# Pullback Attractors for a Nonautonomous Retarded Degenerate Parabolic Equation

Authors: Fahe Miao; Hui Liu; Jie Xin
Journal: Discrete Dynamics in Nature and Society
 (2020)

---

## Abstract

This paper is devoted to a nonautonomous retarded degenerate parabolic equation. We first show the existence and uniqueness of a weak solution for the equation by using the standard Galerkin method. Then we establish the existence of pullback attractors for the equation by proving the existence of compact pullback absorbing sets and the pullback asymptotic compactness.


---

## Body

## 1. Introduction

In this paper, we study the following nonautonomous retarded degenerate parabolic equation defined on an arbitrary domain (bounded or unbounded)D
N
⊂
R
N
,
N
≥
2:(1)
∂
u
∂
t
+
λ
u
−
div
σ
x
∇
u
=
f
u
t
+
g
x
,
t
,
x
∈
D
N
,
t
>
0
,with the initial value and boundary condition(2)
u
x
,
t
=
u
0
x
,
t
,
u
x
,
t
∂
D
N
=
0
,
x
∈
D
N
,
t
∈
−
ν
,
0
,whereλ is a fixed positive constant, and we suppose that σ
:
D
N
→
R satisfies the following assumptions:H
α: when D
N is bounded, we suppose that σ
∈
L
l
o
c
1
D
N and lim
inf
x
→
z
|
x
−
z
|
−
α
σ
x
>
0 for some α
∈
0
,
2 and each z
∈
D
N
¯;H
α
β: when D
N is unbounded, we suppose that σ satisfies condition H
α and lim
inf
x
→
∞
x
−
β
σ
x
>
0 for some β
>
2.Forν
>
0, we denote by F the Banach space which is consisted of all continuous functions ξ
:
−
ν
,
0
→
L
2
D
N endowed with norm ξ
F
=
sup
s
∈
−
ν
,
0
ξ
s. For all real number a
≤
b, t
∈
a
,
b, and any continuous function u
:
a
−
ν
,
b
→
L
2
D
N, we denote by u
t the element of F, and u
t
s
=
u
t
+
s for s
∈
−
ν
,
0.f is a nonlinear function satisfying the following conditions:(i)
f
0
=
0;
(ii)
there exists a positive continuous functionl
f
r with(3)
lim
r
→
∞
l
f
r
r
k
0
=
0
,for some positive integersk
0 such that if ξ
,
η
∈
F, ξ
≤
r and η
≤
r, then we have(4)
f
ξ
−
f
η
≤
l
f
r
ξ
−
η
F
;
(iii)
there exists a positive constantc
f such that, for any t
>
0
,
u
∈
C
−
ν
,
t
;
L
2
D
N, and x
∈
D
N,(5)
∫
0
t
f
u
s
x
2
d
s
≤
c
f
2
∫
−
ν
t
u
s
x
2
d
s
;(iv)
λ
>
c
f.g is a given function satisfying(6)
g
∈
L
l
o
c
2
R
;
L
2
D
N
.It is well known that problem (1) can be seen as a simple model for neutron diffusion (see [1]). For this case, u and σ represent the neutron flux and neutron diffusion, respectively.As we know, the most interesting problem about the partial differential equations is to investigate the asymptotic behavior of the solutions when time tends to infinity. In addition, we can get the useful information about the future of the model from it. Besides, when we want to model some phenomena arising in physics, chemistry, biology and other fields, some hereditary variables such as aftereffect, time-lag, and time delay can be added in the variables. For example, the stochastic retarded reaction-diffusion equation on unbounded domains was considered in [2]. For this kind of equation with the delay term, the reader is referred to [2–4] and the references therein.The theory of pullback attractors is a good choice to investigate the long time behavior of evolution systems. Anh and Bao [5] proved the existence of a pullback attractor for a nonautonomous semi-linear degenerate parabolic equation. Cui and Li [6] investigated the existence and upper semicontinuity of random attractors for stochastic degenerate parabolic equations with multiplicative noises. For nonautonomous stochastic PDEs, the existence of attractors and well-posedness of the problem have been studied in [7–15]. In addition, the existence of a random attractor is generally ensured by the well-posedness (i.e., existence and uniqueness) of the problem combined with a compact absorbing set. When we want to get the compactness of the absorbing sets and get the compactness of the system, Sobolev compact embedding is a good choice, see, e.g., [5, 16].Motivated by [5] and [2], in this article, we devote to study a nonautonomous retarded degenerate parabolic equation. In fact, as we can see, the term of delay makes the phase space not reflexive and the uniform estimates of solutions will be difficult. The main result of the paper is Theorem 2.12.The rest of this paper consists of the following contents. For convenience, in Section2, we first introduce some preliminaries about function spaces and operators as well as dynamical systems and pullback attractors. Then, in Section 3, we get the existence and uniqueness of a weak solution for the problem (1) by using the standard Galerkin method. Section 4 is devoted to prove the existence of the pullback attractors.


