# Upper Semicontinuity of Pullback Attractors for the 3D Nonautonomous Benjamin-Bona-Mahony Equations

Authors: Xinguang Yang; Xiaosong Wang; Juntao Li; Lingrui Zhang
Journal: The Scientific World Journal
 (2014)

---

## Abstract

We will study the upper semicontinuity of pullback attractors for the 3D nonautonomouss Benjamin-Bona-Mahony equations with  external force perturbation terms. Under some regular assumptions, we can prove the pullback attractors𝒜
ε
(
t
) of  equation u
t
-
Δ
u
t
-
ν
Δ
u
+
∇
·
F
→
(
u
)
=
ɛ
g
(
x
,
t
), x
∈
Ω
, converge to the global attractor 𝒜 of the above-mentioned equation with ε
=
0 for any t
∈
ℝ.


---

## Body

## 1. Introduction

In this paper, we will consider the upper semicontinuity of pullback attractors for the following 3D Benjamin-Bona-Mahony equation:(1)
u
t
-
Δ
u
t
-
ν
Δ
u
+
∇
·
F
→
(
u
)
=
ɛ
g
(
x
,
t
)
,
x
∈
Ω
,
(2)
u
(
t
,
x
)
|
∂
Ω
=
0
,
(3)
u
(
τ
,
x
)
=
u
τ
(
x
)
,
τ
∈
ℝ
.
Here Ω
⊂
ℝ
3 is a bounded domain with sufficiently smooth boundary ∂
Ω; u
(
t
,
x
)
=
(
u
1
(
t
,
x
)
,
u
2
(
t
,
x
)
,
u
3
(
t
,
x
)
) is the velocity vector field; ν
>
0 is the kinematic viscosity; F
→ is a nonlinear vector function; ɛ
≥
0 is a small nonnegative parameter; the external force g
(
x
,
t
) is locally square integrable in time for (
x
,
t
)
∈
Ω
×
ℝ, that is, for any t
∈
ℝ, g
(
x
,
t
)
∈
L
loc
⁡
2
(
ℝ
;
H
), where H
=
(
L
2
(
Ω
)
)
3, V
=
(
H
0
1
(
Ω
)
)
3, and (
·
,
·
) and ∥
·
∥ are the inner product and norm of H, respectively.The Benjamin-Bona-Mahony (BBM) equation is a well-known model in physical applications which incorporates dispersive effects for long waves in shallow water that was introduced by Benjamin et al. [1] as an improvement of the Korteweg-de Vries equation (KdV equation) for modeling long waves of small amplitude in two dimensions. Contrasting with the KdV equation, the BBM equation is unstable in its high wave number components. Further, while the KdV equation has an infinite number of integrals of motion, the BBM equation only has three. Both KdV and BBM equations cover cases of surface waves of long wavelength in liquids, acoustic-gravity waves in compressible fluid, hydromagnetic waves in cold plasma, and acoustic waves in harmonic crystals.For the well-posedness of global solutions for BBM equation, we can refer to [2–7]. For the long-time behavior, such as the existence of global attractor and its structure and the dimension of the attractors, we will discuss the known results in details.Biler [8] investigated the long-time behavior of 2D generalized BBM equation
(4)
u
t
-
Δ
u
t
=
(
b
,
∇
u
)
+
u
p
(
a
,
∇
u
)
in ℝ
2, t
∈
ℝ. Here b
≠
0, a
∈
ℝ
2, and p
≥
3 is an integer. The author proved the supremum norms of the solutions with small initial data decay to zero like t
-
2
/
3 as t tends to infinity.By energy equation and weak continuous method, Wang [9] and Wang and Yang [10] investigated the finite-dimensional behavior of solutions and derived the global weak attractor and the strong attractors for BBM equation:
(5)
u
t
-
u
x
x
t
-
ν
u
x
x
+
(
f
(
u
)
)
x
=
g
(
x
)
with period boundary value condition in H
per
2
(
Ω
) and H
per
1
(
Ω
), respectively. Moreover, Wang et al. [11] got the existence of global attractor for the above BBM equation defined in a three-dimensional channel; the asymptotic compactness of the solution operator is obtained by the uniform estimates on the tails of solutions.By the decomposition of the semigroup, Wang [12] studied the regularity of attractors for the BBM equation
(6)
u
t
-
u
x
x
t
-
ν
u
x
x
+
u
x
+
u
u
x
=
g
(
x
)
.
