
B
0
,
where B
0 is the uniform absorbing set defined in Theorem 5.Theorem 7.
Assume that (P0) and (H1)–(H4) hold.(a)
There exists a continuous positive value functionL
t in t such that, for every τ
∈
ℝ,
(67)
∥
U
(
t
+
τ
,
τ
)
φ
τ
-
U
(
t
+
τ
,
τ
)
ψ
τ
∥
E
≤
L
t
·
∥
φ
τ
-
ψ
τ
∥
E
,
aaa
∀
φ
τ
,
ψ
τ
∈
Y
(
τ
)
.
(b)
There exist positive constantsT
*
>
0, γ
∈
[
0
,
1
/
2
), and a 2
(
2
N
+
1
)-dimensional orthogonal projection P
N
:
E
→
E
N
(
N
∈
ℤ
), such that, for every τ
∈
ℝ and φ
τ
,
ψ
τ
∈
Y
(
τ
),
(68)
∥
(
I
-
P
N
)
(
U
(
τ
+
T
*
,
τ
)
φ
τ
-
U
(
τ
+
T
*
,
τ
)
ψ
τ
)
∥
E
≤
γ
∥
φ
τ
-
ψ
τ
∥
E
.
(c)
For each fixedt
∈
ℝ, lim
⁡
t
→
τ
sup
⁡
φ
τ
∈
Y
(
τ
)
∥
U
(
t
,
τ
)
φ
τ
-
φ
τ
∥
E
=
0.Proof.
For anyτ
∈
ℝ, initial data φ
τ
=
(
u
τ
,
v
τ
), ψ
τ
=
(
x
τ
,
y
τ
). Let φ
(
t
)
=
U
(
t
,
τ
)
φ
τ, ψ
(
t
)
=
U
(
t
,
τ
)
ψ
τ, and ϕ
(
t
)
=
φ
(
t
)
-
ψ
(
t
)
=
(
ϕ
i
(
t
)
)
i
∈
ℤ; then, φ
(
t
), ψ
(
t
), ϕ
(
t
)
∈
C
(
[
τ
,
+
∞
)
,
E
).
(a) For anyt
≥
τ, let φ
m
(
t
), ψ
m
(
t
) be two solutions of (29) with initial data φ
τ, ψ
τ and set ϕ
m
(
t
)
=
φ
m
(
t
)
-
ψ
m
(
t
), and then ϕ
m
(
t
)
∈
C
(
[
τ
,
+
∞
)
,
E
)
∩
C
1
(
(
τ
,
+
∞
)
,
E
) satisfies
(69)
ϕ
˙
m
+
C
(
ϕ
m
)
=
F
m
(
φ
m
,
t
)
-
F
m
(
ψ
m
,
t
)
,
ϕ
m
(
τ
)
=
φ
τ
-
ψ
τ
∈
E
,
t
>
τ
.
Taking the inner product(
·
,
·
)
E of (69) with ϕ
m, we have
(70)
1
2
d
d
t
∥
ϕ
m
∥
E
2
+
(
C
(
ϕ
m
)
,
ϕ
m
)
E
=
(
F
m
(
φ
m
,
t
)
-
F
m
(
ψ
m
,
t
)
,
ϕ
m
)
E
.
