3.1. The autonomous case
Consider the case where f and g do not depend on the time t, and let us recall the definition of multi-valued semiflows.
Definition 3.1. 5 Let E be a Banach space. The mapping
G
:
[
0
,
+
∞
)
×
E
→
2
E
is called a multi-valued semiflow if the following conditions are satisfied:
(1) G
(
0
,
w
)
=
w
for arbitrary w ∈ E;
(2)
G
(
t
1
+
t
2
,
w
)
⊂
G
(
t
1
,
G
(
t
2
,
w
)
)
for all w ∈ E, t1, t2 ∈ ℝ+, where G (t, B) = ∪x∈B G (t, x), B ⊂ E.
It is called a strict multi-valued semiflow if
G
(
t
1
+
t
2
,
w
)
=
G
(
t
1
,
G
(
t
2
,
w
)
)
, for all w ∈ E, t1, t2 ∈ ℝ+.
We now consider problem (1.1) with τ = 0. By Theorem 2.1, we construct a multi-valued mapping as follows
G
(
t
,
u
0
)
=
{
u
(
t
)
|
u
(
⋅
)
is
a
global
weak
solution
of
(1 .1)
such
that
u
(
0
)
=
u
0
}
.
Lemma 3.1. ![]()
is a strict multi-valued semiflow in the sense of Definition 3.1.
Proof. Assume that ξ
∈
G
(
t
1
+
t
2
,
u
0
)
, then ξ = u(t1 + t2), where u(t) is a solution of (1.1). Denoting v (t) = u(t + t2), we see that v(.) is also in the set of solutions of (1.1) with respect to initial condition v(0) = u(t2). Therefore, ξ
=
v
(
t
1
)
∈
G
(
t
1
,
u
(
t
2
)
)
⊂
G
(
t
2
,
G
(
t
2
,
u
0
)
)
. It remains to show that
G
(
t
1
,
G
(
t
2
,
u
0
)
)
⊂
G
(
t
1
+
t
2
,
u
0
)
. If ξ
∈
G
(
t
1
,
G
(
t
2
,
u
0
)
)
then ξ = v(t1), where v
(
0
)
∈
G
(
t
2
,
u
0
)
. One can suppose that v(0) = u(t2), where u(0) = u0. Set
w
(
τ
)
=
u
(
τ
)
,
0
≤
τ
<
t
2
,
v
(
τ
-
t
2
)
,
τ
≥
t
2
.
Since u and v are two solutions of (1.1), we obtain that w is a solution of (1.1) with w(0) = u(0) = u0. In addition, since ξ = v(t1) = w(t1 + t2), we have ξ∈G(t1+t2,u0).
Definition 3.2. 5 A set ![]()
is said to be a global attractor of the multi-valued semiflow ![]()
if the following conditions hold:
• ![]()
is an attracting, i.e., d
i
s
t
(
G
(
t
,
B
)
,
A
)
→
0
as t → ∞ for all bounded subsets B ⊂ E,
• ![]()
is negatively semi-invariant: A
⊂
G
(
t
,
A
)
for arbitrary t ≥ 0,
• If ℬ is an attracting of ![]()
then A
⊂
ℬ
̄
,
where d
i
s
t
(
C
,
A
)
=
sup
c
∈
C
inf
a
∈
A
c
-
a
is the Hausdorff semi-distance.
The following theorem gives the sufficient conditions for the existence of a global attractor for the multi-valued semiflow ![]()
.
Theorem 3.2. 57 Suppose that the strict multi-valued semiflow ![]()
has the following properties:
(1) ![]()
is pointwise dissipative, i.e., there exists K > 0 such that for
u
0
∈
E
,
u
(
t
)
∈
G
(
t
,
u
0
)
one has ∥u(t)∥E ≤ K if t ≥ t0 (∥u0∥E);
(2) G
(
t
,
.
