In this section, we will prove the existence of the strong compact uniform attractor of problem (1)~(4) applying Ball et al.’s idea (see [19, 22]). Firstly, we construct a bounded uniformly absorbing set. Next, we show the weak uniform attractor of the system. Lastly, we derive that the weak uniform attractor is actually the strong one.
Proof.
We prove this theorem by three steps.
Step 1.
{
U
σ
∈
∑
(
t
,
τ
)
}
possess a bounded uniformly absorbing set in
E
0
.
Let
B
0
=
{
W
∈
E
0
∥
W
∥
E
0
2
⩽
C
(
α
,
δ
,
β
,
∥
Y
0
∥
L
c
2
(
R
;
∑
0
)
,
∥
f
0
t
∥
L
2
(
R
;
H
α
)
,
∥
W
τ
∥
E
0
)
}
. By Theorem 17,
B
0
is a bounded absorbing set of the process
U
σ
=
Y
0
.
By Assumption 10, we know that, for each
Y
∈
∑
,
∥
Y
∥
L
b
2
(
R
;
∑
0
)
2
⩽
∥
Y
0
∥
L
b
2
(
R
;
∑
0
)
2
holds. So the solution of (1)
~
(4) satisfies
(85)
∥
W
∥
E
0
⩽
C
(
α
,
δ
,
β
,
∥
f
0
t
∥
L
2
(
R
;
H
α
)
,
∥
Y
∥
L
b
2
(
R
;
∑
0
)
,
∥
W
τ
∥
E
0
)
⩽
C
(
α
,
δ
,
β
,
∥
f
0
t
∥
L
2
(
R
;
H
α
)
,
∥
Y
0
∥
L
b
2
(
R
;
∑
0
)
,
∥
W
τ
∥
E
0
)
.
Then we can get that the set
B
0
is a bounded uniformly absorbing set of
{
U
σ
∈
∑
(
t
,
τ
)
}
.
Step 2. we prove the existence of weakly compact uniform attractor
A
∑
in
E
0
.
From Lemma 6, Theorem 17, and Step 1, we only need to prove that
{
U
σ
∈
∑
(
t
,
τ
)
}
is
(
E
0
×
∑
,
E
0
)
-continuous. We denote weak convergence by
⇀
and
*
weak convergence by
⇀
*
.
For any fixed
t
1
⩾
τ
∈
R
, let
(86)
(
W
τ
k
,
σ
k
)
⇀
(
W
τ
,
σ
)
in
E
0
×
∑
.
If we can deduce that
(87)
W
σ
k
(
t
1
)
⇀
W
σ
(
t
1
)
in
E
0
,
where
W
σ
k
(
t
1
)
=
(
u
k
(
t
1
)
,
n
k
(
t
1
)
)
=
U
σ
k
(
t
1
,
τ
)
W
τ
k
,
W
σ
(
t
1
)
=
(
u
(
t
1
)
,
n
(
t
1
)
)
=
U
σ
(
t
1
,
τ
)
W
τ
, we will obtain that
{
U
σ
∈
∑
(
t
,
τ
)
}
is
(
E
0
×
∑
,
E
0
)
-continuous. By (86) and Theorem 17, we can get that
(88)
∥
W
τ
k
∥
E
0
⩽
C
,
(89)
sup
t
∈
[
τ
,
T
]
∥
W
σ
k
(
t
)
∥
E
0
⩽
C
.
Then by Lemmas 12
~
16, we can see that
(90)
∥
W
σ
k
(
t
)
∥
∞
⩽
C
,
∀
0
⩽
t
⩽
T
.
Note that
(91)
i
u
k
t
=
(
-
Δ
)
α
u
k
+
n
k
u
k
-
i
δ
u
k
+
f
k
(
x
,
t
)
,
(92)
n
k
t
=
-
|
u
k
|
x
2
-
β
n
k
+
g
k
(
x
,
t
)
,
and
σ
k
=
(
f
k
(
x
,
t
)
,
g
k
(
x
,
t
)
)
∈
∑
. By (89) and (90), we find that
∂
t
W
σ
k
(
t
)
∈
L
∞
(
τ
,
T
;
L
2
(
Ω
)
×
H
1
(
Ω
)
)
and
(93)
∥
∂
t
W
σ
k
(
t
)
∥
L
∞
(
τ
,
T
;
L
2
(
Ω
)
×
H
1
(
Ω
)
)
⩽
C
.