Since φ
τ, ψ
τ
∈
Y
(
τ
), by (36) and (66), for t
≥
τ, φ
m
(
t
)
=
(
u
m
(
t
)
,
v
m
(
t
)
), ψ
m
(
t
)
=
(
x
m
(
t
)
,
y
m
(
t
)
)
∈
Y
(
t
)
⊆
B
0, thus ∥
φ
m
(
t
)
∥
E
≤
r
0, ∥
ψ
m
(
t
)
∥
E
≤
r
0, and sup
⁡
i
∈
ℤ
{
ρ
i
|
u
i
m
(
t
)
|
,
ρ
i
|
x
i
m
(
t
)
|
}
≤
r
0
/
λ
_ for t
≥
τ. By (33), we have
(71)
2
(
C
(
ϕ
m
)
,
ϕ
m
)
E
≥
ε
0
∥
ϕ
m
∥
E
2
+
α
∥
v
m
-
y
m
∥
ρ
2
,
and, by (H2)-(H3),
(72)
2
(
A
m
(
t
)
(
u
m
-
x
m
)
,
v
m
-
q
m
)
ρ
≤
q
~
(
1
λ
_
+
1
)
η
(
t
)
∥
ϕ
m
∥
E
2
,
∥
f
(
u
m
,
t
)
-
f
(
p
m
,
t
)
∥
ρ
2
=
∑
i
∈
ℤ
‍
[
f
i
′
(
y
i
m
+
ϑ
i
(
u
i
m
-
y
i
m
)
,
t
)
]
2
·
(
u
i
m
-
y
i
m
)
2
≤
Γ
2
(
r
0
λ
_
,
t
)
∥
u
m
-
y
m
∥
ρ
2
≤
1
λ
_
Γ
2
(
r
0
λ
_
,
t
)
∥
ϕ
m
∥
E
2
,
2
(
F
m
(
φ
m
,
t
)
-
F
m
(
ψ
m
,
t
)
,
ϕ
m
)
E
≤
(
1
α
λ
_
Γ
2
(
r
0
λ
_
,
t
)
+
q
~
(
1
λ
_
+
1
)
η
(
t
)
)
∥
ϕ
m
∥
E
2
+
α
∥
v
m
-
q
m
∥
ρ
2
.
Thus,
(73)
d
d
t
∥
ϕ
m
(
t
)
∥
E
2
≤
(
-
ε
0
+
1
α
λ
_
Γ
2
(
r
0
λ
_
,
t
)
+
q
~
(
1
λ
_
+
1
)
η
(
t
)
)
·
∥
ϕ
m
(
t
)
∥
E
2
.
Applying Gronwall’s inequality to (73) on [
τ
,
τ
+
t
], (
t
>
0
), we have
(74)
∥
ϕ
m
(
t
+
τ
)
∥
E
2
≤
e
-
ε
0
t
+
∫
τ
τ
+
t
‍
(
(
1
/
α
λ
_
)
Γ
2
(
(
r
0
/
λ
_
)
,
s
)
+
q
~
(
(
1
/
λ
_
)
+
1
)
η
(
s
)
)
d
s
·
∥
ϕ
m
(
τ
,
τ
;
ϕ
τ
)
∥
E
2
;
that is,
(75)
∥
ϕ
m
(
τ
+
t
)
∥
E
=
∥
φ
m
(
τ
+
t
)
-
ψ
m
(
τ
+
t
)
∥
E
≤
L
t
·
∥
φ
τ
-
ψ
τ
∥
E
,
∀
t
≥
0
,
where
(76)
L
t
=
e
-
ε
0
t
+
∫
τ
τ
+
t
‍
(
(
1
/
α
λ
_
)
Γ
2
(
(
r
0
/
λ
_
)
,
s
)
+
q
~
(
(
1
/
λ
_
)
+
1
)
η
(
s
)
)
d
s
,
∀
t
≥
0
.
By replacing m with m
k in (75) and letting m
k
→
∞, we have
(77)
∥
φ
(
τ
+
t
,
τ
,
φ
τ
)
-
ψ
(
τ
+
t
,
τ
,
ψ
τ
)
∥
E
≤
L
t
·
∥
φ
τ
-
ψ
τ
∥
E
,
aaaaaaaaai
∀
t
≥
0
.
(b) Fori
∈
ℤ, let ω
i
m
=
ξ
(
|
i
|
/
M
)
ϕ
i
m
=
(
ξ
(
|
i
|
/
M
)
(
u
i
m
-
x
i
m
)
,
ξ
(
|
i
|
/
M
)
(
v
i
m
-
y
i
m
)
) and ω
m
=
(
ω
i
m
)
i
∈
ℤ
∈
E, where ξ is as in (52) and M
>
I
0
+
q. Taking the inner product of (69) with ω
m in E, we have that, for t
≥
τ,
(78)
d
d
t
∑
i
∈
ℤ
‍
ξ
(
|
i
|
M
)
∥
ϕ
i
m
∥
E
2

=

-
2
(
C
(
ϕ
m
)
,
ω
m
)
E
+
2
(
F
m
(
φ
m
,
t
)
-
F
m
(
ψ
m
,
t
)
,
ω
m
)
E
.