)
is a closed map for any t ≥ 0, i.e., if ξn → ξ, ηn → η,
ξ
n
∈
G
(
t
,
η
n
)
then ξ
∈
G
(
t
,
η
)
;
(3) ![]()
is asymptotically upper semicompact, i.e., if B is a bounded set in E such that for some T
(
B
)
,
γ
T
(
B
)
+
(
B
)
:
=
⋃
t
≥
T
(
B
)
G
(
t
,
B
)
is bounded, any sequence
ξ
n
∈
G
(
t
n
,
B
)
with tn → ∞ is precompact in E.
Then ![]()
has a compact global attractor ![]()
in E. Moreover, ![]()
is invariant, i.e.,
G
(
t
,
A
)
=
A
for any t ≥ 0.
Lemma 3.3. G
(
t
*
,
.
)
is a compact mapping for each t* ∈ (0, T].
Proof. This lemma is a direct consequence of Lemma 3.8 in Section 3.2 below.
We now can prove the existence of a global attractor.
Theorem 3.4. Under conditions (H1)-(H3), where f andg are assumed to be independent of time t, the strict multi-valued semiflow ![]()
generated by problem (1.1) has an invariant compact global attractor in L2(Ω).
Proof. We will check hypotheses (1)-(3) of Theorem 3.2. First, assume u
(
t
)
∈
G
(
t
,
u
0
)
, we have
1
2
d
d
t
u
(
t
)
L
2
(
Ω
)
2
+
∫
Ω
x
-
p
γ
∇
u
p
+
C
2
u
L
q
(
Ω
)
q
≤
k
2
Ω
+
∫
Ω
u
g
d
x
≤
k
2
Ω
+
ε
u
L
2
(
Ω
)
2
+
C
ε
g
L
2
(
Ω
)
2
.
Noting that
C
2
u
L
q
(
Ω
)
q
≥
λ
u
L
2
(
Ω
)
2
-
C
,
C
=
C
q
,
Ω
>
0
,
we have
1
2
d
d
t
u
L
2
(
Ω
)
2
+
λ
u
L
2
(
Ω
)
2
≤
C
q
,
Ω
,
g
L
2
(
Ω
)
.
Therefore
u
(
t
)
L
2
(
Ω
)
2
≤
u
(
0
)
L
2
(
Ω
)
2
e
-
2
λ
t
+
C
q
,
Ω
,
g
L
2
(
Ω
)
,
Hence one can deduce that ![]()
is pointwise dissipative.
We now check hypothesis (2) of Theorem 3.2. Assume that
ξ
n
∈
G
(
t
,
η
n
)
,
ξ
n
→
ξ
,
η
n
→
η
in L2(Ω). Then there exists a sequence {un} such that
u
n
(
t
)
=
ξ
n
,
u
n
(
0
)
=
η
n
.
Using the same arguments as in the proof of Theorem 2.1, we have
• un → u in L2(Q0,T),
• un(t) ⇀ u(t) in L2(Ω) for arbitrary t ∈ [0, T] (and then u(0) = η),
• f(un)⇀ f(u) in
L
q
′
(
Q
0
,
T
)
,
•
d
u
n
d
t
⇀
d
u
d
t
in V,
• -Δp,γ un ⇀ -Δp,γ u in
L
p
′
0
,
T
;
D
-
γ
-
1
,
p
′
(
Ω
)
,
up to a subsequence. Hence, passing to the limit in the equality
∫
0
T
u
′
n
,
v
+
∫
0
T
d
t
∫
Ω
x
-
p
γ
∇
u
n
p
-
2
∇
u
n
∇
v
+
∫
0
T
d
t
∫
Ω
f
(
u
n
)
v
=
∫
0
T
d
t
∫
Ω
g
v
we conclude that u(t) is a weak solution of (1.1) with the initial condition u(0) = η. Thus,
ξ
∈
G
(
t
,
η
)
.
For hypothesis (3), one observes that for n large enough,
G
(
t
n
,
B
)
=
G
(
t
*
+
t
n
-
t
*
,
B
)
⊂
G
(
t
*
,
G
(
t
n
-
t
*
B
)
)
⊂
G
(
t
*
,
B
*
)
,
where t* > 0 and B* is a bounded set in L2(Ω). Using Lemma 3.3, we conclude that, if
ξ
n
∈
G
(
t
n
,
B
)
, then {ξn} is precompact in L2(Ω).