Because of Theorem 17 and (93), we easily see that there exist a subsequence
{
W
σ
k
l
(
t
)
}
of
{
W
σ
k
(
t
)
}
and
W
~
(
t
)
≜
(
u
~
(
t
)
,
n
~
(
t
)
)
∈
L
∞
(
τ
,
T
;
E
0
)
, such that
(94)
W
σ
k
l
(
t
)
⇀
*
W
~
(
t
)
in
L
∞
(
τ
,
T
;
E
0
)
,
(95)
∂
t
W
σ
k
l
(
t
)
⇀
*
∂
t
W
~
(
t
)
in
L
∞
(
τ
,
T
;
L
2
(
Ω
)
×
H
1
(
Ω
)
)
.
Besides, for any
t
1
∈
[
τ
,
T
]
, by (89) there exists
W
0
≜
(
u
0
(
t
1
)
,
n
0
(
t
1
)
)
∈
E
0
such that
(96)
W
σ
k
(
t
1
)
⇀
W
0
in
E
0
.
By (94) and compactness embedding theorem, we can get that
(97)
u
k
l
(
t
)
⟶
u
~
(
t
)
in
L
2
(
τ
,
T
;
H
α
)
.
Next, we will obtain that
W
~
(
t
)
is a solution of problem (1)
~
(4).
For
all
v
∈
L
2
(
Ω
)
,
∀
ψ
∈
C
0
∞
(
τ
,
T
)
, by (91) we have that
(98)
∫
τ
T
(
i
u
k
l
t
,
ψ
(
t
)
v
)
d
t
-
∫
τ
T
(
(
-
Δ
)
α
u
k
l
,
ψ
(
t
)
v
)
d
t
-
∫
τ
T
(
n
k
l
u
k
l
,
ψ
(
t
)
v
)
d
t
+
∫
τ
T
(
i
δ
u
k
l
,
ψ
(
t
)
v
)
d
t
-
∫
τ
T
(
f
(
x
,
t
)
,
ψ
(
t
)
v
)
d
t
=
0
.
Since
(99)
∫
τ
T
(
n
k
l
u
k
l
,
ψ
(
t
)
v
)
d
t
-
∫
τ
T
(
n
~
u
~
,
ψ
(
t
)
v
)
d
t
=
∫
τ
T
(
(
u
k
l
-
u
~
)
n
k
l
,
ψ
(
t
)
v
)
d
t
+
∫
τ
T
(
u
~
(
n
k
l
-
n
~
)
,
ψ
(
t
)
v
)
d
t
,
by (90), (94), and (97),
(100)
∫
τ
T
(
(
u
k
l
-
u
~
)
n
k
l
,
ψ
(
t
)
v
)
d
t
⩽
sup
0
⩽
t
⩽
T
∥
n
k
l
(
t
)
∥
L
∞
∥
ψ
(
t
)
v
∥
L
2
(
0
,
T
;
L
2
(
Ω
)
)
111111
×
∥
u
k
l
-
u
~
∥
L
2
(
0
,
T
;
L
2
(
Ω
)
)
⟶
0
,
∫
τ
T
(
u
~
(
n
k
l
-
n
~
)
,
ψ
(
t
)
v
)
d
t
=
∫
τ
T
(
(
n
k
l
-
n
~
)
,
ψ
(
t
)
v
u
~
¯
)
d
t
⟶
0
.
Then we have
(101)
∫
τ
T
(
n
k
l
u
k
l
,
ψ
(
t
)
v
)
d
t
⟶
∫
τ
T
(
n
~
u
~
,
v
)
ψ
(
t
)
d
t
.
And by (94), we have that
(102)
∫
τ
T
(
(
-
Δ
)
α
u
k
l
,
ψ
(
t
)
v
)
d
t
-
∫
τ
T
(
(
-
Δ
)
α
u
~
,
ψ
(
t
)
v
)
d
t
⩽
∥
(
-
Δ
)
α
(
u
k
l
-
u
~
)
∥
L
2
(
τ
,
T
;
L
2
(
Ω
)
)
×
∥
ψ
(
t
)
v
∥
L
2
(
τ
,
T
;
L
2
(
Ω
)
)
⟶
0
.
By using the similar methods to the other terms of (98), we have
(103)
∫
τ
T
(
i
u
~
t
,
v
)
ψ
(
t
)
d
t
-
∫
τ
T
(
(
-
Δ
)
α
u
~
,
v
)
ψ
(
t
)
d
t
-
∫
τ
T
(
n
~
u
~
,
v
)
ψ
(
t
)
d
t
+
∫
τ
T
(
i
δ
u
~
,
v
)
ψ
(
t
)
d
t
-
∫
τ
T
(
f
(
x
,
t
)
,
v
)
ψ
(
t
)
d
t
=
0
.