For the terms in the right hand of (78), we have
(79)
-
2
(
C
(
ϕ
m
)
,
ω
m
)
E
≤
∑
i
∈
ℤ
‍
ξ
(
|
i
|
M
)
(
-
ε
0
∥
ϕ
i
m
∥
E
2
-
α
∥
v
i
m
-
q
i
m
∥
ρ
2
)
,
2
(
A
m
(
t
)
(
u
m
-
p
m
)
,
y
¯
m
)
ρ
≤
q
~
(
1
λ
_
+
1
)
η
(
t
)
∑
i
∈
ℤ
‍
ξ
(
|
i
|
M
)
∥
ϕ
i
m
∥
E
2
+
k
1
λ
_
M
∥
ϕ
m
∥
E
2
,
where
(80)
∥
ϕ
i
m
∥
E
2
=
ρ
i
(
λ
i
|
u
i
m
|
2
+
|
v
i
m
|
2
)
for
ϕ
i
m
=
(
u
i
m
,
v
i
m
)
.
By the continuity of Γ
(
u
,
t
) and Γ
(
0
,
t
)
=
0 (see (H3c)), there exists δ
~ such that
(81)
Γ
2
(
δ
~
,
t
)
≤
λ
_
α
ε
0
2
,
t
∈
ℝ
.
By Lemma 6 and for δ
~
>
0, there exist I
(
δ
~
)
∈
ℕ, T
2
(
δ
~
,
B
0
)
>
0 such that, for t
≥
T
2
(
δ
~
,
B
0
),
(82)
sup
⁡
i
>
I
(
δ
~
)
{
λ
i
ρ
i
|
u
i
m
(
t
,
τ
;
u
τ
)
|
2
,
λ
i
ρ
i
|
x
i
m
(
t
,
τ
;
p
τ
)
|
2
}
≤
λ
_
δ
~
2
,
implying that, for |
i
|
>
I
(
δ
~
),
(83)
ρ
i
|
u
i
m
+
θ
i
m
(
u
i
m
-
p
i
m
)
|
≤
δ
~
,
for
θ
i
∈
(
0,1
)
,
and thus
(84)
sup
⁡
|
i
|
>
I
(
δ
~
)
|
f
i
′
(
u
i
m
+
θ
i
m
(
p
i
m
-
u
i
m
)
,
t
)
|
2
≤
Γ
2
(
δ
~
,
t
)
≤
λ
_
α
ε
0
2
,
a
a
a
a
a
a
a
a
a
t
∈
ℝ
.
For |
i
|
≥
M
≥
I
(
δ
~
) and t
>
T
2
(
δ
~
,
B
0
)
>
T
1,
(85)
2
(
f
(
u
m
,
t
)
-
f
(
x
m
,
t
)
,
y
¯
m
)
ρ
≤
∑
i
∈
ℤ
‍
ξ
(
|
i
|
M
)
ρ
i
(
1
α
|
f
′
(
u
i
m
+
θ
i
m
(
y
i
m
-
u
i
m
)
)
|
2
w
w
w
w
w
w
w
w
w
w
w
w
×
|
u
i
m
-
y
i
m
|
2
+
α
|
v
i
m
-
q
i
m
|
2
1
α
)
≤
∑
i
∈
ℤ
‍
ξ
(
|
i
|
M
)
1
λ
_
α
|
f
′
(
u
i
m
+
θ
i
m
(
p
i
m
-
u
i
m
)
)
|
2
∥
ϕ
i
m
∥
E
2
+
α
∑
i
∈
ℤ
‍
ρ
i
|
v
i
m
-
y
i
m
|
2
≤
ε
0
2
∑
i
∈
ℤ
‍
ξ
(
|
i
|
M
)
∥
ϕ
i
m
∥
E
2
+
α
∥
v
m
-
y
m
∥
ρ
2
.