## 2. Preliminaries

### 2.1. Function Spaces and Operators

We recall some basic concepts related to the function spaces and operators which we will use, the reader is referred to [5, 7, 16] for more details.In order to study problem (1), we define the space D
0
1
,
2
D
N
,
σ as the closure of C
0
∞
D
N with respect to the norm(7)
u
D
0
1
,
2
D
N
,
σ
≔
∫
D
N
σ
x
∇
u
2
d
x
1
/
2
<
∞
.This space is a Hilbert space with respect to the inner product(8)
u
,
v
σ
≔
∫
D
N
σ
x
∇
u
∇
v
d
x
.We denote by·, ·
,
·, ·
σ, ·
,
·
σ the norms and inner products in L
2
D
N and D
0
1
,
2
D
N
,
σ, respectively, and ·
F the norm in F. And also, define(9)
ξ
σ
,
F
=
sup
s
∈
−
ν
,
0
ξ
s
σ
.Letc be a normal positive number which might vary from line to line. We have the following lemma which is a generalized version of the Poincaré inequality in [5].Lemma 2.1.
Suppose thatD
N is a bounded (unbounded) domain of R
N
,
N
≥
2 and suppose that the condition H
α (H
α
β) is satisfied. Then there is a positive constant c such that(10)
∫
D
N
u
2
d
x
≤
c
∫
D
N
σ
x
∇
u
2
d
x
,
for
every
u
∈
C
0
∞
D
N
.The following results can be found in [16].Remark 2.2.
LetN
≥
2
,
α
∈
0
,
2, and(11)
2
α
∗
=
4
α
∈
2
,
∞
if
N
=
2
,
2
N
N
−
2
+
α
∈
2
,
2
N
N
−
2
if
N
≥
3
.
The given exponent2
α
∗ has the role of the critical exponent in the classical Sobolev embedding.Lemma 2.3.
Suppose thatD
N is a bounded domain in R
N
,
N
≥
2 and σ satisfies (H
α). We get the following embedding:(1)
D
0
1
,
2
D
N
,
σ
↪
L
2
α
∗
D
N continuously,
(2)
D
0
1
,
2
D
N
,
σ
↪
L
p
D
N compactly for every p
∈
1
,
2
α
∗.Lemma 2.4.
Suppose thatD
N is an unbounded domain in R
N
,
N
≥
2 and σ satisfies (H
α
β). We get the following embedding:(1)
D
0
1
,
2
D
N
,
σ
↪
L
p
D
N continuously, if p
∈
2
β
∗
,
2
α
∗,
(2)
D
0
1
,
2
D
N
,
σ
↪
L
p
D
N compactly, if p
∈
2
β
∗
,
2
α
∗.We now investigate the case whereD
N is a bounded domain (the unbounded case is considered similarly).We define a linear operator determined by the leading term in (1)(12)
A
u
=
−
div
σ
x
∇
u
.Under the conditionsH
α or H
α
β, the operator A is a positive and self-adjoint linear operator with the domain of definition(13)
D
A
=
u
∈
D
0
1
,
2
D
N
,
σ
:
A
u
∈
L
2
D
N
.The eigenvectors of the operatorA construct the complete orthonormal family e
j
j
=
1
∞ in L
2
D
N and the relevant spectrum is discrete and represented by λ
j
j
=
1
∞ such that(14)
e
i
,
e
j
L
2
=
δ
i
j
,
and
A
e
j
=
λ
j
e
j
,
i
,
j
=
1
,
2
,
…
,and(15)
0
<
λ
1
<
λ
2
<
⋯
,
λ
j
→
+
∞
as
j
→
+
∞
.Furthermore(16)
λ
1
=
inf
u
σ
2
u
2
:
u
∈
D
0
1
,
2
D
N
,
σ
,
u
≠
0
.Noting that(17)
u
σ
2
≥
λ
1
u
2
,
for
all
u
∈
D
0
1
,
2
D
N
,
σ
.DefineD
−
1
D
N
,
σ as the conjugate space of the space D
0
1
,
2
D
N
,
σ. Since 2
α
∗
>
2, we get an evolution triple(18)
D
0
1
,
2
D
N
,
σ
↪
L
2
D
N
↪
D
−
1
D
N
,
σ
.