He proved that the global attractor is smooth if the forcing term is smooth. In addition, Wang [13] also obtained the approximate inertial manifolds to the global attractors for the generalized BBM equations.Wang [14] considered the stochastic BBM equations on unbounded domains
(7)
d
u
-
d
(
Δ
u
)
-
ν
Δ
u
d
t
+
∇
·
F
(
u
)
→
d
t
=
g
d
t
+
h
d
w
and concluded the existence of random attractor in H
0
1 under certain assumptions, here w is the two-sided real-valued Wiener process on a probability space. He also proved the random attractor is invariant and attracts every pulled-back tempered random set under the forward flow. The asymptotic compactness of the random dynamical system is established by a tail-estimates method, which shows that the solutions are uniformly asymptotically small when space and time variables approach infinity.Stanislavova et al. [15] first provided a sufficient condition to verify the asymptotic compactness of an evolution equation defined in an unbounded domain, which involves the Littlewood-Paley projection operators, then they proved the existence of an attractor for the Benjamin-Bona-Mahony equation in the phase space H
1
(
ℝ
3
) by showing the solutions are point dissipative and asymptotic compact
(8)
u
t
-
Δ
u
t
-
ν
Δ
u
+
div
⁡
(
f
(
u
)
)
=
g
for g
∈
L
2
(
ℝ
3
) and f
(
u
)
=
u
+
(
1
/
2
)
u
2. Stanislavova [16] investigated the existence of global attractors of (8) in two dimension.By the method of orthogonal decomposition, Zhu [17, 18] obtained the asymptotic attractor, global attractor, and its Hausdorff dimension of the damped BBM equations with periodic boundary conditions in homogeneous periodic space H
˙
per
1
(
Ω
)
(9)
u
t
-
δ
u
x
x
t
-
ν
u
x
x
+
u
u
x
=
f
(
x
)
which overcome difficulty coming from the precision of approximate inertial manifolds. Zhu and Mu [19] deduced the exponential decay estimates of solutions for time-delayed BBM equations.J. Park and S. Park [20] studied the pullback attractors for the nonautonomous BBM equations in unbounded domains
(10)
u
t
-
Δ
u
t
-
ν
Δ
u
+
∇
·
F
(
u
)
→
=
g
(
x
,
t
)
,
by weak continuous method and some priori estimates in H
0
1
(
Ω
). Qin et al. [21] derived the existence of pullback attractor of (10) in H
0
2
(
Ω
) by weak continuous method. Zhao et al. [22] investigated the convergence of corresponding uniform attractors between averaging BBM and state BBM equations.Moreover, Çelebi et al. [23] deduced the existence of attractors with a finite fractal dimension and the existence of the exponential attractor for the corresponding asymptotically compact semigroup for the periodic initial-boundary value problem of a generalized BBM equation. Chueshov et al. [24] studied the regularity of global attractor for a generalized BBM equation.For the upper semicontinuity of corresponding attractors between autonomous and perturb nonautonomous systems, we can refer to Bao [25], Hale and Raugel [26], Carvalho et al. [27], Caraballo and Langa [28], Caraballo et al. [29], Fitzgibbon et al. [30], Kloeden [31], Miyamoto [32], Wang and Qin [33], Younsi [34], Wang [35], and Zhou [36].To our knowledge, there are less results on the upper semicontinuity of pullback attractors for the 3D nonautonomous BBM equations with the nonautonomous perturbation; we will pay attention to this issue in the sequel.This paper is organized as following. In Section2, we will recall some fundamental theory of pullback attractors for nonautonomous dynamical systems and give a method to verify the upper semicontinuity of pullback attractors. In Section 3, the upper semicontinuity of pullback attractors for the problems (1)–(3) will be proved.


## 2. Pullback Attractors of Nonautonomous Dynamical Systems

In this section, we will consider the relationship between pullback attractors𝒜
ɛ
=
{
A
ɛ
(
t
)
}
t
∈
ℝ for the perturbed nonautonomous system with ɛ
>
0 and global attractor 𝒜 for the unperturbed autonomous system with ɛ
=
0 of the following equation:
(11)
∂
u
∂
t
=
𝔸
f
u
(
x
,
t
)
+
ɛ
f
(
x
,
t
)
.
If the global attractor is unique, then the global attractor is the pullback attractor when ɛ
=
0.LetX be a Banach space with norm ∥
·
∥
X. The Hausdorff semidistance dist
⁡
X
(
B
1
,
B
2
) in X between B
1
⊆
X and B
2
⊆
X is defined by
(12)
dist
⁡
X
(
B
1
,
B
2
)
=
sup
⁡
x
∈
B
1
inf
⁡
y
∈
B
2
d
X
(
x
,
y
)
for
B
1
,
B
2
⊂
X
,
where d
X
(
x
,
y
) denotes the distance between two points x and y.For an autonomous system,S
(
t
)
:
X
→
X (t
∈
ℝ) is a C
0-semigroup defined on X. If the global attractor 𝒜 for S
(
t
) exists, then it has the following properties: (1) 𝒜 is an invariant, compact set; (2) 𝒜 attracts every bounded sets in X, that is, lim
⁡
t
→
+
∞
⁡
dist
⁡
(
S
(
t
)
B
,
𝒜
)
=
0 for all bounded subsets B
⊂
X.For a nonautonomous system, the two-parameter mapping class{
U
(
t
,
τ
)
}
t
≥
τ is said to be a process in X if
(13)
U
(
t
,
s
)
U
(
s
,
τ
)
=
U
(
t
,
τ
)
,
∀
t
≥
s
≥
τ
,
τ
∈
ℝ
,
U
(
τ
,
τ
)
=
I
d
,
(
identity
operator
in
X
)
,
∀
τ
∈
ℝ
.
Moreover, throughout the paper, we always assume that the process U
(
·
,
·
) is continuous in X.Now we will recall some definitions and framework on the existence theory of pullback attractors.Definition 1.
A family of compact sets𝒜
=
{
A
(
t
)
}
t
∈
ℝ is said to be a pullback attractor for the continuous process {
U
(
·
,
·
)
} if it satisfies the following:(i)
𝒜 is invariant for all t
≥
τ.
(ii)
𝒜 is pullback attracting, that is, lim
⁡
τ
→
+
∞
⁡
dist
⁡
(
U
(
t
,
t
-
τ
)
B
,
A
(
t
)
)
=
0 for all bounded subsets B
⊂
X.Definition 2.
The family of subsetsℬ
=
{
B
(
t
)
}
t
∈
ℝ is said to be pullback absorbing for the process U
(
·
,
·
), if for every t
∈
ℝ and all bounded subsets B
⊂
X, there exists a time T
(
t
,
B
)
>
0, such that
(14)
U
(
t
,
t
-
τ
)
B
⊂
B
(
t
)
∀
τ
≥
T
(
t
,
B
)
.Definition 3.
Letℬ
=
{
B
(
t
)
}
t
∈
ℝ be a family of subsets in X. A process U
(
·
,
·
) is said to be pullback ℬ-asymptotically compact in X if for all t
∈
ℝ, any sequences τ
n
→
∞ and x
n
∈
B
(
t
-
τ
n
); the sequence {
U
(
t
,
t
-
τ
n
)
x
n
} is precompact in X.Theorem 4.