3.2. The non-autonomous case
Let us recall some definitions and related results. The pair of functions (f(s,⋅),g(⋅,s)) = σ(s) is called a symbol of (1.1). We consider (1.1) with a family of symbols including the shifted forms σ(s + h) = (f(s + h,⋅), g (⋅, s + h)) and the limits of some sequence {σ(s + hn)}n∈N in an appropriate topological space Σ. The family of such symbols is said to be the hull of σ in Σ and is denoted by ℋ
(
σ
)
, i.e.,
ℋ
(
σ
)
=
c
l
∑
{
σ
(
⋅
+
h
)
h
∈
ℝ
}
.
If the hull ℋ(σ) is a compact set in Σ, we say that σ is translation compact in Σ.
Denote ℝd = {(t, τ) ∈ ℝ2 | τ ≤ t}. Let X be a complete metric space, P
(
X
)
and ℬ
(
X
)
be the set of all nonempty subsets and the set of all nonempty bounded subsets of the space X, respectively and let Σ be a subspace of Σ.
Denote
Z
=
{
φ
∈
C
(
ℝ
;
ℝ
)
:
φ
(
u
)
≤
C
φ
(
1
+
u
q
-
1
)
,
for
some
C
φ
>
0
}
,
φ
Z
=
sup
u
∈
ℝ
φ
(
u
)
1
+
u
q
-
1
.
Then Z is a Banach space. We say that fn → f in the space C(ℝ; Z) if
lim
n
→
∞
sup
s
∈
[
t
,
t
+
r
]
f
n
(
s
,
⋅
)
-
f
(
s
,
⋅
)
Z
=
0
for all t ∈ ℝ, r > 0.
Let
f
0
∈
C
(
ℝ
;
Z
)
,
g
0
∈
L
loc
2
,
w
(
ℝ
;
L
2
(
Ω
)
)
, and
ℋ
(
f
0
)
=
c
l
C
(
ℝ
;
Z
)
{
f
0
(
⋅
+
h
)
h
∈
ℝ
}
,
ℋ
(
g
0
)
=
c
l
L
loc
2
,
w
(
ℝ
;
L
2
(
Ω
)
)
{
g
0
(
⋅
+
h
)
h
∈
ℝ
}
,
where the topology in
L
loc
2
,
w
(
ℝ
;
L
2
(
Ω
)
)
is equipped by the local weak convergence, i.e., gn → g in Lloc2,w(ℝ;L2(Ω)) if
lim
n
→
∞
∫
t
t
+
r
∫
Ω
(
g
n
(
s
,
x
)
-
g
(
s
,
x
)
)
ϕ
(
x
,
s
)
d
s
d
x
=
0
for all t ∈ ℝ, r > 0 and ϕ ∈ L2 (Qt,t+r). We define ∑
=
ℋ
(
f
0
)
×
ℋ
(
g
0
)
.
In order to deal with a uniform attractor with respect to the family of symbols, one usually requires the translation compact property. Let us recall some discussions on this requirement. It is known that hypothesis (H2) ensures that g is translation compact in Lloc2,w(ℝ;L2(Ω)) (see 4 for details). In addition, the following statement gives a sufficient condition for the translation compact property in C (ℝ; Z).
Proposition 3.5. 4 The function f ∈ C(ℝ; Z) is translation compact if and only if for all R > 0 one has
(1) |f(t, v)| ≤ C(R) for all t ∈ ℝ, v ∈ [-R, R],
(2) |f(t1, v1)-f(t2, v2)| ≤ α(|t1-t2| + |v1-v2|,R), ∀t1, t2 ∈ ℝ, v1, v2 ∈ [-R, R], here C(R) > 0 and α(.,.) is a function such that α(s, R) → 0 as s → 0+.
From now on, we suppose that f is translation compact. Together with the fact that g is translation compact in Lloc2,w(ℝ;L2(Ω)), one sees that Σ is a compact set in Lloc2,w(ℝ;L2(Ω)). Then it follows from 4 that T(h) : Σ → Σ is continuous and T(h)Σ ⊂ Σ for all h ∈ ℝ.