### 2.2. Dynamical Systems and Pullback Attractors

Next, we will show some theories about the dynamical systems and pullback attractors, the following contents can be found in [2, 5, 7].LetΩ be a nonempty set and X be a metric space endowed with metric d. For any A
,
B
⊂
X, define the Hausdorff semidistance between A and B
(19)
dist
A
,
B
=
sup
x
∈
A
inf
y
∈
B
d
x
,
y
.Definition 2.5.
A family of mappingθ
t
t
∈
R from Ω to itself is said to be a family of shift operates on Ω if for θ
t
t
∈
R, the following two group properties are satisfied:(i)
θ
0
ω
=
ω, for any ω
∈
Ω,
(ii)
θ
t
θ
τ
ω
=
θ
t
+
τ
ω, for any ω
∈
Ω and t
,
τ
∈
R.Definition 2.6.
Suppose thatθ
t
t
∈
R is a family of shift operators on Ω. Then a mapping on X
(20)
ϕ
:
R
+
×
Ω
×
X
→
X
,
t
,
ω
,
x
→
ϕ
t
,
ω
,
x
,
is called a continuousθ-cocycle if for any ω
∈
Ω and t
,
τ
∈
R, it satisfies:(i)
ϕ
0
,
ω
,
· is the identity on X,
(ii)
ϕ
t
+
τ
,
ω
,
·
=
ϕ
t
,
θ
τ
ω
,
·
∘
ϕ
τ
,
ω
,
·,
(iii)
ϕ
t
,
ω
,
·
:
X
→
X is continuous.
In what follows, we will always suppose thatϕ is a continuous θ-cocycle on X and D is a collection of families of subsets of X:(21)
D
=
D
=
D
ω
ω
∈
Ω
:
D
ω
⊂
X
for
every
ω
∈
Ω
.Definition 2.7.
Suppose thatD is a collection of families of subsets of X. Then D is said to be inclusion closed if D
=
D
ω
ω
∈
Ω
∈
D and D
∼
=
D
∼
ω
⊂
X
:
ω
∈
Ω with D
∼
ω
⊂
D
ω for any ω
∈
Ω imply that D
∼
∈
D.Definition 2.8.
Suppose thatD is a collection of families of subsets of X. Then K
ω is said to be a D-pullback absorbing set for ϕ if for any B
∈
D and ω
∈
Ω, there is t
ω
,
B
>
0 such that(22)
ϕ
t
,
θ
−
t
ω
,
B
θ
−
t
ω
⊂
K
ω
,
∀
t
≥
t
ω
,
B
.Definition 2.9.
Suppose thatD is a collection of families of subsets of X. Then ϕ is called D-pullback asymptotically compact in X if for all ω
∈
Ω, ϕ
t
n
,
θ
−
t
n
ω
,
x
n possesses a convergent sub-sequence in X whenever t
n
→
∞ and x
n
∈
B
θ
−
t
n
ω with B
ω
ω
∈
Ω
∈
D.Definition 2.10.
Suppose thatD is a collection of families of subsets of X and A
ω
ω
∈
Ω
∈
D. Then A
ω
ω
∈
Ω
∈
D is said to be a D-pullback attractor for ϕ if for any ω
∈
Ω:(i)
A
ω is compact,
(ii)
A
ω
ω
∈
Ω is invariant, that is,(23)
ϕ
t
,
ω
,
A
ω
=
A
θ
t
ω
,
∀
t
≥
0
,(iii)
A
ω
ω
∈
Ω attracts every set in D, that is, for any B
=
B
ω
ω
∈
Ω
∈
D,(24)
lim
t
→
∞
d
ϕ
t
,
θ
−
t
,
B
θ
−
t
ω
,
A
ω
=
0
.Proposition 2.11.
Suppose thatD is an inclusion-closed collection of families of subsets of X and ϕ is a continuous θ-cocycle in X. Assume that K
ω
ω
Ω
∈
D is a closed absorbing set for ϕ in D and ϕ is D-pullback asymptotically compact in X. Then ϕ possesses a unique D-pullback attractor A
ω
ω
∈
Ω
∈
D such that(25)
A
ω
=
⋂
τ
≥
0
⋃
t
≥
τ
ϕ
t
,
θ
−
t
ω
,
K
θ
−
t
ω
¯
.In this article, letD be the collection of families of subsets of F. Then, we will give the main theorem of this article.Theorem 2.12.
(Main Theorem) Assume that (i)–(iv) hold and (6) satisfies. Then the problem (1) and (2) has a D-pullback attractor A
ω
ω
∈
Ω.