So, we can get that
(104)
i
u
~
t
-
(
-
Δ
)
α
u
~
-
u
~
n
~
+
i
δ
u
~
=
f
(
x
,
t
)
,
which shows that (
u
~
,
n
~
)
satisfies (1).
For any
v
∈
L
2
(
Ω
)
,
∀
ψ
∈
C
0
∞
(
τ
,
T
)
with
ψ
(
T
)
=
0
,
ψ
(
τ
)
=
1
, by (91) we find that
(105)
-
∫
τ
T
(
i
u
k
l
,
v
)
ψ
′
(
t
)
d
t
-
∫
τ
T
(
(
-
Δ
)
α
u
k
l
,
v
)
ψ
(
t
)
d
t
-
∫
τ
T
(
n
k
l
u
k
l
,
v
)
ψ
(
t
)
d
t
+
∫
τ
T
(
i
δ
u
k
l
,
v
)
ψ
(
t
)
d
t
-
∫
τ
T
(
f
(
x
,
t
)
,
v
)
ψ
(
t
)
d
t
=
i
(
u
k
l
(
τ
)
,
v
)
.
We know that Assumption (86) implies that
(106)
u
k
l
(
τ
)
=
u
τ
k
l
⇀
u
τ
in
H
2
α
.
Then from (105) and (106), we have
(107)
-
∫
τ
T
(
i
u
~
,
v
)
ψ
′
(
t
)
d
t
-
∫
τ
T
(
(
-
Δ
)
α
u
~
,
v
)
ψ
(
t
)
d
t
-
∫
τ
T
(
n
~
u
~
,
v
)
ψ
(
t
)
d
t
+
∫
τ
T
(
i
δ
u
~
,
v
)
ψ
(
t
)
d
t
-
∫
τ
T
(
f
(
x
,
t
)
,
v
)
ψ
(
t
)
d
t
=
i
(
u
τ
,
v
)
,
while by (104) we know that
(108)
-
∫
τ
T
(
i
u
~
,
v
)
ψ
′
(
t
)
d
t
-
∫
τ
T
(
(
-
Δ
)
2
u
~
,
v
)
ψ
(
t
)
d
t
-
∫
τ
T
(
n
~
u
~
,
v
)
ψ
(
t
)
d
t
+
∫
τ
T
(
i
δ
u
~
,
v
)
ψ
(
t
)
d
t
-
∫
τ
T
(
f
(
x
,
t
)
,
v
)
ψ
(
t
)
d
t
=
i
(
u
~
(
τ
)
,
v
)
.
So by (107) and (108), we have that
(109)
(
u
τ
,
v
)
=
(
u
~
(
τ
)
,
v
)
,
∀
v
∈
L
2
(
Ω
)
,
(110)
u
~
(
τ
)
=
u
τ
.
By (104) and (110), we have
(111)
u
~
(
t
)
=
u
(
t
)
.
For any
v
∈
L
2
(
Ω
)
,
∀
ψ
∈
C
0
∞
(
τ
,
t
1
)
, with
ψ
(
τ
)
=
0
,
ψ
(
t
1
)
=
1
, then we repeat the procedure of proofs of (105)
~
(108) by (96) having
(112)
u
0
(
t
1
)
=
u
~
(
t
1
)
.
From (96), (111), and (112), we have that
(113)
u
k
(
t
1
)
⇀
u
(
t
1
)
in
H
2
α
(
Ω
)
.
Similarly, we can also derive that
(114)
n
k
(
t
1
)
⇀
n
(
t
1
)
in
H
1
(
Ω
)
.
From (113) and (114), we deduce (87). We complete the proof of the step.
Step 3. We show the weakly compact uniform attractor
A
∑
is actually the strong one.
