Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 272, pp. 1–12.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
PULLBACK ATTRACTOR OF A NONAUTONOMOUS
FOURTH-ORDER PARABOLIC EQUATION MODELING
EPITAXIAL THIN FILM GROWTH
NING DUAN, XIAOPENG ZHAO
Abstract. We study a nonautonomous fourth-order parabolic equation modeling epitaxial thin film growth. It is shown that a pullback attractor of the
model exists when the external force has exponential growth.
1. Introduction
The study of pullback attractors for nonautonomous infinite dimensional dynamical systems has attracted much attention and made fast progress in recent years;
see for instance, [1, 3, 4, 10, 19]. Recently, Caraballo et al [2] introduced the notion
of the pullback D-attractor for nonautonomous dynamical systems and gave a general method to prove the existence of pullback D-attractor. In [12], Li and Zhong
proposed the concept of norm-to-weak continuous process and proved the existence
of pullback attractors for the nonautonomous reaction-diffusion equation. The authors in [17] considered the existence of pullback attractor for a nonautonomous
modified Swift-Hohenberg equation when its external force has exponential growth.
Suppose that Ω is an open connected bounded domain in R3 with a smooth
boundary ∂Ω, p ∈]2, 10
3 [. We are concerned with the existence of pullback attractor
for the following non-autonomous equation
ut + ∆2 u − ∇ · |∇u|p−2 ∇u − ∇u = g(x, t), in Ω × [τ, ∞),
(1.1)
with the boundary value conditions
∂u
u=
= 0, on ∂Ω × [τ, ∞),
(1.2)
∂n
and the initial condition
u(x, τ ) = uτ (x), in Ω.
(1.3)
Equation (1.1) arises in epitaxial growth of nanoscale thin films (see [8, 22]),
where u(x, t) denotes the height from the surface of the film in epitaxial growth, the
term ∆2 u denotes the capillarity-driven surface diffusion (see [16]), ∇·(|∇u|p−2 ∇u)
corresponding to the upward hopping of atoms (see [5]) and ∆u denotes the diffusion
due to the evaporation-condensation (see [7]). Recently, Equation (1.1) was studied
2010 Mathematics Subject Classification. 35Q35, 35B40, 35B41.
Key words and phrases. Pullback attractor; asymptotic compactness;
nonautonomous fourth-order parabolic equation.
c
2015
Texas State University.
Submitted July 30, 2015. Published October 21, 2015.
1
2
N. DUAN, X. ZHAO
EJDE-2015/272
by several authors. By numerical simulations and heuristic arguments, Kohn and
Yan [9] proved that the standard deviation of u in Equation (1.1) with g ≡ 0
1
1
grows as t− 3 , and the energy per unit area decays as t− 3 . Based on Schauder type
estimates and Campanato spaces, Liu [13, 14] Studied the regularity of solutions
for Equation (1.1) with nonlinear principal part and g ≡ 0 in 1D and 2D case. The
analysis of the long-time behavior of Equation (1.1) with g ≡ 0 has been developed
by [6, 23]. There are also some papers which has done with the numerical analysis
for the discrete scheme of Equation (1.1), such as [15, 18, 21] and so on. However,
the existence of pullback attractor for the fourth-order evolution equation modeling
epitaxial thin film growth has not been considered yet.
In this article, we are interested in the existence of pullback attractor for the
nonautonomous problem (1.1)-(1.3). This article is organized as follows. In Section
2, we recall some abstract results on pullback attractors and give the main result.
In Section 3, we prove the existence of pullback attractor for problem (1.1)-(1.3).
Throughout this paper, we denote (·, ·) as the inner product of L2 (Ω) and k · k
as the induced norm. k · kX denotes the norm of a Banach space X. For simplicity,
we denote k · kLp (Ω) by k · kp , respectively. In the following, c will represent generic
positive constants that may change from line to line even if in the same inequality.
2. Preliminaries
In this section, we give some basic definitions and results on the existence of
pullback attractor. Suppose that X is a complete metric space and {U (t, τ )} =
{U (t, τ ) : t ≥ τ, τ ∈ R} is a two-parameter family of mappings act on X: U (t, τ ) :
X → X, t ≥ τ, τ ∈ R.
Definition 2.1 ([11]). A two-parameter family of mappings {U (t, τ )} is said to be
norm-to-weak continuous process in X if
• U (t, s)U (s, τ ) = U (t, τ ) for all t ≥ s ≥ τ ,
• for all τ ∈ R, U (t, τ ) = Id is the identity operator,
• U (t, τ )xn * U (t, τ )x, if xn * x in X.