## 2.1. Function Spaces and Operators

We recall some basic concepts related to the function spaces and operators which we will use, the reader is referred to [5, 7, 16] for more details.In order to study problem (1), we define the space D
0
1
,
2
D
N
,
σ as the closure of C
0
∞
D
N with respect to the norm(7)
u
D
0
1
,
2
D
N
,
σ
≔
∫
D
N
σ
x
∇
u
2
d
x
1
/
2
<
∞
.This space is a Hilbert space with respect to the inner product(8)
u
,
v
σ
≔
∫
D
N
σ
x
∇
u
∇
v
d
x
.We denote by·, ·
,
·, ·
σ, ·
,
·
σ the norms and inner products in L
2
D
N and D
0
1
,
2
D
N
,
σ, respectively, and ·
F the norm in F. And also, define(9)
ξ
σ
,
F
=
sup
s
∈
−
ν
,
0
ξ
s
σ
.Letc be a normal positive number which might vary from line to line. We have the following lemma which is a generalized version of the Poincaré inequality in [5].Lemma 2.1.
Suppose thatD
N is a bounded (unbounded) domain of R
N
,
N
≥
2 and suppose that the condition H
α (H
α
β) is satisfied. Then there is a positive constant c such that(10)
∫
D
N
u
2
d
x
≤
c
∫
D
N
σ
x
∇
u
2
d
x
,
for
every
u
∈
C
0
∞
D
N
.The following results can be found in [16].Remark 2.2.
LetN
≥
2
,
α
∈
0
,
2, and(11)
2
α
∗
=
4
α
∈
2
,
∞
if
N
=
2
,
2
N
N
−
2
+
α
∈
2
,
2
N
N
−
2
if
N
≥
3
.
The given exponent2
α
∗ has the role of the critical exponent in the classical Sobolev embedding.Lemma 2.3.
Suppose thatD
N is a bounded domain in R
N
,
N
≥
2 and σ satisfies (H
α). We get the following embedding:(1)
D
0
1
,
2
D
N
,
σ
↪
L
2
α
∗
D
N continuously,
(2)
D
0
1
,
2
D
N
,
σ
↪
L
p
D
N compactly for every p
∈
1
,
2
α
∗.Lemma 2.4.
Suppose thatD
N is an unbounded domain in R
N
,
N
≥
2 and σ satisfies (H
α
β). We get the following embedding:(1)
D
0
1
,
2
D
N
,
σ
↪
L
p
D
N continuously, if p
∈
2
β
∗
,
2
α
∗,
(2)
D
0
1
,
2
D
N
,
σ
↪
L
p
D
N compactly, if p
∈
2
β
∗
,
2
α
∗.We now investigate the case whereD
N is a bounded domain (the unbounded case is considered similarly).We define a linear operator determined by the leading term in (1)(12)
A
u
=
−
div
σ
x
∇
u
.Under the conditionsH
α or H
α
β, the operator A is a positive and self-adjoint linear operator with the domain of definition(13)
D
A
=
u
∈
D
0
1
,
2
D
N
,
σ
:
A
u
∈
L
2
D
N
.The eigenvectors of the operatorA construct the complete orthonormal family e
j
j
=
1
∞ in L
2
D
N and the relevant spectrum is discrete and represented by λ
j
j
=
1
∞ such that(14)
e
i
,
e
j
L
2
=
δ
i
j
,
and
A
e
j
=
λ
j
e
j
,
i
,
j
=
1
,
2
,
…
,and(15)
0
<
λ
1
<
λ
2
<
⋯
,
λ
j
→
+
∞
as
j
→
+
∞
.Furthermore(16)
λ
1
=
inf
u
σ
2
u
2
:
u
∈
D
0
1
,
2
D
N
,
σ
,
u
≠
0
.Noting that(17)
u
σ
2
≥
λ
1
u
2
,
for
all
u
∈
D
0
1
,
2
D
N
,
σ
.DefineD
−
1
D
N
,
σ as the conjugate space of the space D
0
1
,
2
D
N
,
σ. Since 2
α
∗
>
2, we get an evolution triple(18)
D
0
1
,
2
D
N
,
σ
↪
L
2
D
N
↪
D
−
1
D
N
,
σ
.


## 2.2. Dynamical Systems and Pullback Attractors

Next, we will show some theories about the dynamical systems and pullback attractors, the following contents can be found in [2, 5, 7].LetΩ be a nonempty set and X be a metric space endowed with metric d. For any A
,
B
⊂
X, define the Hausdorff semidistance between A and B
(19)
dist
A
,
B
=
sup
x
∈
A
inf
y
∈
B
d
x
,
y
.Definition 2.5.
A family of mappingθ
t
t
∈
R from Ω to itself is said to be a family of shift operates on Ω if for θ
t
t
∈
R, the following two group properties are satisfied:(i)
θ
0
ω
=
ω, for any ω
∈
Ω,
(ii)
θ
t
θ
τ
ω
=
θ
t
+
τ
ω, for any ω
∈
Ω and t
,
τ
∈
R.Definition 2.6.
Suppose thatθ
t
t
∈
R is a family of shift operators on Ω. Then a mapping on X
(20)
ϕ
:
R
+
×
Ω
×
X
→
X
,
t
,
ω
,
x
→
ϕ
t
,
ω
,
x
,
is called a continuousθ-cocycle if for any ω
∈
Ω and t
,
τ
∈
R, it satisfies:(i)
ϕ
0
,
ω
,
· is the identity on X,
(ii)
ϕ
t
+
τ
,
ω
,
·
=
ϕ
t
,
θ
τ
ω
,
·
∘
ϕ
τ
,
ω
,
·,
(iii)
ϕ
t
,
ω
,
·
:
X
→
X is continuous.
In what follows, we will always suppose thatϕ is a continuous θ-cocycle on X and D is a collection of families of subsets of X:(21)
D
=
D
=
D
ω
ω
∈
Ω
:
D
ω
⊂
X
for
every
ω
∈
Ω
.Definition 2.7.
Suppose thatD is a collection of families of subsets of X. Then D is said to be inclusion closed if D
=
D
ω
ω
∈
Ω
∈
D and D
∼
=
D
∼
ω
⊂
X
:
ω
∈
Ω with D
∼
ω
⊂
D
ω for any ω
∈
Ω imply that D
∼
∈
D.Definition 2.8.
Suppose thatD is a collection of families of subsets of X. Then K
ω is said to be a D-pullback absorbing set for ϕ if for any B
∈
D and ω
∈
Ω, there is t
ω
,
B
>
0 such that(22)
ϕ
t
,
θ
−
t
ω
,
B
θ
−
t
ω
⊂
K
ω
,
∀
t
≥
t
ω
,
B
.Definition 2.9.
Suppose thatD is a collection of families of subsets of X. Then ϕ is called D-pullback asymptotically compact in X if for all ω
∈
Ω, ϕ
t
n
,
θ
−
t
n
ω
,
x
n possesses a convergent sub-sequence in X whenever t
n
→
∞ and x
n
∈
B
θ
−
t
n
ω with B
ω
ω
∈
Ω
∈
D.Definition 2.10.
Suppose thatD is a collection of families of subsets of X and A
ω
ω
∈
Ω
∈
D. Then A
ω
ω
∈
Ω
∈
D is said to be a D-pullback attractor for ϕ if for any ω
∈
Ω:(i)
A
ω is compact,
(ii)
A
ω
ω
∈
Ω is invariant, that is,(23)
ϕ
t
,
ω
,
A
ω
=
A
θ
t
ω
,
∀
t
≥
0
,(iii)
A
ω
ω
∈
Ω attracts every set in D, that is, for any B
=
B
ω
ω
∈
Ω
∈
D,(24)
lim
t
→
∞
d
ϕ
t
,
θ
−
t
,
B
θ
−
t
ω
,
A
ω
=
0
.Proposition 2.11.
Suppose thatD is an inclusion-closed collection of families of subsets of X and ϕ is a continuous θ-cocycle in X. Assume that K
ω
ω
Ω
∈
D is a closed absorbing set for ϕ in D and ϕ is D-pullback asymptotically compact in X. Then ϕ possesses a unique D-pullback attractor A
ω
ω
∈
Ω
∈
D such that(25)
A
ω
=
⋂
τ
≥
0
⋃
t
≥
τ
ϕ
t
,
θ
−
t
ω
,
K
θ
−
t
ω
¯
.In this article, letD be the collection of families of subsets of F. Then, we will give the main theorem of this article.Theorem 2.12.
(Main Theorem) Assume that (i)–(iv) hold and (6) satisfies. Then the problem (1) and (2) has a D-pullback attractor A
ω
ω
∈
Ω.