Definition 3.3. 6 The map U
:
ℝ
d
×
X
→
P
(
X
)
is called an multi-valued semiprocess (MSP) if
(1)U (τ, τ,.) = Id (the identity map),
(2)U (t, τ, x) ⊂ U(t, s, U(s, τ, x)), for all x ∈ X, t, s, τ ∈ ℝ,τ ≤ s ≤ t.
It is called a strict multi-valued semiprocess if U(t, τ, x) = U(t, s, U(s, τ, x)).
We denote by
D
τ
,
σ
(
u
τ
)
the set of all global weak solutions (defined for all t ≥ τ) of the problem (1.1) with data (fσ, gσ) instead of (f, g) such that u(τ) = uτ. For each σ = (f, g) ∈ Σ, we consider the family of MSP {Uσ : σ ∈ Σ} defined by
U
σ
(
t
,
τ
,
u
τ
)
=
{
u
=
u
(
t
)
u
(
⋅
)
∈
D
τ
,
σ
(
u
τ
)
}
.
Lemma 3.6. Uσ (t, τ, uτ) is a multi-valued semiprocess. Moreover,
U
σ
(
t
+
s
,
τ
+
s
,
u
τ
)
=
U
T
(
s
)
σ
(
t
,
τ
,
u
τ
)
f
o
r
a
l
l
u
τ
∈
L
2
(
Ω
)
,
(
t
,
τ
)
∈
ℝ
d
,
s
∈
ℝ
.
Proof. Given z ∈ Uσ(t, τ, uτ)) we have to prove that z ∈ Uσ(t, s, Uσ(s, τ, uτ)). Take y
(
.
)
∈
D
τ
,
σ
(
u
τ
)
such that y(τ) = uτ and y(t) = z. Clearly, y(s) ∈ Uσ(s, τ, uτ). Then if we define z(t) = y(t) for t ≥ s we have that z(s) = y(s) and obviously z
(
.
)
∈
D
s
,
σ
(
y
(
s
)
)
. Consequently, z(t) ∈ Uσ(t, s, Uσ(s, τ, uτ)).
Let z ∈ Uσ(t + s, τ + s, uτ). Then there exists u
(
⋅
)
∈
D
τ
+
s
,
σ
(
u
τ
)
such that z = u(t + s) and v
(
⋅
)
=
u
(
⋅
+
s
)
∈
D
τ
,
T
(
s
)
σ
(
u
τ
)
, so that z = v(t) ∈ uτ,T (s)σ (uτ).
Conversely, if z ∈ Uτ,T(s)σ (uτ), then there is z
∈
D
τ
,
T
(
s
)
σ
(
u
τ
)
such that z = u(t) and
v
(
⋅
)
=
u
(
-
s
+
⋅
)
∈
D
τ
+
s
,
σ
(
u
τ
)
so that z = v(t + s) ∈ Uσ(t + s, τ + s, uτ).
Denote by
U
∑
(
t
,
τ
,
x
)
=
⋃
σ
∈
∑
U
σ
(
t
,
τ
,
x
)
.
Definition 3.4. 6 A set ![]()
is called a uniform global attractor for the family of multi-valued semiprocesses UΣ if:
(1) it is negatively semiinvariant, i.e., A
⊂
U
∑
(
t
,
τ
,
A
)
for all t ≥ τ;
(2) it is uniformly attracting, i.e., d
i
s
t
(
U
∑
(
t
,
τ
,
B
)
,
A
)
→
0
, as t → ∞ , for all B
∈
ℬ
(
X
)
and τ ∈ ℝ;
(3) for any closed uniformly attracting set Y, we have A
⊂
Y
(minimality).