Let B be a bounded subset of X. The Kuratowski measure of noncompactness
α(B) of B is defined by
α(B) = inf{δ > 0 : B has a finite open cover of sets of diameter ≤ δ}.
Suppose D is a nonempty class of parameterised sets D̂ = {D(t) : t ∈ R} ⊂ B(X).
Definition 2.2 ([12]). A process {U (t, τ )} is called pullback ω-D-limit compact if
for any ε > 0 and D̂ ∈ D, there exists a τ0 (t, D̂ ≤ t such that α(Uτ ≥τ0 U (t, τ )D(τ )) ≤
ε.
Definition 2.3 ([12]). The family  = {A(t) : t ∈ R} ⊂ B(X) is said to be a
pullback D-attractor for U (t, τ ) if
• for all t ∈ R, A(t) is compact,
• Â is invariant, i.e., U (t, τ )A(τ ) = A(τ ) for all t ≥ τ ,
• Â is pullback D-attracting, i.e.,
lim dist(U (t, τ )D(τ ), A(t)) = 0,
τ →−∞
∀D̂ ∈ D, t ∈ R.
• if {C(t)}t∈R is another family of closed attracting sets, then A(t) ⊂ C(t)
for all t ∈ R.
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PULLBACK ATTRACTOR
3
In the following, we give the result on the existence of pullback D-attractor for
nonautonomous systems which can be seen in [12].
Lemma 2.4. Let {U (t, τ )}τ ≤t be a norm-to-weak continuous process such that
{U (t, τ )}τ ≤t is pullback ω-D-limit compact. If there exists a family of pullback Dabsorbing sets {B(t) : t ∈ R} ∈ D, i.e., for any t ∈ R and D̂ ∈ D there is a
τ0 (t, D̂) ≤ t such that U (t, τ )D(τ ) ⊂ B(t) for all τ ≤ τ0 . Then, there is a pullback
D-attractor {A(t) : t ∈ R} and
A(t) = ∩s≤t ∪τ ≤s U (t, τ )D(τ ).
To study the existence of pullback attractor for problem (1.1)-(1.3), we suppose
that g(x, t) is translation bounded in L2loc (R; L2 (Ω)); that is,
g(x, t) ∈
L2loc (R; L2 (Ω))
Z
and
sup
t∈R
t+1
|g(x, t)|2 ds < ∞.
t
We also suppose that, for any t ∈ R, there exist β ≥ 0 and 0 ≤ α <
such that
kg(t)k2 ≤ βeα|t| ,
20−6p 2
λ,
6p−4
(2.1)
2
where λ is the first eigenvalue of A = ∆ . By (2.1), we have the following properties:
Z t
G1 (t) :=
eλs kg(s)k2 ds < ∞, ∀t ∈ R,
Z
t
−∞
s
Z
G2 (t) :=
−∞
Z
t
eλr kg(r)k2 dr ds < ∞,
∀t ∈ R,
−∞
6p−4 6p−4
24−12p
e 20−6p λs [G1 (s)] 20−6p + [G2 (s)] 20−6p ds < ∞,
∀t ∈ R.
−∞
Using a slight modification of the classical results in the autonomous framework,
mainly of the Faedo-Galerkin method (see [20]), we obtain the following result on
the existence and uniqueness of solutions for problem (1.1)-(1.3) (see [8, 23]).
Lemma 2.5. Suppose that g ∈ L2loc (R; L2 (Ω)). There is a unique solution u(x, t)
such that
• if u0 = uτ ∈ L2 (Ω) ⇒ u(x, t) ∈ C 0 ([τ, ∞); L2 (Ω));
• if u0 = u(τ ) ∈ H02 (Ω) ⇒ u(x, t) ∈ C 0 ([τ, ∞); H02 (Ω)).
Based on Lemma 2.5, we obtain that the solution u(x, t) is continuous with
respect to the initial value condition uτ in the space H02 (Ω). In order to construct
a process {U (t, τ )} for problem (1.1)-(1.3), we define U (t, τ ) : H02 (Ω) → H02 (Ω) by
U (t, τ )uτ . Thus, the process {U (t, τ )} is a norm-to-weak continuous process in the
space H02 (Ω).