## 3. Existence and Uniqueness of the Solution

In this section, we will prove the existence and uniqueness of the solution for problem (1) and (2).Theorem 3.1.
Suppose thatf satisfies (i)–(iv) and g
∈
L
l
o
c
2
R
;
L
2
D
N, then for any given T
>
0, u
0
∈
F, the problem (1) and (2) has a unique weak solution(26)
u
∈
L
∞
0
,
T
;
L
2
D
N
∩
L
2
−
ν
,
T
;
L
2
D
N
∩
L
2
0
,
T
;
D
0
1
,
2
D
N
,
σ
,
and the equation is satisfied in the sense of distribution.Proof.
Consider the approximating solutionu
n
t in the form(27)
u
n
t
=
∑
k
=
1
n
u
n
k
t
e
k
,
wheree
j
j
=
1
∞ are eigenvectors of the linear operator A. We have the following equations for u
n
(28)
∂
u
n
t
∂
t
,
e
k
+
λ
u
n
t
,
e
k
+
A
u
n
t
,
e
k
=
f
u
n
t
,
e
k
+
g
x
,
t
,
e
k
,
(29)
u
n
t
,
e
k
=
u
n
0
t
,
e
k
,
t
∈
−
ν
,
0
,
k
=
1
,
…
,
n
.
Hence, we get the local existence ofu
n. We now give some priori estimates for u
n. We have(30)
1
2
d
d
t
u
n
2
+
λ
u
n
2
+
u
n
σ
2
=
∫
D
N
f
u
n
t
u
n
d
x
+
∫
D
N
g
x
,
t
u
n
d
x
.
By the Young inequality, we get(31)
∫
D
N
f
u
n
t
u
n
d
x
≤
c
f
2
u
n
2
+
1
2
c
f
f
u
n
t
2
,
∫
D
N
g
x
,
t
u
n
d
x
≤
1
2
α
g
x
,
t
2
+
α
2
u
n
t
2
,
wherec
f is given by (5), α is a positive constant. By the condition (iv) of f, we can choose α small enough such that 2
λ
>
2
c
f
+
α, then(32)
d
d
t
u
n
2
+
2
λ
−
c
f
−
α
u
n
2
+
2
u
n
σ
2
≤
1
α
g
x
,
t
2
+
1
c
f
f
u
n
t
2
.
Integrating (32) on 0
,
t, 0
≤
t
≤
T, then from the conditions (iii) and (iv) of f, we can get(33)
u
n
t
2
+
2
λ
−
c
f
−
α
∫
0
t
u
n
s
2
d
s
+
2
∫
0
t
u
n
s
σ
2
d
s
≤
u
n
0
2
+
1
α
∫
0
t
g
x
,
s
2
d
s
+
c
f
∫
−
ν
t
u
n
s
2
d
s
=
u
n
0
2
+
1
α
∫
0
t
g
x
,
s
2
d
s
+
c
f
∫
−
ν
0
u
n
s
2
d
s
+
c
f
∫
0
t
u
n
s
2
d
s
≤
u
n
0
2
+
c
f
ν
u
n
0
F
2
+
1
α
∫
0
t
g
x
,
s
2
d
s
+
c
f
∫
0
t
u
n
s
2
d
s
.
Noting that2
λ
>
2
c
f
+
α, we obtain(34)
u
n
t
2
+
2
∫
0
t
u
n
s
σ
2
d
s
≤
u
n
0
2
+
c
f
ν
u
n
0
F
2
+
1
α
∫
0
t
g
x
,
s
2
d
s
.
By (34), we also have(35)
u
n
t
2
≤
1
+
c
f
ν
u
n
0
F
2
+
1
α
∫
0
t
g
x
,
s
2
d
s
,
for fixedρ
∈
−
ν
,
0, we obtain that, for t
∈
−
ρ
,
T,(36)
u
n
t
+
ρ
2
≤
1
+
c
f
ν
u
n
0
F
2
+
1
α
∫
0
t
+
ρ
g
x
,
s
2
d
s
,
and fort
∈
0
,
−
ρ,(37)
u
n
t
+
ρ
2
≤
u
n
0
F
2
≤
1
+
c
f
ν
u
n
0
F
2
.
Hence, from (36) and (38), we get for all t
∈
0
,
T
(38)
u
n
t
F
2
≤
1
+
c
f
ν
u
n
0
F
2
+
1
α
∫
0
T
g
x
,
s
2
d
s
,
The (34) and (38) imply that for each u
n
0
∈
F,(39)
u
n
is
bounded
in
L
∞
0
,
T
;
L
2
D
N
,
(40)
u
n
is
bounded
in
L
2
−
ν
,
T
;
L
2
D