Let the family of setsℬ
=
{
B
(
t
)
}
t
∈
ℝ be pullback absorbing set for the process U
(
·
,
·
) and U
(
·
,
·
) is pullback ℬ-asymptotically compact in X. Then, the family 𝒜
=
{
A
(
t
)
}
t
∈
ℝ that is defined by A
(
t
)
=
Λ
(
ℬ
,
t
) is a pullback attractor for U
(
·
,
·
) in X for the process {
U
(
·
,
·
)
}, where
(15)
Λ
(
ℬ
,
t
)
=
⋂
s
≥
0
⋃
τ
≥
s
‍
U
(
t
,
t
-
τ
)
B
(
t
-
τ
)
¯
f
o
r
e
a
c
h
t
∈
ℝ
.In the following, we will characterize the pullbackℬ-asymptotic compactness in terms of the noncompact measure.Definition 5.
LetB
⊂
X, ℬ
=
{
B
(
t
)
}
t
∈
ℝ be a family of sets in X. A process U
(
·
,
·
) is said to be pullback ℬ-κ contracting, if for any t
∈
ℝ, ɛ
>
0, there exists a time T
ℬ
(
t
,
ɛ
)
>
0, such that
(16)
κ
(
U
(
t
,
t
-
τ
)
B
(
t
-
τ
)
)
≤
ɛ
∀
τ
≥
T
ℬ
(
t
,
ɛ
)
.
Here κ
(
B
) is the Kuratowski noncompact measure defined as
(17)
κ
(
B
)
=
inf
⁡
{
δ
>
0
∣
B
admits
a
finite
cover
by
sets
of
diameter
<
δ
}
.Lemma 6.
Letℬ
=
{
B
(
t
)
}
t
∈
ℝ, ℬ
^
=
{
B
^
(
t
)
}
t
∈
ℝ be two families of sets in X and satisfy that for any t
∈
ℝ, there exists a time T
ℬ
,
ℬ
^
(
t
)
>
0, such that
(18)
U
(
t
,
t
-
τ
)
B
(
t
-
τ
)
⊂
B
^
(
t
)
∀
τ
≥
T
ℬ
,
ℬ
^
(
t
)
.
Then U
(
·
,
·
) is pullback ℬ-asymptotically compact, if it is pullback ℬ
^-κ contracting.Proof.
See, for example, Wang and Qin [33].Theorem 7.
Assume that the assumptions in Lemma6 hold. If the process U
(
·
,
·
) is pullback ℬ
^-κ contracting and the family of sets ℬ
=
{
B
(
t
)
}
t
∈
ℝ is pullback absorbing for U
(
·
,
·
), then the process U
(
·
,
·
) possesses a pullback attractor.Proof.
See, for example, Wang  and  Qin [33].Theorem 8.
Letℬ
=
{
B
(
t
)
}
t
∈
ℝ be a family of sets in X. Suppose U
(
·
,
·
)
=
U
1
(
·
,
·
)
+
U
2
(
·
,
·
)
:
ℝ
×
ℝ
×
X
→
X satisfies(i)
for anyt
∈
ℝ,
(19)
∥
U
1
(
t
,
t
-
τ
)
x
t
-
τ
∥
X
≤
Φ
(
t
,
τ
)
∀
x
t
-
τ
∈
B
(
t
-
τ
)
,
τ
>
0
,
whereΦ
(
·
,
·
)
:
ℝ
×
ℝ
→
ℝ
+ satisfies lim
⁡
τ
→
+
∞
⁡
Φ
(
t
,
τ
)
=
0 for each t
∈
ℝ;
(ii)
for anyt
∈
ℝ and T
≥
0, ⋃
0
≤
τ
≤
T
‍
U
2
(
t
,
t
-
τ
)
B
(
t
-
τ
) is bounded and U
2
(
t
,
t
-
τ
)
B
(
t
-
τ
) is precompact in X for any τ
>
0.Then the process U
(
·
,
·
) is pullback ℬ-κ contracting in X.Proof.
See, for example, Wang and Qin [33].We now perturb the nonautonomous term with a small parameterɛ
∈
(
0
,
ε
0
]; thus we obtain a nonautonomous dynamical system driven by the process U
ɛ
(
·
,
·
).For eacht
∈
ℝ,  τ
∈
ℝ, and x
∈
X, we have
(20)
(
H
1
)
lim
⁡
ɛ
→
0
d
X
(
U
ɛ
(
t
,
t
-
τ
)
x
,
S
(
t
)
x
)
=
0
,
uniformly on bounded sets of X.Theorem 9 (Caraballo et al. [28, 29]).
Assume that(
H
1
) holds, and for any ɛ
∈
(
0
,
ε
0
], there exist pullback attractors 𝒜
ɛ
=
{
A
ɛ
(
t
)
}
t
∈
ℝ for all ɛ
>
0. If there exists a compact set K
⊂
X, such that
(21)
(
H
2
)
lim
⁡
ɛ
→
0
d
i
s
X
(
A
ɛ
(
t
)
,
K
)
=
0
f
o
r
a
n
y
t
∈
ℝ
.
Then 𝒜
ɛ and 𝒜 have the upper semicontinuity, that is,
(22)
lim
⁡
ɛ
→
0
d
i
s
X
(
A
ɛ
(
t
)
,
𝒜
)
=
0
f
o
r
a
n
y
t
∈
ℝ
.In order to apply Theorem9 to obtain the upper semicontinuity of pullback attractors 𝒜
ɛ and global attractor 𝒜, we now present a technique to verify (
H
2
) for the process generated by the nonautonomous dissipative system.Lemma 10.