Now, we give the main result of this article, which will be proved in the next
section.
Theorem 2.6. The process corresponding to problem (1.1)-(1.3) possesses a unique
pullback D-attractor in the space H02 (Ω).
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N. DUAN, X. ZHAO
EJDE-2015/272
3. Proof of main results
In this section, we study the existence of pullback D-attractors for non-autonomous
problem (1.1)-(1.3). First of all, we derive uniform estimates of solutions for problem (1.1)-(1.3), which are necessary for proving the existence of absorbing set of
{u(t, τ )} associated with the problem.
Lemma 3.1. Consider the problem (1.1)-(1.3), for all t ≥ τ , we have
2M
1
ku(t)k2 ≤ e−λ(t−τ ) kuτ k2 +
+ e−λt G1 (t),
λ
λ
and
Z t
4M λt 1
eλs k∆u(s)k2 ds ≤ [1 + λ(t − τ )]eλτ kuτ k2 +
e + G1 (t) + G2 (t),
λ
λ
τ
where M =
−2
p p−2
p
p−2 ( 2 )
(3.1)
is a positive constant.
Proof. Multiplying (1.1) by u, integrating it over Ω, using Young’s inequality, we
obtain
1 d
ku(t)k2 + k∆u(t)k2 + k∇u(t)kpp
2 dt
= k∇u(t)k2 + (g(t), u(t))
(3.2)
1
≤ k∇u(t)kpp + M + kg(t)kku(t)k
2
1
1
1
≤ k∇u(t)kpp + M +
kg(t)k2 + k∆u(t)k2 ,
2
2λ
2
−2
2p p p−2
where M = p−2
(2)
. It then follows from (3.2) that
kg(t)k2
d
ku(t)k2 + k∆u(t)k2 + k∇u(t)kpp ≤ 2M +
,
dt
λ
d
kg(t)k2
ku(t)k2 + λku(t)k2 ≤ 2M +
,
dt
λ
Multiplying (3.4) by eλ(t−τ ) and integrating it over (τ, t), we derive that
Z t
1
2M
+ e−λt
eδs kg(s)k2 ds.
ku(t)k2 ≤ e−λ(t−τ ) kuτ k2 +
λ
λ
−∞
(3.3)
(3.4)
(3.5)
Multiplying (3.5) by eλt and integrating it over (τ, t), we deduce that
Z t
Z Z s
2M
1 t
eλs ku(s)k2 ds ≤ (t − τ )eλτ kuτ k2 + 2 eλt +
eλr kg(r)k2 dr ds. (3.6)
λ
λ −∞ −∞
τ
Similarly, multiplying (3.3) by eλt and integrating it over (τ, t), we obtain
Z t
Z t
λs
2
e k∆u(s)k ds +
eλs k∇u(s)kpp ds
τ
τ
Z t
Z
2M λt
1 t λs
λτ
2
λs
2
≤ e kuτ k +
e +λ
e ku(s)k ds +
e kg(s)k2 ds
λ
λ −∞
τ
Z t Z s
4M λt
e +
eλr kg(r)k2 dr ds
≤ [1 + λ(t − τ )] eλτ kuτ k2 +
λ
−∞ −∞
Z
1 t λs
+
e kg(s)k2 ds.
λ −∞
(3.7)
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PULLBACK ATTRACTOR
The proof is complete.
5
Lemma 3.2. In problem (1.1)-(1.3), for all t ≥ τ , we have
k∇u(t)k2
n
1
o
2M
≤ c e−λ(t−τ ) (kuτ k2 + k∇uτ k2 ) + 2 + e−λt G2 (t) + G1 (t) .
λ
λ
(3.8)
Proof. Multiplying (1.1) by ∆u, integrating it over Ω, we derive that
1 d
k∇u(t)k2 + k∇∆u(t)k2 + (∇ · (|∇u(t)|p−2 ∇u(t), ∆u(t))
2 dt
= k∆u(t)k2 − (g(t), ∆u(t)).