From the proof of Lemma 16, we know each solution trajectory for problem (1)–(4) satisfies
(115)
d
d
t
(
∥
u
x
x
∥
2
+
F
)
+
2
δ
(
∥
u
x
x
∥
2
+
F
)
=
G
,
(116)
d
d
t
∥
n
x
∥
2
+
2
β
∥
n
x
∥
2
=
G
1
,
where
(117)
F
=
2
Re
∫
n
u
(
-
Δ
)
α
u
¯
d
x
+
2
Re
∫
f
(
x
,
t
)
(
-
Δ
)
α
u
¯
d
x
,
G
=
2
Re
∫
u
g
(
x
,
t
)
(
-
Δ
)
α
u
¯
d
x
-
2
Re
∫
β
u
n
(
-
Δ
)
α
u
¯
d
x
-
2
Re
∫
u
2
u
¯
x
(
-
Δ
)
α
u
¯
d
x
-
2
Re
∫
|
u
|
2
u
x
(
-
Δ
)
α
u
¯
d
x
-
2
Re
∫
i
n
f
(
x
,
t
)
(
-
Δ
)
α
u
¯
d
x
-
2
Re
∫
i
n
2
u
(
-
Δ
)
α
u
¯
d
x
+
2
Re
∫
f
t
(
x
,
t
)
(
-
Δ
)
α
u
¯
d
x
+
2
Re
∫
δ
f
(
x
,
t
)
(
-
Δ
)
α
u
¯
d
x
,
G
1
=
-
2
∫
g
(
x
,
t
)
n
x
x
d
x
+
2
∫
|
u
|
x
2
n
x
x
d
x
G
1
=
2
∫
g
x
(
x
,
t
)
n
x
d
x
-
2
Re
∫
u
u
¯
x
x
n
x
d
x
+
2
∫
(
u
x
x
u
¯
x
+
u
x
u
¯
x
x
)
n
d
x
.
By the uniform boundedness and the compactness embedding, we have that
F
,
G
, and
G
1
are all weakly continuous in
E
0
×
Σ
.
From Step 2, we can see that the point
(
w
,
m
)
∈
A
Σ
if and only if there exist two sequences
{
w
k
0
,
m
k
0
}
k
=
1
∞
and
{
t
k
}
k
=
1
∞
such that for all
σ
(
t
)
∈
Σ
, it uniformly satisfies that
(118)
U
σ
(
t
k
,
τ
)
(
w
k
0
,
m
k
0
)
⇀
(
w
,
m
)
in
E
0
,
k
⟶
∞
,
where
t
k
→
∞
as
k
→
∞
. If the weak convergence implies strong one, we obtain
A
∑
is actually the strong compact attractor. For each fixed
h
>
τ
, because of
t
k
→
∞
, we consider it as
h
<
t
k
,
k
∈
N
+
. By Lemma 16 and Theorem 17,
U
σ
(
t
k
-
h
,
τ
)
(
w
k
0
,
m
k
0
)
is bounded in
E
0
. Then there exists a subsequence
U
σ
(
t
k
l
-
h
,
τ
)
(
w
k
l
0
,
m
k
l
0
)
of
U
σ
(
t
k
-
h
,
τ
)
(
w
k
0
,
m
k
0
)
and a point
(
v
,
p
)
∈
E
0
, such that
(119)
U
σ
(
t
k
l
-
h
,
τ
)
(
w
k
l
0
,
m
k
l
0
)
⇀
(
v
,
p
)
in
E
0
.
Let
(120)
(
w
k
l
(
t
)
,
m
k
l
(
t
)
)
=
U
T
(
t
k
l
-
h
-
τ
)
σ
(
t
+
τ
,
τ
)
U
σ
(
t
k
l
-
h
,
τ
)
(
w
k
l
0
,
m
k
l
0
)
=
U
σ
(
t
+
t
k
l
-
h
,
t
k
l
-
h
)
U
σ
(
t
k
l
-
h
,
τ
)
(
w
k
l
0
,
m
k
l
0
)
=
U
σ
(
t
+
t
k
l
-
h
,
τ
)
(
w
k
l
0
,
m
k
l
0
)
,
where
T
(
·
)
is the translation operator on
Σ
. Since
σ
(
t
)
is translation compact symbol, there exists a symbol
σ
*
∈
Σ
such that
(121)
T
(
t
k
l
-
h
-
τ
)
σ
⟶
σ
*
in
Σ
.
Then by (118), (119), and the weak
(
E
×
Σ
)
-continuity of
U
σ
∈
Σ
(
t
,
τ
)
, we can get that
(122)
(
w
k
l
(
t
)
,
m
k
l
(
t
)
)
⇀
U
σ
*
(
t
,
τ
)
(
v
,
p
)
=
(
w
,
m
)
in
E
0
,
∀
t
>
τ
.