Assume that the familyℬ
=
{
B
(
t
)
}
t
∈
ℝ is pullback absorbing for U
(
·
,
·
), and for each ɛ
∈
(
0
,
ε
0
], 𝒦
ɛ
=
{
K
ɛ
(
t
)
}
t
∈
ℝ is a family of compact sets in X. Suppose U
ɛ
(
·
,
·
)
=
U
1
,
ɛ
(
·
,
·
)
+
U
2
,
ɛ
(
·
,
·
)
:
ℝ
×
ℝ
×
X
→
X satisfies(i)
for anyt
∈
ℝ and any ɛ
∈
(
0
,
ε
0
],
(23)
∥
U
1
,
ɛ
(
t
,
t
-
τ
)
x
t
-
τ
∥
X
≤
Φ
(
t
,
τ
)
∀
x
t
-
τ
∈
B
(
t
-
τ
)
,
τ
>
0
,
whereΦ
(
·
,
·
)
:
ℝ
×
ℝ
→
ℝ
+ satisfies lim
⁡
τ
→
+
∞
⁡
Φ
(
t
,
τ
)
=
0 for each t
∈
ℝ;
(ii)
for anyt
∈
ℝ and any T
≥
0, ∪
0
≤
τ
≤
T
U
2
,
ɛ
(
t
,
t
-
τ
)
B
(
t
-
τ
) is bounded, and for any t
∈
ℝ, there exists a time T
ℬ
(
t
)
>
0, which is independent of ɛ, such that
(24)
U
2
,
ɛ
(
t
,
t
-
τ
)
B
(
t
-
τ
)
⊂
K
ɛ
(
t
)
∀
τ
≥
T
ℬ
(
t
)
,
ɛ
∈
(
0
,
ε
0
]
and there exists a compact setK
⊂
X, such that
(25)
(
H
2
′
)
lim
⁡
ɛ
→
0
d
i
s
t
X
(
K
ɛ
(
t
)
,
K
)
=
0
,
f
o
r
a
n
y
t
∈
ℝ
.
Then for eachɛ
∈
(
0
,
ε
0
], there exists a pullback attractor 𝒜
ɛ
=
{
A
ɛ
(
t
)
}
t
∈
ℝ and (
H
2
) holds.Proof.
See, for example, Wang and Qin [33].


## 3. Upper Semicontinuity of Pullback Attractors

In this section, firstly, we recall some notations about the functional spaces which will be used later to discuss the regularity of pullback attracting set.The operatorA is denoted by A
=
-
Δ with domain D
(
A
)
=
H
2
(
Ω
)
⋂
‍
H
0
1
(
Ω
) and λ is the first eigenvalue of A; we consider the family of Hilbert spaces
(26)
ℋ
α
=
D
(
A
α
/
2
)
,
α
∈
ℝ
generated by the Laplacian operator with the Dirichlet boundary value conditions equipped with the standard inner product and norm
(27)
(
·
,
·
)
ℋ
α
=
(
A
α
/
2
·
,
A
α
/
2
·
)
,
∥
·
∥
ℋ
α
=
∥
A
α
/
2
·
∥
respectively, then we have D
(
A
s
/
2
)
↪
D
(
A
r
/
2
) for any s
>
r and the continuous embedding
(28)
ℋ
s
≡
D
(
A
s
/
2
)
↪
(
L
6
/
(
3
-
2
s
)
(
Ω
)
)
3
for all s
∈
[
0
,
3
/
2
), ℋ
2
=
H
2
(
Ω
)
⋂
‍
H
0
1
(
Ω
).Then, applying the Helmholtz-Leray projector𝒫 to the systems (1)–(3), we obtain the following problem which is equivalent to the original problems (1)–(3)
(29)
u
t
+
ν
A
u
+
A
u
t
+
B
(
u
)
=
ɛ
f
(
x
,
t
)
,
(
x
,
t
)
∈
Ω
×
[
τ
,
∞
)
,
u
(
x
,
t
)
=
0
,
(
x
,
t
)
∈
∂
Ω
×
[
τ
,
∞
)
,
u
(
τ
,
x
)
=
u
τ
(
x
)
,
x
∈
Ω
.
Here A
=
-
𝒫
Δ, B
(
u
)
=
𝒫
(
∇
·
F
→
(
u
)
), and f
(
x
,
t
)
=
𝒫
g
(
x
,
t
).Assume thatu
τ
∈
H
0
1
(
Ω
), the external force g
∈
L
loc
⁡
2
(
ℝ
,
H
). Also we assume that there exist constants β
>
0, 0
≤
α
<
σ
/
2, and σ
=
2
ν
/
(
(
2
/
λ
)
+
2
), such that
(30)
∥
g
(
t
)
∥
2
≤
β
e
α
|
t
|
,
which implies that
(31)
∫
-
∞
t
‍
e
σ
s
∥
g
(
s
)
∥
2
d
s
<
+
∞
,
∀
t
∈
ℝ
,
∫
-
∞
τ
‍
(
∫
-
∞
t
‍
e
σ
s
/
2
∥
g
(
s
)
∥
2
d
s
)
d
t
<
+
∞
,
∀
τ
∈
ℝ
.
Moreover, we assume that
(32)
lim
⁡
k
→
∞
∫
-
∞
τ
‍
∫
|
x
3
|
≥
k
‍
e
σ
t
|
g
(
x
,
t
)
|
2
d
x
d
t
=
0
,
∀
τ
∈
ℝ
.
From (31), we can easily derive that the term f
(
x
,
t
) is locally square integrable in time; that is, f
(
x
,
t
)
∈
L
loc
⁡
2
(
ℝ
,
H
) and satisfies
(33)
∫
-
∞
t
‍
e
η
s
∥
f
(
x
,
t
)
∥
H
2
d
s
<
+
∞
for 0
<
η
≤
min
⁡
{
λ
ν
,
ν
,
ν
/
(
(
2
/
λ
)
+
2
)
} and any t
∈
ℝ.For the nonlinear vector functionF
→
(
s
)
=
(
F
1
(
s
)
,
F
2
(
s
)
,
F
3
(
s
)
)
(
s
∈
ℝ
), we denote
(34)
f
i
(
s
)
=
F
i
′
(
s
)
,
ℱ
i
(
s
)
=
∫
0
s
‍
F
i
(
r
)
d
r
,
where
(35)
f
→
(
s
)
=
(
f
1
(
s
)
,
f
2
(
s
)
,
f
3
(
s
)
)
,
ℱ
→
(
s
)
=
(
ℱ
1
(
s
)
,
ℱ
2
(
s
)
,
ℱ
3
(
s
)
)
.