(3.9)
Note that
(∇ · (|∇u(t)|p−2 ∇u(t), ∆u(t))
Z
Z
=
|∇u|p−2 |∆u|2 dx +
∇u(|∇u|p−2 )∆udx
Ω
Z
ZΩ
p−2
p−2
2
∇u(|∇u|2 )|∇u|p−4 ∆u dx
=
|∇u| |∆u| dx +
2
Ω
Ω
Z
Z X
3
3
X
p−2
2
ukk )|∇u|p−4 dx
uij ui uj )(
=
|∇u| |∆u| dx + (p − 2) (
Ω
Z
Ω i,j=1
|∇u|p−2 |∆u|2 dx + (p − 2)
=
Z
(
Ω
k=1
uii u2i )(
Ω i=1
3
X
Z
+ (p − 2)
(
uij ui uj )(
Ω i,j=1,i6=j
Z
|∇u|
=
3
X
p−2
3
X
|∆u| dx + (p − 2)
Ω
+ (p − 2)
i,j=1,i6=j
Z
Ω
ukk )|∇u|
(3.10)
dx
|∇u|p−4 u2i u2ii dx
Ω
|∇u|p−4 u2i uii ujj dx
Ω
|∇u|p−4 ∆u(
+ (p − 2)
k=1
p−4
3 Z
X
i=1
Z
ukk )|∇u|p−4 dx
k=1
2
3
X
3
X
3
X
uij ui uj )dx .
i,j=1,i6=j
Using Hölder’s inequality, we have
(p − 2)
3
X
Z
Ω
i,j=1,i6=j
≥−
p−2
2
3
X
Z
i,j=1,i6=j
3
=−
|∇u|p−4 u2i uii ujj dx
p−2X
2 i=1
Z
Ω
|∇u|p−4 u2i (u2ii + u2jj )dx
(3.11)
Ω
|∇u|p−4 u2i u2ii dx −
p−2
2
3
X
i,j=1,i6=j
Z
Ω
|∇u|p−4 u2i u2jj dx.
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N. DUAN, X. ZHAO
EJDE-2015/272
Based on the regularity theorem of elliptic equation, we have
Z
3
X
(p − 2)
|∇u|p−4 ∆u
uij ui uj dx
Ω
p−2
≥−
2
(3.12)
i,j=1,i6=j
Z
p−2
|∇u|
2
|∆u| dx.
Ω
Combining (3.10), (3.11) and (3.12) together gives
(∇ · (|∇u(t)|p−2 ∇u(t), ∆u(t))
Z
3 Z
p−2X
≥
|∇u|p−2 |∆u|2 dx +
|∇u|p−4 u2i u2ii dx
2
Ω
i=1 Ω
Z
p−2
−
|∇u|p−2 |∆u|2 dx ≥ 0.
2
Ω
By Nirenberg’s inequality, we obtain
k∆u(t)k ≤ ck∇∆u(t)k2/3 ku(t)k1/3 .
(3.13)
(3.14)
It then follows from (3.9), (3.13) and (3.14) that
1 d
k∇u(t)k2 + k∇∆u(t)k2
2 dt
≤ k∆u(t)k2 + kg(t)kk∆u(t)k
4
≤ ck∇∆u(t)k 3 ku(t)k2/3 + ckg(t)kk∇∆u(t)k2/3 ku(t)k1/3
1
≤ k∇∆u(t)k2 + c(ku(t)k2 + kg(t)k2 );
2
(3.15)
that is,
d
k∇u(t)k2 + k∇∆u(t)k2 ≤ c(ku(t)k2 + kg(t)k2 ),
dt
(3.16)
and
d
k∇u(t)k2 + λk∇u(t)k2 ≤ c(ku(t)k2 + kg(t)k2 ).
dt
Multiplying (3.17) by eλ(t−τ ) and integrating over (τ, t), we derive that
(3.17)
k∇u(t)k2
≤ e−λ(t−τ ) k∇uτ k2 + ce−λt
Z
τ
t
eλs ku(s)k2 ds + ce−λt
Z
t
eλs kg(s)k2 ds
(3.18)
τ
Combining (3.6) and (3.18) gives
n
1
o
2M
k∇u(t)k2 ≤ c e−λ(t−τ ) (kuτ k2 + k∇uτ k2 ) + 2 + e−λt G2 (t) + G1 (t) .
λ
λ
Hence, the proof is complete.
Lemma 3.3. Consider the problem (1.1)-(1.3), for all t ≥ τ , we have
n
6p−4
6p−4 1 −λ(t−τ ) k∆u(t)k2 ≤ c 1 + (t − τ ) +
e
kuτ k2 + kuτ k 10−3p + k∇uτ k 10−3p
t−τ
1 + 1+
{1 + e−λt [G1 (t) + G2 (t)]}
t−τ
Z t
h
i o
24−12p
6p−4
6p−4
+ e−λt
e 20−6p λs [G1 (s)] 20−6p + [G2 (s)] 20−6p ds .