N
,
(41)
u
n
is
bounded
in
L
2
0
,
T
;
D
0
1
,
2
D
N
,
σ
.
Noting that from the condition off, we conclude f
u
n
t is bounded in L
2
0
,
T
;
L
2
D
N
, then we get(42)
f
u
n
t
⇀
ξ
in
L
2
0
,
T
;
L
2
D
N
.
Hence we get(43)
u
n
⇀
u
in
L
2
0
,
T
;
D
0
1
,
2
D
N
,
σ
,
(44)
f
u
n
t
⇀
ξ
in
L
2
0
,
T
;
L
2
D
N
,
(45)
A
u
n
⇀
A
u
in
L
2
0
,
T
;
D
−
1
D
N
,
σ
.
We also have the following equation which has another form(46)
∂
u
n
∂
t
=
−
λ
u
n
−
A
u
n
+
f
u
n
t
+
g
,
we obtain that∂
u
n
/
∂
t is bounded in L
2
0
,
T
;
D
−
1
D
N
,
σ
⋃
L
2
D
N. We also see the following triple(47)
D
0
1
,
2
D
N
,
σ
⊂
⊂
L
2
D
N
⊂
D
−
1
D
N
,
σ
∪
L
2
D
N
,
applying the compactness lemma we can suppose thatu
n
→
u strongly in L
2
−
ν
,
T
;
L
2
D
N. Thus u
n
→
u
a
.
e
. in D
N
×
−
ν
,
T and u
n
t
→
u
t
a
.
e
. in D
N
×
0
,
T. Since f is continuous, f
u
n
t
→
f
u
t
a
.
e
. in D
N
×
0
,
T, we get(48)
f
u
n
t
→
f
u
t
in
L
2
0
,
T
;
L
2
D
N
.
Therefore by (46) we get(49)
∂
u
∂
t
=
−
λ
u
−
A
u
+
f
u
t
+
g
in
L
2
0
,
T
;
D
−
1
D
N
,
σ
∪
L
2
D
N
.
By a standard argument, and using the strong convergenceu
n
→
u in C
0
,
T
;
L
2
D
N and the Doubinskii’s theorem [4], we get that any weak-∗limit is a solution of problem (1) subject to the initial conditions. Hence we obtain the existence of the solution.
Next we investigate the uniqueness of the solution. Letu
1
,
u
2 be the two solutions of problem (1) with the initial value u
1
0
,
u
2
0, respectively. Thus from (1) we get(50)
∂
∂
t
u
1
−
u
2
+
λ
u
1
−
u
2
+
A
u
1
−
A
u
2
=
f
u
1
t
−
f
u
2
t
.
By the condition (ii) off we conclude that(51)
1
2
∂
∂
t
u
1
−
u
2
2
≤
f
u
1
t
−
f
u
2
t
u
1
−
u
2
≤
l
f
r
T
u
1
t
−
u
2
t
F
u
1
−
u
2
≤
l
f
r
T
u
1
t
−
u
2
t
F
2
.
Integrating on0
,
t, 0
<
t
≤
T,(52)
u
1
t
−
u
2
t
2
≤
u
1
0
−
u
2
0
2
+
l
f
r
T
∫
0
t
u
1
s
−
u
2
s
F
2
d
s
.
Therefore, for fixedρ
∈
−
ν
,
0, we have that, for t
∈
−
ρ
,
T,(53)
u
1
t
+
ρ
−
u
2
t
+
ρ
2
≤
u
1
0
−
u
2
0
2
+
l
f
r
T
∫
0
t
+
ρ
u
1
s
−
u
2
s
2
d
s
≤
u
1
0
−
u
2
0
F
2
+
+
l
f
r
T
∫
0
t
u
1
s
−
u
2
s
F
2
d
s
,
fort
∈
0
,
−
ρ,(54)
u
1
t
+
ρ
−
u
2
t
+
ρ
2
≤
u
1
0
−
u
2
0
F
2
.
Thus, from (53) and (54) we get that, for all t
∈
0
,
T,(55)
u
1
t
−
u
2
t
F
2
≤
u
1
0
−
u
2
0
F
2
+
l
f
r
T
∫
0
t
u
1
s
−
u
2
s
F
2
d
s
.
By using the Gronwall inequality we obtain that, for allt
∈
0
,
T,(56)
u
1
t
−
u
2
t
F
2
≤
u
1
0
−
u
2
0
F
2
1
+
l
f
r
T
t
e
l
f
r
T
t
.
Hence we obtain the uniqueness (ifu
1
0
=
u
2
0) of the solution.