Assume that F
i (i
=
1,2
,
3) are smooth functions satisfying
(36)
F
i
(
0
)
=
0
,
|
F
i
(
s
)
|
≤
C
1
|
s
|
+
C
2
|
s
|
2
,
C
1
(
1
+
σ
2
|
s
|
)
≤
|
f
i
(
s
)
|
≤
C
2
(
1
+
σ
2
|
s
|
)
,
|
ℱ
i
(
s
)
|
≤
C
1
|
s
|
2
+
C
2
|
s
|
3
,
for all s
∈
ℝ, where C
1, C
2, and σ are positive constants.At last, we will state the main result and the proof of this paper as the following.Theorem 11.
Assume that (30)–(36) hold, and u
τ
∈
V, then the pullback attractors 𝒜
ɛ
=
{
𝒜
ɛ
(
t
)
}
t
∈
ℝ for (29) (which is equivalent to (1)) with ɛ
>
0 and the global attractor 𝒜 for (29) with ɛ
=
0 satisfy
(37)
lim
⁡
ɛ
→
0
+
d
i
s
V
(
𝒜
ɛ
(
t
)
,
𝒜
)
=
0
f
o
r
a
n
y
t
∈
ℝ
.
The Hausdorff semidistance d
i
s
t
(
·
,
·
) is defined on the Banach space V.In order to apply Theorem9 and Lemma 10 to prove Theorem 11, we will introduce the existence of global attractor for autonomous system (1) with ɛ
=
0 and pullback attractors for nonautonomous system (1) with ɛ
>
0 in the following lemmas.Lemma 12.
Assume that (34)–(36) hold, and u
τ
∈
V, then the semigroup S
(
t
) (t
∈
ℝ) generated by problem (29) (or problems (1)–(3)) with ɛ
=
0 possesses a global attractor 𝒜 in V.Proof.
Using similar technique as in [9–11, 17, 18], we only need to consider the Dirichlet boundary value condition instead of the periodic boundary value condition in these papers which investigated the existence of global attractors. This means that we can obtain our lemma easily, here we omit the details.Lemma 13.
Assume that (30)–(36) hold, and u
τ
∈
V, then problem (29) possesses a unique global solution u
ɛ
(
x
,
t
) (ɛ
≥
0) satisfying
(38)
u
ɛ
(
x
,
t
)
∈
C
(
[
τ
,
+
∞
)
,
V
)
∩
L
∞
(
0
,
+
∞
;
V
)
,
u
t
∈
L
2
(
0
,
T
;
V
)
.
Moreover, the process {
U
ɛ
(
t
,
τ
)
} generated by the global solutions possess pullback attractors 𝒜
ɛ for all ɛ
≥
0 in V.Proof.
See, for example, [20].Now we decompose the solutionu
ɛ
(
t
)
=
U
ɛ
(
t
,
τ
)
u
τ of (29) with initial data u
τ
∈
V as
(39)
u
ɛ
=
U
ɛ
(
t
,
τ
)
u
τ
=
U
1
,
ɛ
(
t
,
τ
)
u
τ
+
U
2
,
ɛ
(
t
,
τ
)
u
τ
,
where
(40)
U
1
,
ɛ
(
t
,
τ
)
u
τ
=
v
(
t
)
,
U
2
,
ɛ
(
t
,
τ
)
u
τ
=
w
(
t
)
solve the following problems:
(41)
v
t
+
A
v
t
+
ν
A
v
+
B
(
v
)
=
0
,
in
(
x
,
t
)
∈
Ω
×
[
τ
,
∞
)
,
v
(
x
,
t
)
=
0
,
on
⁡
∂
Ω
×
[
τ
,
∞
)
,
v
(
τ
,
x
)
=
v
τ
(
x
)
,
x
∈
Ω
,
(42)
w
t
+
A
w
t
+
ν
A
w
=
-
B
(
u
)
+
B
(
v
)
+
ɛ
f
(
x
,
t
)
,
in
(
x
,
t
)
∈
Ω
×
[
τ
,
∞
)
,
w
(
x
,
t
)
=
0
,
on
⁡
∂
Ω
×
[
τ
,
∞
)
,
w
(
τ
,
x
)
=
0
,
x
∈
Ω
,
respectively.Lemma 14.
Suppose that (34)–(36) hold. For any bounded set B
⊂
V and t
∈
ℝ, there exists a time T
(
B
,
t
)
>
0, such that
(43)
∥
U
ɛ
(
t
,
t
-
τ
)
u
t
-
τ
∥
2
≤
R
ɛ
(
t
)
∀
τ
≥
T
(
B
,
t
)
,
a
l
l
u
t
-
τ
∈
B
,
where R
ɛ
(
t
)
=
C
ɛ
e
-
η
t
∫
-
∞
t
‍
e
η
s
∥
f
(
s
)
∥
H
2
d
s, and C is a positive constant independent of B, t, τ.Proof.
We chooseσ
=
2
ν
/
(
(
2
/
λ
)
+
2
), R
σ
=
{
r
:
R
→
(
0
,
+
∞
)
∣
lim
⁡
t
→
-
∞
e
σ
t
r
2
(
t
)
=
0
} and denote by 𝒟
σ the class of families D
^
=
{
D
(
t
)
:
t
∈
R
}
⊂
𝒟
(
H
) such that D
(
t
)
⊂
B
¯
(
0
,
r
D
^
(
t
)
) for some r
D
^, where B
¯
(
0
,
r
D
^
(
t
)
) denotes the closed ball in V centered at zero with radius r
D
^
(
t
).
Lett
∈
ℝ,  τ
∈
ℝ, and u
τ
∈
V be fixed, and denote
(44)
u
(
r
)
=
u
(
r
;
t
-
τ
,
u
0
)
=
U
(
r
-
t
+
τ
,
t
-
τ
,
u
0
)
for
r
≥
t
-
τ
.