−∞
EJDE-2015/272
PULLBACK ATTRACTOR
7
Proof. Multiplying (1.1) by ∆2 u, integrating it over Ω, using Nirenberg’s inequality
and Young’s inequality, we derive that
1 d
k∆u(t)k2 + k∆2 u(t)k2
2 dt
= ∇ · (|∇u(t)|p−2 ∇u(t)), ∆2 u(t) + (g(t), ∆2 u(t)) − (∆u(t), ∆2 u(t))
1
≤ k∆2 u(t)k2 + c(k∆u(t)k2 + kg(t)k2 + k∇ · (|∇u(t)|p−2 ∇u(t))k2 )
4
1
≤ k∆2 u(t)k2 + c(k∆u(t)k2 + kg(t)k2 + k∆u(t)k2∞ k∇u(t)k2p−4
2p−4 )
4
1
≤ k∆2 u(t)k2 + c(k∆u(t)k2 + kg(t)k2
2
5
+ k∆2 u(t)k 3 k∇u(t)k1/3 k∆2 u(t)kp−3 k∇u(t)kp−1 )
6p−4
1
≤ k∆u(t)k2 + c(k∆u(t)k2 + kg(t)k2 + k∇u(t)k 10−3p );
2
(3.19)
that is,
6p−4
d
k∆u(t)k2 + k∆2 u(t)k2 ≤ c(k∆u(t)k2 + kg(t)k2 + k∇u(t)k 10−3p ),
dt
(3.20)
6p−4
d
k∆u(t)k2 + λk∆u(t)k2 ≤ c(k∆u(t)k2 + kg(t)k2 + k∇u(t)k 10−3p ),
dt
(3.21)
and
Multiplying this by (t − τ )eλt and integrating it over (τ, t), we obtain
(t − τ )eλt k∆u(t)k2
Z t
hZ t
λs
2
≤c
[1 + (s − τ )]e k∆u(s)k ds +
(s − τ )eλs kg(s)k2 ds
τ
τ
Z t
i
6p−4
+
(s − τ )eλs k∇u(t)k 10−3p .
τ
Therefore,
Z
1 −λt t λs
e
e k∆u(s)k2 ds + ce−λt G1 (t)
t−τ
τ
Z t
6p−4
+ ce−λt
eλs k∇u(t)k 10−3p ds
k∆u(t)k ≤ c 1 +
2
(3.22)
τ
=: I1 + I2 + I3 .
It then follows from (3.1) that
i
4M λt 1
1 −λt h
e
[1 + λ(t − τ )]eλτ kuτ k2 +
e + G1 (t) + G2 (t)
I1 ≤ c 1 +
t−τ
λ
λ
1 −λ(t−τ )
1
≤ c 1 + (t − τ ) +
e
kuτ k2 + c 1 +
t−τ
t−τ
1 −λt
+c 1+
e [G1 (t) + G2 (t)].
t−τ
(3.23)
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N. DUAN, X. ZHAO
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On the other hand, using Hölder’s inequality and (3.8) for sum, we obtain
Z t
n
2M
−λt
eλs e−λ(t−τ ) (kuτ k2 + k∇uτ k2 ) + 2
I3 ≤ ce
λ
τ
6p−4
o 20−6p
1
+ e−λs G2 (s) + G1 (s)
ds
λ
Z t
Z t
6p−4
6p−4
eλs e− 20−6p λ(s−τ ) (kuτ k2 + k∇uτ k2 ) 20−6p ds
eλs ds + ce−λt
≤ ce−λt
τ
τ
Z t
6p−4
6p−4
6p−4
eλs e− 20−6p λs {[G1 (t)] 20−6p + [G2 (t)] 20−6p }ds
+ ce−λt
τ
−λt
6p−4
− eλτ ] + ce−λ(t−τ ) (kuτ k 10−3p
Z t
6p−4
6p−4
10−3p
+ k∇uτ k
)
e− 20−6p λ(s−τ ) ds
τ
Z t
6p−4
6p−4
24−12p
+ ce−λt
e 20−6p λs {[G1 (t)] 20−6p + [G2 (t)] 20−6p }ds
≤ ce
[e
λt
(3.24)
τ
6p−4
6p−4
≤ c + c(t − τ )e−λ(t−τ ) (kuτ k 10−3p + k∇uτ k 10−3p )
Z t
6p−4
6p−4
24−12p
−λt
+ ce
e 20−6p λs {[G1 (t)] 20−6p + [G2 (t)] 20−6p }ds.