Theorem 3.7. [6, Theorem 2] Suppose that the family of multi-valued semiprocesses UΣ satisfies the following conditions:
(1) On Σ is defined the continuous shift operator T(s)σ(t) = σ(t + s), s ∈ ℝ such that T(h)Σ ⊂ Σ, and for any (t, τ) ∈ ℝd, σ ∈ Σ, s ∈ ℝ, x ∈ X, we have
U
σ
(
t
+
s
,
τ
+
s
,
x
)
=
U
T
(
s
)
σ
(
t
,
τ
,
x
)
;
(2) Uσ is uniformly asymtopically upper semicompact;
(3) Uσ is pointwise dissipative;
(4) The map (x, σ) ↦ Uσ(t, 0, x) has closed values and is w-upper semicontinuous.
Then the family of multi-valued semiprocesses UΣ has a uniform global compact attractor ![]()
.
The following is the key point of this subsection.
Lemma 3.8. Let conditions (H1)-(H3) hold and let {un}n∈ℕ is a sequence of weak solutions of (1.1) with respect to the sequence of symbols {σn} ⊂ Σ such that
(
1
)
u
n
(
τ
)
→
u
τ
i
n
L
2
(
Ω
)
,
(
2
)
σ
n
→
σ
i
n
∑
.
Then there exists a solution u of (1.1) with respect to the symbol σ such that u(τ) = uT and un(t*) → u(t*) in L2(Ω) for any t* > τ, up to a subsequence.
Proof. Let σn = (fn, gn). Since f satisfies (H1) for all t ∈ ℝ and
f
n
∈
ℋ
(
f
)
, one sees that fn also satisfies (H1). On the other hand, noting that {un(τ)} is bounded in L2(Ω) and
g
n
L
b
2
≤
g
L
b
2
. Thus, repeating the arguments in the proof of Theorem 2.1, we obtain that
{
u
n
}
is
bounded
in
V
=
L
p
τ
,
T
;
D
0
,
γ
1
,
p
(
Ω
)
∩
L
q
(
τ
,
T
;
L
q
(
Ω
)
)
,
{
u
′
n
}
is
bounded
in
V
′
=
L
p
′
τ
,
T
;
D
-
γ
-
1
,
p
′
(
Ω
)
+
L
q
′
τ
,
T
;
L
q
′
(
Ω
)
,
{
u
n
}
is
bounded
in
C
(
[
τ
,
T
]
;
L
2
(
Ω
)
)
,
{
f
n
(
t
,
u
n
)
}
is
bounded
in
L
q
′
(
Q
τ
,
T
)
,
{
-
Δ
p
,
γ
u
n
}
is
bounded
in
L
p
′
τ
,
T
;
D
-
γ
-
1
,
p
′
(
Ω
)
.
In particular, we have
u
n
(
t
)
⇀
u
(
t
)
in
L
2
(
Ω
)
for
all
t
∈
[
τ
,
T
]
,
up to a subsequence. Let
σ
n
→
σ
=
(
f
¯
,
g
¯
)
in Σ, to show that u is a solution of (1.1) with respect to the symbol σ such that u(τ) = uT, we need to pass to the limits in the following relation
∫
τ
T
∫
Ω
u
′
n
v
+
x
-
p
γ
∇
u
n
p
-
2
∇
u
n
∇
v
+
f
n
(
t
,
u
n
)
v
d
x
d
t
=
∫
τ
T
∫
Ω
g
n
v
d
x
d
t
for all v ∈ V. Since
g
n
⇀
ḡ
in L2(τ,T; L2(Ω)), it remains to prove that
f
n
(
t
,
u
n
)
⇀
f
̄
(
t
,
u
)
in
L
q
′
(
Q
τ
,
T
)
. We first show that
f
n
(
t
,
u
n
)
→
f
̄
(
t
,
u
n
)
in Lq′(Qτ,T). Indeed,
∫
τ
T
∫
Ω
f
n
(
t
,
u
n
)
-
f
̄
(
t
,
u
n
)
q
′
d
x
d
t
=
∫
τ
T
∫
Ω
f
n
(
t
,
u
n
)
-
f
̄
(
t
,
u
n
)
q
′
(
1
+
u
n
q
-
1
)
q
′
(
1
+
u
n
q
-
1
)
q
′
d
x
d
t
≤
sup
[
τ
,
T
]
f
n
-
f
̄
Z
q
′
∫
τ
T
∫
Ω
(
1
+
u
n
q
)
d
x
d
t
→
0
because
f
n
→
f
̄
in Z and {un} is bounded in Lq(Qτ,T). On the other hand, since
{
f
̄
(
t
,
u
n
)
}
is bounded in Lq′(Qτ,T), by using Lemma 1.3 in [21, Chapter 1] and the continuity of
f
̄
as in the proof of Theorem 2.1, we can conclude that
f
̄
(
t
,
u
n
)
→
f
̄
(
t
,
u
)
weakly in Lq′(Qτ,T). Hence, we have
f
n
(
t
,
u
n
)
-
f
̄
(
t
,
u
)
=
(
f
n
(
t
,
u
n
)
-
f
̄
(
t
,
u
n
)
)
+
(
f
̄
(
t
,
u
n
)
-
f
̄
(
t
,
u
)
)
→
0
weakly
in
L
q
′
(
Q
τ
,
T
)
.