Since u
∈
C
(
(
τ
,
T
)
;
V
), then for all u
∈
V, we derive that
(45)
d
d
t
(
e
σ
t
∥
u
(
t
)
∥
2
+
e
σ
t
∥
∇
u
(
t
)
∥
2
)
+
2
ν
e
σ
t
∥
∇
u
(
t
)
∥
2
=
-
2
e
σ
t
(
∇
F
→
(
u
)
,
∇
u
)
+
σ
(
e
σ
t
∥
u
(
t
)
∥
2
+
e
σ
t
∥
∇
u
∥
2
)
+
2
e
σ
t
(
ɛ
f
(
t
)
,
u
(
t
)
)
=
σ
(
e
σ
t
∥
u
(
t
)
∥
2
+
e
σ
t
∥
∇
u
∥
2
)
+
2
e
σ
t
(
ɛ
f
(
t
)
,
u
(
t
)
)
≤
σ
(
1
λ
+
1
)
e
σ
t
∥
∇
u
∥
2
+
σ
λ
e
σ
t
∥
∇
u
∥
2
+
ɛ
σ
e
σ
t
∥
f
(
t
)
∥
H
2
≤
σ
(
2
λ
+
1
)
e
σ
t
∥
∇
u
(
t
)
∥
2
+
ɛ
σ
e
σ
t
∥
f
(
t
)
∥
H
2
≤
2
ν
e
σ
t
∥
∇
u
(
t
)
∥
2
+
ɛ
σ
e
σ
t
∥
f
(
t
)
∥
H
2
,
that is,
(46)
d
d
t
(
e
σ
t
∥
u
(
t
)
∥
2
+
e
σ
t
∥
∇
u
(
t
)
∥
2
)
≤
ɛ
σ
e
σ
t
∥
f
(
t
)
∥
H
2
,
which gives
(47)
∥
u
(
t
)
∥
2
+
∥
∇
u
(
t
)
∥
2
≤
e
-
σ
(
t
-
τ
)
(
∥
u
τ
∥
2
+
∥
∇
u
τ
∥
2
)
+
ɛ
σ
∫
τ
t
‍
e
-
σ
(
t
-
ξ
)
∥
f
(
ξ
)
∥
H
2
d
ξ
for all τ
∈
ℝ.
LetD
^
∈
𝒟
σ be given, choosing appropriate parameter σ, we easily get
(48)
∥
U
(
t
,
τ
,
u
τ
)
∥
V
2
≤
e
-
σ
(
t
-
τ
)
r
D
^
2
+
ɛ
σ
∫
-
∞
t
‍
e
-
σ
(
t
-
ξ
)
∥
f
(
ξ
)
∥
H
2
d
ξ
,
for all u
τ
∈
D
(
τ
),  t
≥
τ.
Settinge
-
σ
(
t
-
τ
)
r
D
^
2
≤
(
ɛ
/
σ
)
∫
-
∞
t
‍
e
-
σ
(
t
-
ξ
)
∥
f
(
ξ
)
∥
H
2
d
ξ, then we denote R
ɛ
(
t
) the nonnegative number given for each t
∈
ℝ by
(49)
(
R
ɛ
(
t
)
)
2
=
2
ɛ
σ
∫
-
∞
t
‍
e
-
η
(
t
-
ξ
)
∥
f
(
ξ
)
∥
H
2
d
ξ
,
and consider the family B
^
ɛ of closed balls in V defined by
(50)
B
ɛ
(
t
)
=
{
v
∈
V
∣
∥
v
∥
V
≤
2
R
ɛ
(
t
)
}
.
It is straightforward to check that B
^
ɛ
∈
𝒟
σ and hence B
^
σ is the 𝒟
σ-pullback absorbing for the process {
U
(
t
,
τ
,
u
τ
)
}.Setting(51)
B
ɛ
=
{
u
∈
V
∣
∥
u
∥
V
≤
R
ɛ
(
t
)
}
,
then we can check that family ℬ
ɛ
=
{
B
ɛ
(
t
)
}
t
∈
ℝ is pullback absorbing in V easily. Moreover,
(52)
lim
⁡
t
→
-
∞
e
η
t
R
ɛ
(
t
)
=
0
for
any
ɛ
>
0
.Lemma 15.
LetR
ɛ
(
t
), B
ɛ
(
t
) be given as above. For any t
∈
ℝ, the solution v
(
t
)
=
U
1
,
ɛ
(
t
,
t
-
τ
)
u
(
t
-
τ
) of (41) satisfies
(53)
∥
U
1
,
ɛ
(
t
,
t
-
τ
)
u
t
-
τ
∥
V
2
≤
e
-
η
τ
R
ɛ
(
t
-
τ
)
,
for all τ
≥
0 and u
t
-
τ
∈
D
ɛ
(
t
-
τ
).Proof.
Multiplying equation in (41) with v and integrating over Ω, we derive
(54)
1
2
d
d
t
(
∥
v
(
t
)
∥
2
+
∥
∇
v
(
t
)
∥
2
)
+
ν
∥
∇
v
∥
2
≤
0
.
Here we use the property of operator B
(
·
) and ℱ
i
(
0
)
=
0 as
(55)
∫
Ω
‍
(
∇
·
F
→
(
u
)
)
u
d
x
=
-
∫
Ω
‍
F
→
(
u
)
·
∇
u
d
x
=
-
∫
Ω
‍
∇
·
ℱ
→
(
u
)
d
x
=
-
∫
∂
Ω
‍
ℱ
→
(
u
)
·
n
→
d
x
=
0
,
where n
→ is the outer unit normal vector.
Using Poincaré’s inequality, it follows(56)
d
d
t
(
∥
v
(
t
)
∥
2
+
∥
∇
v
(
t
)
∥
2
)
+
η
(
∥
v
∥
2
+
∥
∇
v
∥
2
)
≤
0
,
where we set 0
<
η
≤
min
⁡
{
λ
1
ν
,
ν
}.
Integrating (56) from t
-
τ to t, we get
(57)
∥
U
1
,
ɛ
(
t
,
t
-
τ
)
u
t
-
τ
∥
V
2
≤
∥
v
(
t
)
∥
2
+
∥
∇
v
(
t
)
∥
2
≤
(
∥
v
t
-
τ
∥
2
+
∥
∇
v
t
-
τ
∥
2
)
e
-
η
(
t
-
τ
)
≤
e
-
η
τ
R
ɛ
(
t
-
τ
)
for all t
≥
τ, which completes our proof.Lemma 16.