## 4. Existence of Pullback Attractors

In this section, we will prove the existence ofD-pullback attractors of this nonautonomous retarded degenerate parabolic equation by proving the existence of the D-pullback absorbing sets and the D-pullback asymptotic compactness.We will first construct aθ-cocycle for the nonautonomous retarded degenerate parabolic equation defined on D
N. As in the case of bounded domains, the Theorem 3.1 implies that the problem (1) and (2) is well defined. In order to construct a cocycle ϕ for the problem (1) and (2), let Ω
=
D
N and define a shift operator θ
t on Ω, for any t
∈
R,(57)
θ
t
τ
=
t
+
τ
,
∀
τ
∈
R
.Then we can defineϕ as a mapping from R
+
×
Ω
×
F to F such that(58)
ϕ
t
,
τ
,
u
τ
=
u
t
+
τ
,
τ
,
u
τ
,wheret
≥
0, τ
∈
R, u
τ
∈
F, and u is the solution of the problem (1) and (2). It is not difficult to see that ϕ is a continuous θ-cocycle on F.Then, we will derive uniform estimates on the weak solutions of problem (1) and (2) when t
→
∞, which is essential to prove the existence of a bounded pullback absorbing set and the pullback asymptotic compactness for ϕ.Lemma 4.1.
Suppose that (i)–(iv) hold and (6) satisfies. Then there exists T
>
0 such that for any t
≥
T,(59)
u
t
F
2
≤
K
1
,
which implies the existence ofD-pullback absorbing set in F.Proof.
Taking the inner product of (1) with u over D
N, we have(60)
1
2
∂
∂
t
u
2
+
λ
u
2
+
u
σ
2
+
∫
D
N
f
u
t
u
d
x
=
∫
D
N
g
x
,
t
u
d
x
.
By using the Young inequality and replacingu
n with u in (35) and (38), we can get for all t
∈
0
,
T,(61)
u
t
2
≤
1
+
c
f
ν
u
0
F
2
+
1
α
∫
0
t
g
x
,
s
2
d
s
,
(62)
u
t
F
2
≤
1
+
c
f
ν
u
0
F
2
+
1
α
∫
0
T
g
x
,
s
2
d
s
.
Sinceg
∈
L
l
o
c
2
R
;
L
2
D
N, let K
1 be(63)
K
1
=
1
+
c
f
ν
u
0
F
2
+
1
α
∫
0
T
g
x
,
s
2
d
s
,
then for anyt
≥
T, K
1 is bounded. Hence we get the lemma immediately.Lemma 4.2.
Suppose that (i)–(iv) hold and (6) satisfies. Then for any t
≥
T,(64)
∫
0
t
+
1
u
s
σ
2
d
s
≤
K
2
.Proof.
Noting that from (32) we obtain that(65)
d
d
t
u
2
+
2
λ
−
c
f
−
α
u
2
+
2
u
σ
2
≤
1
α
g
x
,
t
2
+
1
c
f
f
u
t
2
.
Integrating (65) over the interval t
,
t
+
1, we get that(66)
u
t
+
1
2
−
u
t
2
+
2
λ
−
c
f
−
α
∫
t
t
+
1
u
s
2
d
s
+
2
∫
t
t
+
1
u
s
σ
2
d
s
≤
1
α
∫
t
t
+
1
g
x
,
s
2
+
1
c
f
∫
t
t
+
1
f
u
s
2
d
s
.
It follows from the Young inequality and (iii) off that(67)
1
c
f
∫
t
t
+
1
f
u
s
2
d
s
≤
c
f
∫
t
−
ν
t
+
1
u
s
2
d
s
≤
c
f
∫
t
−
ν
t
u
s
2
d
s
+
c
f
∫
t
t
+
1
u
s
2
d
s
.
By (66) and (67), we find that(68)
u
t
+
1
2
+
2
λ
−
2
c
f
−
α
∫
t
t
+
1
u
s
2
d
s
+
2
∫
t
t
+
1
u
s
σ
2
d
s
≤
u
t
2
+
1
α
∫
t
t
+
1
g
x
,
s
2
+
c
f
∫
t
−
ν
t
u
s
2
d
s
.
Thus, thanks to2
λ
>
2
c
f
+
α, we have that(69)
2
∫
t
t
+
1
u
s
σ
2
d
s
≤
u
t
2
+
1
α
∫
t
t
+
1
g
x
,
s
2
+
c
f
∫
t
−
ν
t
u
s
2
d
s
≤
1
+
c
f
ν
u
t
F
2
+
1
α
∫
t
t
+
1
g
x
,
s
2
≤
1
+
c
f
ν
K
1
+
1
α
∫
t
t
+
1
g
x
,
s
2
.
Therefore,(70)
∫
t
t
+
1
u
s
σ
2
d
s
≤
1
+
c
f
ν
2
K
1
+
1
2
α
∫
t
t
+
1
g
x
,
s
2
≔
K
2
,
whereK