We now have to show that un(t*) → u(t*) in L2(Ω) for any t* > τ. Taking into account of (3.3), we have to check that
u
n
(
t
*
)
L
2
(
Ω
)
→
u
(
t
*
)
L
2
(
Ω
)
.
Putting
J
n
(
t
)
=
u
n
(
t
)
L
2
(
Ω
)
2
-
2
∫
τ
t
(
g
n
(
s
)
,
u
n
(
s
)
)
d
s
-
(
2
k
2
Ω
+
2
λ
)
(
t
-
τ
)
,
J
(
t
)
=
u
(
t
)
L
2
(
Ω
)
2
-
2
∫
τ
t
(
g
(
s
)
,
u
(
s
)
)
d
s
-
(
2
k
2
Ω
+
2
λ
)
(
t
-
τ
)
.
It is easy to check that the functions Jn(t), J(t) are continuous and non-increasing on [τ, T]. We first show that
J
n
(
t
)
→
J
(
t
)
for
a .e
.
t
∈
[
τ
,
T
]
.
Indeed,
|
J
n
(
t
)
−
J
(
t
)
|
≤
|
‖
u
n
(
t
)
‖
L
2
(
Ω
)
2
−
‖
u
(
y
)
‖
L
2
(
Ω
)
2
|
+
2
|
∫
τ
t
[
(
g
n
(
s
)
,
u
n
(
s
)
)
−
(
g
(
s
)
,
u
(
s
)
)
]
d
s
|
≤
‖
u
n
(
t
)
−
u
(
t
)
‖
L
2
(
Ω
)
(
‖
u
n
(
t
)
‖
L
2
(
Ω
)
+
‖
u
(
t
)
‖
L
2
(
Ω
)
)
+
2
|
∫
τ
t
[
(
g
n
(
s
)
,
u
n
(
s
)
−
u
(
s
)
)
d
s
|
+
2
|
∫
τ
t
[
(
g
n
(
s
)
−
g
(
s
)
,
u
(
s
)
)
d
s
|
,
and
|
∫
τ
t
[
(
g
n
(
s
)
,
u
n
(
s
)
−
u
(
s
)
)
d
s
|
≤
‖
g
n
‖
L
2
(
Q
τ
,
t
)
‖
u
n
(
t
)
−
u
(
t
)
‖
L
2
(
Ω
)
→
0
as n → ∞ since un → u strongly in L2(Qτ,t) and {gn} is bounded in L2(Qτ,t). In addition,
∫
τ
t
(
g
n
(
s
)
-
g
(
s
)
,
u
(
s
)
)
d
s
→
0
as n → ∞ since gn ⇀ g in L2(Qτ,t). Then (3.4) is proved due to the fact that un(t) → u(t) in L2(Ω) for a.e. t ∈ [τ, T].
We choose an increasing sequence {tm} ⊂ [τ, T], tm → t* such that Jn(tm) ⇀ J(tm) as n → ∞. Then, by the continuity,
J
n
(
t
m
)
⇀
J
n
(
t
*
)
,
as
m
→
∞
.