Letℬ
ɛ
(
t
)
=
{
B
ɛ
(
t
)
}
t
∈
ℝ be given by (51) and (52). For any t
∈
ℝ, there exist a time T
ɛ
(
t
,
ℬ
)
>
0 and a function I
ɛ
(
t
)
>
0, such that the solution U
2
,
ɛ
(
t
,
τ
)
u
τ
=
w
(
t
) of (42) satisfies
(58)
∥
U
2
,
ɛ
(
t
,
t
-
τ
)
u
t
-
τ
∥
V
2
≤
I
ɛ
(
t
)
,
for all τ
≥
T
ɛ
(
t
,
ℬ
) and any u
t
-
τ
∈
B
ɛ
(
t
-
τ
).Proof.
Taking the inner product of equation in (42) with A
σ
w
(
t
) in H, we derive
(59)
1
2
d
d
t
(
∥
A
σ
/
2
w
(
t
)
∥
2
+
∥
A
(
σ
+
1
)
/
2
w
(
t
)
∥
2
)
+
ν
∫
Ω
‍
|
A
(
σ
+
1
)
/
2
w
(
t
)
|
2
d
x
=
-
〈
B
(
u
)
,
A
σ
w
〉
+
〈
B
(
v
)
,
A
σ
w
〉
+
ɛ
〈
f
(
t
,
x
)
,
A
σ
w
〉
.
By Poincaré’s inequality, Lemma 15, (51), and (34)–(36), we obtain
(60)
-
〈
B
(
u
)
,
A
σ
w
〉
+
〈
B
(
v
)
,
A
σ
w
〉
≤
|
〈
∇
·
(
F
→
(
v
)
-
F
→
(
u
)
)
,
A
σ
w
〉
|
≤
|
(
sup
⁡
i
=
1,2
,
3
(
|
F
→
i
′
(
v
)
|
+
|
F
→
i
′
u
|
)
×
(
∇
v
-
∇
u
)
,
A
σ
w
sup
⁡
i
=
1,2
,
3
)
|
=
|
(
sup
⁡
i
=
1,2
,
3
(
|
F
→
i
′
(
v
)
|
+
|
F
→
i
′
u
|
)
∇
w
,
A
σ
w
)
|
≤
∥
sup
⁡
i
=
1,2
,
3
(
|
F
→
i
′
(
v
)
|
+
|
F
→
i
′
u
|
)
∥
∥
A
(
σ
+
(
1
/
2
)
)
/
2
w
∥
≤
C
(
ɛ
)
λ
∥
sup
⁡
i
=
1,2
,
3
(
|
F
→
i
′
(
v
)
|
+
|
F
→
i
′
u
|
)
∥
2
+
ν
λ
∥
A
(
σ
+
(
1
/
2
)
)
/
2
w
∥
2
≤
C
(
ɛ
)
λ
∥
sup
⁡
i
=
1,2
,
3
(
|
F
→
i
′
(
v
)
|
+
|
F
→
i
′
u
|
)
∥
2
+
ν
∥
A
(
σ
+
1
)
/
2
w
∥
2
≤
C
λ
(
2
+
σ
2
∥
u
∥
2
+
σ
2
∥
v
∥
2
)
+
ν
∥
A
(
σ
+
1
)
/
2
w
∥
2
≤
C
(
2
+
σ
2
R
ɛ
(
t
)
+
σ
2
e
-
η
τ
R
ɛ
(
t
-
τ
)
)
+
ν
∥
A
(
σ
+
1
)
/
2
w
∥
2
,
〈
f
(
t
,
x
)
,
A
σ
w
〉
≤
1
λ
∥
A
(
σ
+
1
)
/
2
w
(
t
)
∥
2
+
λ
∥
f
(
t
)
∥
H
2
.
Hence according to (59)–(60) and (31), we have
(61)
d
d
t
(
∥
A
σ
/
2
w
(
t
)
∥
2
+
∥
A
(
σ
+
1
)
/
2
w
(
t
)
∥
2
)
≤
C
(
1
+
ɛ
∥
A
(
σ
+
1
)
/
2
w
(
t
)
∥
2
+
ɛ
∥
f
(
t
)
∥
H
2
)
,
where the constant C depends on ∥
u
t
-
τ
∥
V
2, σ, and the first eigenvalue λ of the operator A.
Integrating (61) from t
-
τ to t, we conclude that
(62)
∥
A
σ
/
2
w
(
t
)
∥
2
+
∥
A
(
σ
+
1
)
/
2
w
(
t
)
∥
2
≤
C
e
C
t
∫
t
-
τ
t
‍
(
1
+
ε
2
∥
f
(
x
,
s
)
∥
2
)
e
-
C
s
d
s
=
I
ɛ
(
t
)
for all t
>
τ. This completes the proof of desiring lemma.Lemma 17.
For anyt
∈
ℝ, any τ
>
0, if u
0 varies in bounded sets, then the solution u
ɛ
(
t
)
=
U
ɛ
(
t
,
t
-
τ
)
u
0 of problem (1) converges to the solution u
(
t
)
=
S
(
t
)
u
0 of the unperturbed problem (1) with ɛ
=
0 uniformly in V as ɛ
→
0
+, which means
(63)
lim
⁡
ɛ
→
0
+
sup
⁡
u
0
∈
B
∥
u
ɛ
(
t
)
-
u
(
t
)
∥
V
=
0
,
where B is a bounded subset in V.Proof.
Denote(64)
y
ɛ
(
t
)
=
u
ɛ
(
t
)
-
u
(
t
)
,
then we can verify that y
ɛ
(
t
) satisfies
(65)
y
t
ϵ
+
A
y
t
ɛ
+
ν
A
y
ɛ
=
-
B
(
u
ɛ
)
+
B
(
u
)
+
ɛ
f
(
x
,
t
)
,
(66)
y
ɛ
|
∂
Ω
=
0
,
(67)
y
ɛ
|
t
=
τ
=
(
u
ɛ
)
τ
-
u
τ
.