Therefore, we obtain that, for all M
≥
max
⁡
{
I
(
δ
~
)
,
I
0
+
q
}, t
>
T
2
(
δ
~
,
B
0
),
(86)
d
d
t
∑
i
∈
ℤ
‍
ξ
(
|
i
|
M
)
∥
ϕ
i
m
∥
E
2
≤
∑
i
∈
ℤ
‍
ξ
(
|
i
|
M
)
(
-
ε
0
2
+
q
~
(
1
λ
_
+
1
)
η
(
t
)
)
∥
ϕ
i
m
∥
E
2
+
k
1
λ
_
M
∥
ϕ
m
(
t
,
τ
;
ϕ
τ
)
∥
E
2
.
Applying Gronwall’s inequality to (86) from τ
+
T
2
(
T
2
=
T
2
(
δ
~
,
B
0
)
) to τ
+
t
(
t
≥
T
2
+
T
1
), we have
(87)
∑
i
∈
ℤ
ξ
(
|
i
|
M
)
∥
ϕ
i
m
(
τ
+
t
)
∥
E
2
≤
e
-
(
ε
0
/
2
)
(
t
-
T
2
)
+
∫
τ
+
T
2
τ
+
t
‍
q
~
(
(
1
/
λ
_
)
+
1
)
η
(
s
)
d
s
∥
ϕ
m
(
τ
+
T
2
,
τ
;
ϕ
τ
)
∥
E
2
+
k
1
λ
_
M
∫
τ
+
T
2
τ
+
t
‍
e
-
(
ε
0
/
2
)
(
τ
+
t
-
r
)
+
∫
r
τ
+
t
‍
q
~
(
(
1
/
λ
_
)
+
1
)
η
(
s
)
d
s
w
w
w
w
w
w
w
w
w
w
×
∥
ϕ
m
(
r
,
τ
;
ϕ
τ
)
∥
E
2
d
r
.
Thus, for t
≥
T
2
+
T
1, we have(88)
e
-
(
ε
0
/
2
)
(
t
-
T
2
)
+
∫
τ
+
T
2
τ
+
t
‍
q
~
(
(
1
/
λ
_
)
+
1
)
η
(
s
)
d
s
∥
ϕ
m
(
τ
+
T
2
,
τ
;
ϕ
τ
)
∥
E
2
≤
e
-
(
ε
0
/
2
)
(
t
-
T
2
)
+
∫
τ
+
T
2
τ
+
t
‍
q
~
(
(
1
/
λ
_
)
+
1
)
η
(
s
)
d
s
-
ε
0
T
2
+
(
1
/
α
λ
_
)
Γ
~
2
(
(
r
0
/
λ
_
)
,
T
2
)
+
q
~
(
(
1
/
λ
_
)
+
1
)
∫
τ
τ
+
T
2
‍
η
(
s
)
d
s
∥
ϕ
m
(
τ
,
τ
;
ϕ
τ
)
∥
E
2
≤
e
-
k
~
t
-
ε
0
T
2
+
(
1
/
α
λ
_
)
Γ
~
2
(
(
r
0
/
λ
_
)
,
T
2
)
+
q
~
(
(
1
/
λ
_
)
+
1
)
η
-
T
2
·
∥
ϕ
m
(
τ
,
τ
;
ϕ
τ
)
∥
E
2
=
k
2
e
-
k
~
t
·
∥
ϕ
m
(
τ
,
τ
;
ϕ
τ
)
∥
E
2
,where(89)
k
2
=
e
-
ε
0
T
2
+
(
1
/
α
λ
_
)
Γ
~
2
(
(
r
0
/
λ
_
)
,
T
2
)
+
q
~
(
(
1
/
λ
_
)
+
1
)
η
-
T
2
<
∞
,
k
1
λ
_
∫
τ
+
T
2