From (119), we can see that the solution trajectory
(
w
k
l
(
t
)
,
m
k
l
(
t
)
)
is created by
U
T
(
t
k
l
-
h
-
τ
)
σ
(
t
+
τ
,
τ
)
starting at
U
σ
(
t
k
l
-
h
,
τ
)
(
w
k
l
0
,
m
k
l
0
)
. By (115), (119), and (122), we have that
(123)
∥
w
k
l
(
t
)
∥
H
per
2
α
2
+
F
(
w
k
l
(
t
)
,
m
k
l
(
t
)
)
=
e
-
2
δ
(
t
-
τ
)
(
∥
U
σ
(
t
k
l
-
h
,
τ
)
w
k
l
0
∥
H
per
2
α
2
∥
U
σ
(
t
k
l
-
h
,
τ
)
w
k
l
0
∥
H
p
e
r
2
α
2
+
F
(
U
σ
(
t
k
l
-
h
,
τ
)
w
k
l
0
,
U
σ
(
t
k
l
-
h
,
τ
)
m
k
l
0
)
)
+
∫
τ
t
e
-
2
δ
(
t
-
τ
)
G
(
w
k
l
(
s
)
,
m
k
l
(
s
)
)
d
s
=
e
-
2
δ
(
t
-
τ
)
(
∥
U
σ
(
t
k
l
-
h
,
τ
)
w
k
l
0
∥
H
per
2
α
2
+
F
(
v
,
p
)
)
+
∫
τ
t
e
-
2
δ
(
t
-
τ
)
G
(
U
σ
*
(
s
+
τ
,
τ
)
(
v
,
p
)
)
d
s
.
Let
t
=
h
in (122). Since
F
and
G
are weakly continuous in
E
0
,
∥
U
σ
(
t
k
l
-
h
,
τ
)
(
w
k
l
0
,
m
k
l
0
)
∥
H
per
2
α
2
⩽
C
, and the Lebesgue dominated convergence theorem, we can obtain that
(124)
limsup
k
l
→
∞
∥
U
σ
(
t
k
l
,
τ
)
w
k
l
0
∥
H
per
2
α
2
+
F
(
U
σ
*
(
h
+
τ
,
τ
)
(
v
,
p
)
)
⩽
e
-
2
δ
(
h
-
τ
)
(
C
+
F
(
v
,
p
)
)
+
∫
τ
h
e
-
2
δ
(
h
-
τ
)
G
(
U
σ
*
(
s
+
τ
,
τ
)
(
v
,
p
)
)
d
s
.
Since
(
w
,
m
)
=
U
σ
*
(
S
+
τ
,
τ
)
(
v
,
p
)
, we can see the solution
(
w
,
m
)
as at
h
corresponding to the initial data
(
v
,
p
)
and the symbol
σ
*
. Similarly to (122), we have
(125)
∥
w
∥
H
per
2
α
2
+
F
(
w
,
m
)
=
e
-
2
δ
(
h
-
t
)
(
∥
v
∥
H
per
2
α
2
+
F
(
v
,
p
)
)
+
∫
τ
h
e
-
2
δ
(
h
-
t
)
G
(
U
σ
*
(
s
+
τ
,
τ
)
(
v
,
p
)
)
d
s
.
Deducting (125) from (124), we can get that
(126)
limsup
k
l
→
∞
∥
U
σ
(
t
k
l
,
τ
)
w
k
l
0
∥
H
per
2
α
2
⩽
∥
w
∥
H
per
2
α
2
+
C
e
-
2
δ
(
h
-
t
)
-
e
-
2
δ
(
h
-
t
)
∥
v
∥
H
per
2
α
2
⩽
∥
w
∥
H
per
2
α
2
+
C
e
-
2
δ
(
h
-
t
)
.
As
h
→
∞
, we can get that
(127)
limsup
k
l
→
∞
∥
U
σ
(
t
k
l
,
τ
)
w
k
l
0
∥
H
per
2
α
2
⩽
∥
w
∥
H
per
2
α
2
.
On the other hand, the weak convergence
U
σ
(
t
k
,
τ
)
w
k
0
⇀
w
implies that
(128)
liminf
k
→
∞
∥
U
σ
(
t
k
,
τ
)
w
k
0
∥
H
per
2
α
2
⩾
∥
w
∥
H
per
2
α
2
.
From the above two inequalities, we get that
(129)
lim
k
→
∞
∥
U
σ
(
t
k
,
τ
)
w
k
0
∥
H
per
2
α
2
=
∥
w
∥
H
per
2
α
2
.
Similarly to the above arguments, by using (116) we can derive that
(130)
lim
k
→
∞
∥
U
σ
(
t
k
,
τ
)
m
k
0
∥
H
per
1
2
=
∥
m
∥
H
per
1
2
.
Then we get that
U
σ
(
t
k
,
τ
)
(
w
k
0
,
m
k
0
)
→
(
w
,
m
)
in
E
0
. We complete the proof of the theorem.