1 is given by (59). It follows from (6) and Lemma 4.1 we find that K
2 is bounded. Thus we finish the proof of the Lemma 4.2.Lemma 4.3.
Suppose that (i)–(iv) hold and (6) satisfies. Then for any t
≥
T
+
ν
+
1 and ρ
1
,
ρ
2
∈
−
ν
,
0,(71)
u
t
σ
2
≤
K
3
,
(72)
∫
t
+
ρ
1
t
+
ρ
2
A
u
s
2
d
s
≤
K
4
,
whereK
3, K
4 are two positive constants.Proof.
Taking the inner product of (1) with A
u we get(73)
1
2
d
d
t
u
σ
2
+
λ
u
σ
2
+
A
u
2
=
f
u
t
,
A
u
+
g
x
,
t
,
A
u
.
Applying the Young inequality, we find that(74)
f
u
t
,
A
u
≤
f
u
t
A
u
≤
f
u
t
2
+
1
4
A
u
2
,
(75)
g
x
,
t
,
A
u
≤
g
x
,
t
A
u
≤
g
x
,
t
2
+
1
4
A
u
2
.
It follows from (73)–(75) that(76)
1
2
d
d
t
u
σ
2
+
λ
u
σ
2
+
1
2
A
u
2
≤
f
u
t
2
+
g
x
,
t
2
.
Thus,(77)
d
d
t
u
σ
2
+
A
u
2
≤
2
f
u
t
2
+
2
g
x
,
t
2
.
LetT
>
0 be the constant in Lemma 4.1, then take t
≥
T and s
∈
t
,
t
+
1. Integrating (77) over the interval s
,
t
+
1 leads to(78)
u
t
+
1
σ
2
≤
u
s
σ
2
+
2
∫
s
t
+
1
f
u
τ
2
d
τ
+
2
∫
s
t
+
1
g
x
,
τ
2
d
τ
≤
u
s
σ
2
+
2
∫
t
t
+
1
f
u
τ
2
d
τ
+
2
∫
t
t
+
1
g
x
,
τ
2
d
τ
.
Integrating the above inequality with respect tos over the integral t
,
t
+
1, we obtain that(79)
u
t
+
1
σ
2
≤
∫
t
t
+
1
u
s
σ
2
d
s
+
2
∫
t
t
+
1
f
u
τ
2
d
τ
+
2
∫
t
t
+
1
g
x
,
τ
2
d
τ
≤
K
2
+
2
l
f
2
∫
t
t
+
1
u
s
F
2
d
s
+
2
∫
t
t
+
1
g
x
,
τ
2
d
τ
≤
K
2
+
2
l
f
2
K
1
+
2
∫
t
t
+
1
g
x
,
τ
2
d
τ
.
Thus we get, for allt
≥
T
+
1,(80)
u
t
σ
2
≤
K
2
+
2
l
f
2
K
1
+
2
∫
t
t
+
1
g
x
,
τ
2
d
τ
≔
K
3
.
From Lemma4.1, Lemma 4.2, and (6) we know K
3 is bounded. Let t
≥
ν
,
−
ν
≤
ρ
1
≤
ρ
2
≤
0. Integrating (77) over the interval t
+
ρ
1
,
t
+
ρ
2 we get(81)
u
t
+
ρ
2
σ
2
+
∫
t
+
ρ
1
t
+
ρ
2
A
u
s
2
d
s
≤
u
t
+
ρ
1
σ
2
+
2
∫
t
+
ρ
1
t
+
ρ
2
f
u
s
2
d
s
+
2
∫
t
+
ρ
1
t
+
ρ
2
g
x
,
s
2
d
s
.
Thus, for allt
≥
T
+
ν
+
1 and ρ
1
,
ρ
2
∈
−
ν
,
0,(82)
∫
t
+
ρ
1
t
+
ρ
2
A
u
s
2
d
s
≤
u
t
+
ρ
1
σ
2
+
2
l
f
2
∫
t
−
ν
t
u
s
F
2
d
s
+
2
∫
t
−
ν
t
g
x
,
s
2
d
s
≤
u
t
+
ρ
1
σ
2
+
2
l
f
2
ν
K
1
+
2
∫
t
−
ν
t
g
x
,
s
2
d
s
≤
K
3
+
2
l
f
2
ν
K
1
+
2
∫
t
−
ν
t
g
x
,
s
2
d
s
,
whereK
1 is given by (59), K
3 is given by (71). Then we let K
4 be(83)
K
4
=
K
3
+
2
l
f
2
ν
K
1
+
2
∫
t
−
ν
t
g
x
,
s
2
d
s
.
It is not difficult to find thatK
4 is bounded. Thus we finish the proof.Lemma 4.4.
ϕ is D-pullback asymptotically compact in F; that is, for any ω
∈
Ω, the sequence ϕ
t
n
,
θ
−
t
n
ω
,
u
n
0
θ
−
t
n
ω
n
=
1
∞ possesses a convergence subsequence in F provided t
n
→
∞, B
∈
D, and u
n
0
θ
−
t
n
ω
∈
B
θ
−
t
n
ω.Proof.
The proof is quite similar to the Lemma 18 in [2], more details can be found in [7, 9, 17, 18] and thus we omit the proof here.Proof of Theorem 2.12.
Noting that from Lemma4.1, we get that ϕ has a D-pullback absorbing set. In addition, ϕ is D-pullback asymptotically compact in F by Lemma 4.4. Therefore, the existence of a unique D-pullback attractor A
ω
ω
∈
Ω for ϕ follows from Proposition 2.11.



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*Source: 4718496-2020-01-25.xml*