τ
+
t
‍
e
-
(
ε
0
/
2
)
(
τ
+
t
-
r
)
+
∫
r
τ
+
t
‍
q
~
(
(
1
/
λ
_
)
+
1
)
η
(
s
)
d
s
∥
ϕ
m
(
r
,
τ
;
ϕ
τ
)
∥
E
2
d
r
≤
k
1
λ
_
∥
ϕ
m
(
τ
,
τ
;
ϕ
τ
)
∥
E
2
∫
τ
+
T
2
τ
+
t
‍
e
-
(
ε
0
/
2
)
t
+
∫
τ
τ
+
t
‍
q
~
(
(
1
/
λ
_
)
+
1
)
η
(
s
)
d
s
-
(
ε
0
/
2
)
(
r
-
τ
)
+
(
1
/
α
λ
_
)
∫
τ
r
‍
Γ
2
(
(
r
0
/
λ
_
)
,
s
)
d
s
d
r
≤
k
1
λ
_
∥
ϕ
m
(
τ
,
τ
;
ϕ
τ
)
∥
E
2
∫
τ
+
T
2
τ
+
t
‍
e
-
(
ε
0
/
2
)
t
+
q
~
(
(
1
/
λ
_
)
+
1
)
η
-
t
-
(
ε
0
/
2
)
(
r
-
τ
)
+
(
1
/
α
λ
_
)
Γ
~
2
(
(
r
0
/
λ
_
)
,
t
)
d
r
≤
∥
ϕ
m
(
τ
,
τ
;
ϕ
τ
)
∥
E
2
·
k
1
λ
_
2
ε
0
e
-
(
ε
0
/
2
)
t
+
q
~
(
(
1
/
λ
_
)
+
1
)
η
-
t
-
(
ε
0
/
2
)
T
2
+
(
1
/
α
λ
_
)
Γ
~
2
(
(
r
0
/
λ
_
)
,
t
)
=
k
3
(
t
)
·
∥
ϕ
m
(
τ
,
τ
;
ϕ
τ
)
∥
E
2
,where
(90)
k
3
(
t
)
=
k
1
λ
_
2
ε
0
e
-
(
ε
0
/
2
)
t
+
q
~
(
(
1
/
λ
_
)
+
1
)
η
-
t
-
(
ε
0
/
2
)
T
2
+
(
1
/
α
λ
_
)
Γ
~
2
(
(
r
0
/
λ
_
)
,
t
)
.
Thus, for M
≥
max
⁡
{
I
(
δ
~
)
,
I
0
+
q
}, t
≥
T
2
(
δ
~
,
B
0
)
+
T
1, we have
(91)
∑
|
i
|
>
2
M
‍
∥
ϕ
i
m
(
t
+
τ
)
∥
E
2
≤
∑
i
∈
ℤ
‍
ξ
(
|
i
|
M
)
∥
ϕ
i
m
(
t
+
τ
)
∥
E
2
≤
(
k
2
e
-
k
~
t
+
1
M
k
3
(
t
)
)
·
∥
ϕ
m
(
τ
,
τ
;
ϕ
τ
)
∥
E
2
.
Taking
(92)
T
*
=
max
⁡
{
T
2
(
δ
~
,
B
0
)
+
T
1
,
ln
⁡
(
8
k
2
)
k
2
}
,
N
=
{
2
I
(
δ
~
)
+
1,2
I
0
+
2
q
,
8
k
3
(
T
*
)
}
,
then
(93)
γ
=
k
2
e
-
k
~
T
*
+
1
N
k
3
(
T
*
)
<
1
2
.
Thus, by (91), we have
(94)
∥
(
I
-
P
N
)
(
U
(
τ
+
T
*
,
τ
)
φ
τ
-
U
(
τ
+
T
*
,
τ
)
ψ
τ
)
∥
E
≤
γ
∥
φ
τ
-
ψ
τ
∥
E
.