τ
Combining (3.23) and (3.24) with (3.22), we complete the proof.
Remark 3.4. In this article, motivated by the ideas in [12, 17], we establish the
existence of pullback attractors for the nonautomous fourth-order parabolic problem (1.1)-(1.3) in 3D case. To overcome the difficulty caused by the nonlinearity
term ∇ · (|∇u|p−2 ∇u), we impose the exponential growth condition (2.1) on the
external forcing term g(x, t). On the other hand, in order to obtain the suitable
a prior estimates, we have to restrict the parameter p and take advantage of the
Nirenberg’s inequality more times. For example, in the six line of (3.19), the exponent of k∆2 u(t)k is 53 + p − 3, which should be less than 2, so we have to let
the parameter p ∈]2, 10
3 [. Furthermore, if the problem is studied in nD case, where
n ≤ 3, we need only restrict the parameter p ∈]2, 2 + n4 [. Using the same method
as this article, we can obtain the result on the existence of pullback attractor for
nD problem (1.1)-(1.3) when p ∈]2, 2 + n4 [ and n ≤ 3.
Let R be the set of all function r : R → (0, ∞) such that
6p−4
lim teδt r 10−3p (t) = 0.
t→−∞
Denote by D the class of all families D̂ := {D(t) : t ∈ R} ⊂ B(H02 (Ω)) such that
D(t) ⊂ B0 (r(t)) for some r(t) ∈ R, B0 (r(t)) denote the closed ball in H02 (Ω) with
radius r(t). Let
h
r02 (t) = 2c 1 + e−λt G1 (t) + e−λt G2 (t)
Z t
i
24−12p
6p−4
6p−4 + e−λt
e 20−6p λs [G1 (s)] 20−6p + [G2 (s)] 20−6p ds .
−∞
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PULLBACK ATTRACTOR
9
Using the continuity of embedding H02 (Ω) ,→ L2 (Ω), for any D̂ ∈ D and t ∈ R there
exists τ0 (D̂, t) < t such that
k∆u(t)k ≤ r0 (t),
∀τ < τ0 .
(3.25)
2
Furthermore, since 0 ≤ α < 20−6p
λ, we obtain B0 (r0 (t)) ∈ D. Therefore,
6p−4
B0 (r0 (t)) is a family of bounded pullback D-absorbing sets in H02 (Ω).
Now, we present the proof of the main result.
Proof of Theorem 2.6. To prove the existence of pullback attractor for problem
(1.1)-(1.3), we need only prove that the process {u(t, τ )} is pullback ω − D-limit
compact (PDC). Thanks to A−1 is a continuous compact operator in L2 (Ω), there
is a sequence {λj }∞
j=1 satisfying
λ1 ≤ λ2 ≤ · · · ≤ λj ≤ . . . ,
λj → ∞,
as j → ∞,
2
2
and a family of elements {wj }∞
j=1 of H0 (Ω) which are orthonormal in L (Ω) such
that
Awj = λj wj ,
for j = 1, 2, . . . .
Write Xn = span{w1 , w2 , . . . , wn } ⊂ H02 (Ω) and Pn : H02 (Ω) → Xn is an orthogonal
projector. Hence
u = Pn u + (I − Pn )u := u1 + u2 .
Taking the scalar product of (1.1) with ∆2 u2 , using Young’s inequality, we obtain
6p−4
1
1 d
k∆u2 (t)k2 +k∆2 u2 (t)k2 ≤ k∆2 u2 (t)k2 +c(k∆u(t)k2 +kg(t)k2 +k∇u(t)k 10−3p ),
2 dt
2
which means
6p−4
d
k∆u2 (t)k2 + λn k∆u2 (t)k2 ≤ c(k∆u(t)k2 + kg(t)k2 + k∇u(t)k 10−3p ).
dt
(3.26)
Multiplying (3.26) by (t − τ )eλn t and integrating it over (τ, t), we deduce that
(t − τ )eλn t k∆u2 (t)k2
Z t
Z t
≤
eλn s k∆u2 (s)k2 dx + c
(s − τ )eλn s k∆u(s)k2 ds
τ
τ
Z t
Z t
6p−4
λn s
2
(s − τ )e kg(s)k ds + c
(s − τ )eλn s k∇u(s)k 10−3p ds
+c
τ
τ
Z t
Z t
≤
eλn s k∆u(s)k2 dx + c(t − τ )
eλn s k∆u(s)k2 ds
τ
τ
Z t
Z t
6p−4
λn s
2
+ c(t − τ )
e kg(s)k ds + c
(s − τ )eλn s k∇u(s)k 10−3p ds.