Multiplying (65) byy
ɛ
(
t
), using (34)–(36) and noting the boundary value condition (66), we have
(68)
1
2
d
d
t
(
∥
y
ɛ
∥
2
+
∥
∇
y
ɛ
∥
2
)
+
ν
∥
∇
y
ɛ
∥
2
=
〈
B
(
u
)
,
y
ɛ
〉
-
〈
B
(
u
ɛ
)
,
y
ɛ
〉
+
〈
ɛ
f
,
y
ɛ
〉
≤
|
〈
B
(
u
ɛ
)
-
B
(
u
)
,
y
ɛ
)
〉
|
+
〈
ɛ
f
,
y
ɛ
〉
≤
|
〈
∇
·
(
F
→
(
u
ɛ
)
-
F
→
(
u
)
)
,
y
ɛ
〉
|
+
ε
2
4
ν
∥
f
(
t
)
∥
H
2
+
λ
ν
2
∥
y
ɛ
∥
2
=
|
〈
F
→
(
u
ɛ
)
-
F
→
(
u
)
,
∇
y
ɛ
〉
|
+
ε
2
4
λ
ν
∥
f
(
t
)
∥
H
2
+
λ
ν
2
∥
y
ɛ
∥
2
.
Using (34)–(36) and the Sobolev compact embedding theorem V
↪
L
6
↪
L
4
↪
L
2, we get
(69)
|
〈
F
→
(
u
ɛ
)
-
F
→
(
u
)
,
∇
y
ɛ
〉
|
≤
|
(
C
1
|
u
ɛ
|
+
C
2
|
u
ɛ
|
2
+
C
1
|
u
|
+
C
2
|
u
|
2
,
∇
y
ɛ
)
|
≤
C
ν
(
∥
u
ɛ
∥
2
+
∥
u
ɛ
∥
L
4
2
+
∥
u
∥
2
+
∥
u
∥
L
4
2
)
+
ν
2
∥
∇
y
ɛ
∥
2
≤
C
λ
ν
(
∥
u
ɛ
∥
V
2
+
∥
u
∥
V
2
)
+
ν
2
∥
∇
y
ɛ
∥
2
.
Hence
(70)
1
2
d
d
t
(
∥
y
ɛ
∥
2
+
∥
∇
y
ɛ
∥
2
)
+
ν
∥
∇
y
ɛ
∥
2
≤
C
λ
ν
(
∥
u
ɛ
∥
V
2
+
∥
u
∥
V
2
)
+
ν
2
∥
∇
y
ɛ
∥
2
+
ε
2
4
λ
ν
∥
f
(
t
)
∥
H
2
+
ν
2
∥
∇
y
ɛ
∥
2
,
that is,
(71)
d
d
t
(
∥
y
ɛ
∥
2
+
∥
∇
y
ɛ
∥
2
)
≤
C
λ
ν
(
∥
u
ɛ
∥
V
2
+
∥
u
∥
V
2
)
+
ε
2
4
λ
ν
∥
f
(
t
)
∥
H
2
.
Using Lemmas 13, 14, 15, 16, and (31), noting that η
=
min
⁡
{
λ
ν
,
ν
,
ν
/
(
(
1
/
λ
)
+
2
)
}, we know
(72)
u
ɛ
,
u
∈
C
(
[
τ
,
+
∞
)
,
V
)
,
∫
t
-
τ
t
‍
(
∥
u
ɛ
(
s
)
∥
V
2
+
∥
u
(
s
)
∥
V
2
)
d
s
≤
C
ɛ
∫
t
-
τ
t
‍
e
-
η
t
∫
-
∞
t
‍
e
η
s
∥
f
(
s
)
∥
H
2
d
s
d
t
+
C
ɛ
∫
t
-
τ
t
‍
e
-
η
t
∫
-
∞
t
-
τ
‍
e
η
s
∥
f
(
s
)
∥
H
2
d
s
d
t
≤
C
ɛ
∬
-
∞
t
e
η
s
∥
f
(
s
)
∥
H
2
d
s
d
t
+
C
ɛ
∬
-
∞
t
e
η
s
∥
f
(
s
)
∥
H
2
d
s
d
t
≤
C
ɛ
.
Applying the Gronwall inequality to (71) and noting that f
∈
L
loc
⁡
2
(
ℝ
,
H
), using Lemmas 14, 15, and 16, we conclude
(73)
∥
y
ɛ
∥
V
2
≤
C
(
∥
y
ɛ
∥
2
+
∥
∇
y
ɛ
∥
2
)
≤
C
ɛ
[
∫
t
-
τ
t
‍
(
∥
u
ɛ
(
s
)
∥
V
2
+
∥
u
(
s
)
∥
V
2
)
d
s
+
∫
t
-
τ
t
‍
∥
f
(
s
)
∥
H
2
d
s
]
≤
C
ɛ
[
∫
t
-
τ
t
‍
e
-
η
t
∫
-
∞
t
‍
e
η
s
∥
f
(
s
)
∥
H
2
d
s
d
t
+
∫
t
-
τ
t
‍
e
-
η
t
∫
-
∞
t
-
τ
‍
e
η
s
∥
f
(
s
)
∥
H
2
d
s
d
t
]
+
C
ε
2
[
∫
t
-
τ
t
‍
∥
f
(
s
)
∥
H
2
d
s
]
≤
C
′
ɛ
⟶
0
as ɛ
→
0
+, which implies (63).Proof of Theorem11.
Since the embeddingD
(
A
s
/
2
)
↪
(
L
6
/
(
3
-
2
s
)
(
Ω
)
)
3 is compact, combining Lemmas 13–17 with Theorem 9 and Lemma 10, we can obtain Theorem 11 easily.



---

*Source: 853139-2014-03-26.xml*