So
J
n
(
t
*
)
-
J
(
t
*
)
≤
J
n
(
t
m
)
-
J
(
t
*
)
=
J
n
(
t
m
)
-
J
(
t
m
)
+
J
(
t
m
)
-
J
(
t
*
)
<
ε
for n ≥ n0(ε) and any ε > 0. Hence, lim sup Jn(t*) ≤ J(t*) and then lim sup ∥un(t*)∥ ≤ ∥u(t*)∥. From the weak convergence un(t*) ⇀ u(t*) we have then ∥un(t*)∥ → ∥u(t*)∥, so un(t*) → u(t*) strongly in L2(Ω) as n → ∞. This completes the proof.
Theorem 3.9. Let conditions (H1)-(H3) hold. Then the family of multi-valued semipro-cesses {Uσ (t, τ)} has a uniform global compact attractor ![]()
.
Proof. We know that each symbol σn = (fn, gn) ∈ Σ satisfies the same conditions as in (H1)-(H2). Furthermore, since
g
n
∈
ℋ
(
g
)
, we have
g
n
L
b
2
≤
g
L
b
2
. Hence if un is a weak solution of (1.1) with respect to the symbol σn, one has
u
n
(
t
)
L
2
(
Ω
)
2
≤
u
n
(
τ
)
L
2
(
Ω
)
2
e
-
λ
(
t
-
τ
)
+
1
λ
(
1
-
e
-
λ
)
g
L
b
2
2
+
2
k
2
Ω
λ
+
2
.
The last inequality ensures the existence of a positive number R0 such that if un(τ) ∈ BR, the ball in L2(Ω) centered at 0 with radius R, then there exists T0 = T0(τ, R) such that
u
n
(
t
)
∈
B
R
0
for
all
t
≥
T
0
,
that is,
U
∑
(
t
,
τ
,
B
R
)
⊂
B
R
0
, for all t ≥ T0(τ, R). Thus, {Uσ(t, τ)} fulfills condition (3) in Theorem 3.7.
We now define the set K
=
U
∑
(
1
,
0
,
B
R
0
)
¯
. Lemma 3.8 implies that K is compact. Moreover, since
B
R
0
is an absorbing set, we have
U
σ
n
(
t
,
τ
,
B
R
)
=
U
σ
n
(
t
,
t
−
1
,
U
σ
n
(
t
−
1
,
τ
,
B
R
)
=
U
T
(
t
−
1
)
σ
n
(
1
,
0
,
U
T
(
τ
)
σ
n
(
t
−
1
−
τ
,
0
,
B
R
)
)
⊂
U
∑
(
1
,
0
,
B
R
0
)
⊂
K
for all
σ
n
∈
∑
,
B
R
∈
ℬ
(
L
2
(
Ω
)
)
, and t ≥ T0(τ, BR). It follows that any sequence {ξn} such that
{
ξ
n
}
∈
U
σ
n
(
t
n
,
τ
,
B
R
0
)
,
σ
n
∈
∑
,
t
n
→
+
∞
,
B
R
∈
ℬ
(
L
2
(
Ω
)
)
, is precompact in L2(Ω). It is a consequence of Lemma 3.8 that the map Uσ has compact values for any σ ∈ Σ.
Finally, let us prove that the map (σ, x) ↦ Uσ(t, τ, x) is upper semicontinuous for each fixed t ≥ τ. Suppose that it is not true, that is, there exist ū
∈
L
2
(
Ω
)
,
t
≥
τ
,
σ
̄
∈
∑
,
ε
>
0
,
δ
n
→
0
,
u
n
∈
B
δ
n
(
ū
)
,
σ
n
→
σ
̄
, and
ξ
n
∈
U
σ
n
(
t
,
τ
,
u
n
)
such that
{
ξ
n
}
∉
B
ε
(
U
σ
¯
(
t
,
τ
,
u
¯
)
. But Lemma 3.8 implies (up to a subsequence) that
ξ
n
→
ξ
∈
U
σ
̄
(
t
,
τ
,
ū
)
, which is a contracdition. Thus, the existence of the uniform global compact attractor follows then from Theorem 3.7.