τ
τ
(3.27)
10
N. DUAN, X. ZHAO
EJDE-2015/272
It then follows from (3.25) and (3.27) that
k∆u2 (t)k2
Z t
Z t
eλn s k∆u(s)k2 dx + ce−λn t
eλn s k∆u(s)k2 ds
≤ (t − τ )−1 e−λn t
τ
τ
Z t
Z t
6p−4
+ ce−λn t
eλn s kg(s)k2 ds + ce−λn t
eλn s k∇u(s)k 10−3p ds
τ
τ
Z t
Z t
λn s 2
−λn t
−1 −λn t
eλn s r02 (s)ds
e r0 (s)dx + ce
≤ c(t − τ ) e
τ
τ
Z t
Z t
6p−4
λn s
2
−λn t
−λn t
eλn s r010−3p (s)ds, ∀τ ≤ τ0 .
e kg(s)k ds + ce
+ ce
(3.28)
τ
τ
Note that
Z t
Z t
e−λn t
eλn s r02 (s)dx = ce−λn t
eλn s [1 + e−λs G1 (s) + e−λs G2 (s)]ds
τ
τ
≤
cλ−1
n
−1 −λt
+ c(λn − λ)
e
(3.29)
[G1 (t) + G2 (t)].
and
−λn t
Z
t
6p−4
eλn s r010−3p (s)ds
τ
Z t
h
= ce−λn t
eλn s 1 + e−λs G1 (s) + e−λs G2 (s)
e
τ
+ e−λs
Z
s
6p−4
i 20−6p
6p−4
6p−4 24−12p
ds
e 20−6p λr [G1 (r)] 20−6p + [G2 (r)] 20−6p dr
−∞
6p−4
6p−4
6p−4 6p − 4 −1 − 20−6p
λt
≤ cλ−1
+
c
λ
−
e
λ
[G1 (t)] 20−6p + [G2 (t)] 20−6p
n
n
20 − 6p
6p−4
h Z t 24−12p
20−6p
6p−4
6p−4
6p − 4 −1 − 20−6p
λt
λ
e
+ c λn −
e 20−6p λs [G1 (s)] 20−6p
20 − 6p
−∞
6p−4 i
Z t 24−12p
20−6p
6p−4
+
e 20−6p λs [G2 (s)] 20−6p
.
(3.30)
−∞
On the other hand, simple calculations show that
( −λ t R t λ s −αs
Z t
βe n τ e n e
ds, t ≤ 0,
−λn t
λn s
2
e
e kg(s)k ds ≤
R
t
τ
βe−λn t eλn s eα|s| ds, t ≥ 0,
 −λ t τ
 βe n ,
t ≤ 0,
λn −α
≤
 βe−λn t + βeαt , t ≥ 0.
λn −α
λn +α
(3.31)
Adding (3.28), (3.29), (3.30) and (3.31), we can obtain for any ε > 0, there exist
τ0 < t and N ∈ N such that
k∆u2 (t)k ≤ ε,
∀τ < τ0 .
It indicates that the process {U (t, τ )} is pullback ω − D-limit compact. Then, by
Lemma 2.4, the process corresponding to problem (1.1)-(1.3) possesses a unique
pullback D-attractor in H02 (Ω).
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PULLBACK ATTRACTOR
11
Acknowledgements. The authors would thank the referees for their valuable suggestions for the revision and improvement of the original manuscript. This research
is supported by the Natural Science Foundation of China for Young Scholar (No.
11401258), and by the Natural Science Foundation of Jiangsu Province for Young
Scholar (No. BK20140130).
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12
N. DUAN, X. ZHAO
EJDE-2015/272
Ning Duan
School of Science, Jiangnan University, Wuxi 214122, China
E-mail address: 123332453@qq.com
Xiaopeng Zhao
Department of Mathematics, Southeast University, Nanjing 210096, China;
School of Science, Jiangnan University, Wuxi 214122, China
E-mail address: zhaoxiaopeng@sina.cn