(c) It follows from the following estimate:(95)
∥
U
(
t
,
τ
)
φ
τ
-
φ
τ
∥
E
2
≤
(
t
-
τ
)
(
∫
τ
t
‍
∥
F
(
φ
(
s
)
,
s
)
-
C
(
φ
(
s
)
)
∥
E
2
d
s
)
1
/
2
≤
(
t
-
τ
)
∫
τ
t
‍
[
k
4
+
3
(
2
q
+
1
)
2
r
0
2
η
2
(
s
)
+
r
0
2
Γ
2
(
r
0
λ
_
,
s
)
]
d
s
≤
(
t
-
τ
)
[
k
4
(
t
-
τ
)
+
3
(
2
q
+
1
)
2
r
0
2
∫
τ
t
‍
η
2
(
s
)
d
s
w
w
w
w
w
w
w
w
i
+
r
0
2
∫
τ
t
‍
Γ
2
(
r
0
λ
_
,
s
)
d
s
]
≤
(
t
-
τ
)
[
(
r
0
λ
_
,
t
-
τ
)
k
4
(
t
-
τ
)
+
3
(
2
q
+
1
)
2
r
0
2
Q
(
t
-
τ
)
wwwwwwww
+
r
0
2
Γ
~
(
r
0
λ
_
,
t
-
τ
)
]
.
and the continuity of Q
(
·
), Γ
~
(
·
,
·
).As a direct consequence of Theorem2, Theorem 4, Theorem 5, Theorem 7, and Theorem  3.1 of [15], we have the following result.Theorem 8.
Assume that (P0) and (H1)–(H4) hold. The process{
U
(
t
,
τ
)
}
t
≥
τ associated with (19) possesses a pullback exponential attractor {
𝒜
(
t
)
}
t
∈
ℝ with the following properties: (i)
for anyt
∈
ℝ, 𝒜
(
t
)
⊆
Y
(
t
)
⊆
B
0;
(ii)
dim
⁡
f
𝒜
(
t
)
≤
ln
⁡
N
θ
*
/
-
ln
⁡
a
θ
*;
(iii)
and for any bounded setB
⊂
E, d
h
(
U
(
t
,
τ
)
Y
(
τ
), 𝒜
(
t
)
)
≤
(
L
(
T
*
)
r
0
e
α
*
T
B
/
(
a
θ
*
)
2
)
e
-
α
*
(
t
-
τ
), -
∞
<
τ
+
T
B
<
t
≤
+
∞;
(iv)
lim
⁡
t
→
τ
d
h
(
𝒜
(
t
)
,
𝒜
(
τ
)
)
=
0;where
(96)
a
θ
*
=
2
(
γ
+
θ
L
T
*
)
,
θ
=
1
-
2
γ
4
L
T
*
,
α
*
=
-
ln
⁡
a
θ
*
T
*
,
and N
θ
* is the minimal number of closed balls of E with radius θ covering the closed unit ball B
N
(
0,1
) of E
N centered at 0.Remark 9.
It should be pointed out that, for some special cases, (H2) and (27) can be removed. For example, if the coupled linear operator A
(
t
)
≡
A in (10) is a constant operator A satisfying -
A
=
D
*
D
=
D
D
*, where (
D
u
)
i
=
∑
j
=
-
m
0
j
=
m
0
‍
d
j
u
i
+
j, for all u
=
(
u
i
)
i
∈
ℤ
,
|
d
j
|
⩽
a
~
(constant),-
m
0
⩽
j
⩽
m
0, and D
* is the adjoint of D. A simple example of such an operator A is (
A
u
)
i
=
u
i
-
1
+
u
i
+
1
-
2
u
i and (
D
u
)
i
=
u
i
+
1
-
u
i. Choose a positive weight function ρ
:
ℤ
→
ℝ
+ satisfying the properties (P0) and |
ρ
(
i
±
1
)
-
ρ
(
i
)
|
≤
b
ρ
(
i
), for all i
∈
ℤ for some small b
≥
0. Two examples of such weight functions are ρ
(
i
)
=
1
/
(
1
+
κ
2
i
2
)
q, q
>
1
/
2, and ρ
(
i
)
=
e
-
κ
|
i
|, i
∈
ℤ, with κ
>
0 being a parameter. If the number b satisfies
(97)
0
≤
b
<
min
⁡
{
ε
0
2
a
~
c
2
m
0
,
α
4
a
~
c
2
m
0
2
(
2
m
0
+
1
)
2
}
,
where c
2
m
0
=
c
0
m
0
-
1
+
c
0
m
0
-
2
+
⋯
+
c
0
+
1, then conditions (H2) and (27) can be removed.



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*Source: 237027-2014-04-03.xml*