Pullback attractors for nonclassical diffusion equations

MINISTRY OF EDUCATION AND TRAINING
HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY
--------------------------------------TANG QUOC BAO

TANG QUOC BAO

MATHEMATICS AND
INFORMATICS

PULLBACK ATTRACTORS FOR NONCLASSICAL
DIFFUSION EQUATIONS

MASTER OF SCIENCE THESIS
Mathematics and Informatics

2010B
Hanoi - 2011


MINISTRY OF EDUCATION AND TRAINING
HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY
--------------------------------------TANG QUOC BAO

PULLBACK ATTRACTORS FOR NONCLASSICAL
DIFFUSION EQUATIONS

Major: Mathematics and Informatics

MASTER OF SCIENCE THESIS
Mathematics and Informatics

Supervisor:
1. Dr.Cung The Anh

Hanoi - 2011


Contents
Acknowledgements

2

Introduction

3

1 Existence and uniqueness of solutions
1.1 Setting of the problem . . . . . . . . . . . . . . . . . . . . .
1.2 Existence and uniqueness of solutions . . . . . . . . . . . . .

6
6
7

2 Existence and upper semicontinuity of pullback attractors 13
2.1 Existence of pullback attractors . . . . . . . . . . . . . . . . 13
2.2 The upper semicontinuity of pullback attractors at ε = 0 . . 30
Conclusions

38

1


Acknowledgements
I wish to express my thanks to Dr. Cung The Anh for suggesting the
problem and for many stimulating conversations.
I’m also very thankful for Faculty of Applied Mathematics and Informatics, University of Science and Technology, where the thesis was written, for
financial support.
Last, but not least, I am grateful to my family, my friends for their
encouragement, which helps me very much in completing the thesis.

2


Introduction
In this thesis we consider the Cauchy problem for a class of non-autonomous
nonclassical diffusion equations of the form
ut − ε∆ut − ∆u + f (x, u) + λu = g(x, t), x ∈ Rn , t > τ,
u|t=τ = uτ (x), x ∈ Rn ,

(0.1)

where λ > 0, ε ∈ [0, 1], the nonlinearity f and the external force g satisfy
some specified conditions later.
Nonclassical diffusion equations arise as models to describe physical phenonmena, such as non-Newtonian flows, soil mechanics, and heat conduction (see e.g. [1, 12, 18]). In the last few years, the existence and long-time
behavior of solutions to nonclassical difussion equations has attracted the
attention of many mathematicians.
Let us review some recent results on nonclassical diffusion equations.
For autonomous case, that is the case g independent of time t, in [25], the
author considered equation (0.1) in a bounded domain Ω with ε = 1, the

time-independent external force g ∈ L2 (Ω) and the nonlinearity f satisfy
the following conditions
lim sup
|s|→∞

f (s)
< λ1 ,
s

(0.2)

where λ1 is the first eigenvalue of the operator −∆ in Ω with Dirichlet
condition, and
|f (s)| ≤ C(1 + |s|4 ),
(0.3)
and
|f (s)| ≤ C(1 + |s|γ ), γ < 5.
3

(0.4)


The growth condition (0.4) of f is usually called subcritical case. Under
conditions (0.2)-(0.4), the author proved the existence of a global attractor
in H01 (Ω).
Also consider autonomous nonclassical diffusion equations in a bounded
domain, the authors in [22] assumed that the initial uτ ∈ H 2 (Ω) ∩ H01 (Ω)
and the nonlinearity f satisfies the subcritical growth condition. Under
those assumptions, the authors proved that for each ε ∈ [0, 1], there exists
a global attractor Aε ⊂ H 2 (Ω) ∩ H01 (Ω) for problem (0.1). Moreover, they

showed that when ε → 0, Aε tends to A0 in the sense that
lim distH01 (Ω) (Aε , A0 ) = 0 as ε → 0,

ε→0

where distH01 (Ω) is the Hausdorff semi-distance in H01 (Ω).
For non-autonomous case, under a Sobolev growth condition of f , the
authors in [2] proved that there exists a pullback attractor Aε = {Aε (t) :
t ∈ R} ⊂ H01 (Ω) of (0.1) for each ε ∈ [0, 1] and
lim sup distL2 (Ω) (Aε (t), A0 (t)) = 0 as ε → 0,

ε→0 t∈I

for any interval I ⊂ R.
We refer the reader to [14, 25, 15, 24] for other results. It is noticed
that all existing results are devoted in bounded domains. The dynamic of
nonclassical diffusion equations in unbounded domains is not well understood. In this thesis, we consider the existence and long-time behavior of
solutions to problem (0.1) in the case of unbounded domains, the nonlinearity of polynomial type, and the unbounded external force g depending on
time t. The main aims of this thesis are to prove the existence of pullback
attractors Aˆε = {Aε (t) : t ∈ R}, ε ∈ [0, 1], in H 1 (Rn ) ∩ Lp (Rn ) for problem
(0.1) and to show the upper semicontinuity of Aˆε at ε = 0. The results
obtained in this thesis has been accepted for publication in the journal
Communications on Pure and Applied Analysis [3].
The existence of a pullback attractor for problem (0.1) (on the entire
space Rn ) in the case ε = 0 has been proved recently in [21].
In the case ε > 0, since equation (0.1) contains the term −ε∆ut , it
is different from the classical reaction-diffusion equation essentially. For
4

example, the reaction-diffusion equation has some kind of ”regularity”, e.g.,
although the initial datum only belongs to a weaker topology space, the
solution will belong to a stronger topology space with higher regularity.
However, for problem (0.1) when ε > 0, because of −∆ut , if the initial
datum uτ belongs to H 1 (Rn ) ∩ Lp (Rn ), the solution u(t) with intial datum
u(τ ) = uτ is always in H 1 (Rn ) ∩ Lp (Rn ) and has no higher regularity,
which is similar to hyperbolic equations. This brings some difficulty in
establishing the existence of pullback attractors for nonclassical diffusion
equations. On the other hand, notice that the domain Rn for (1.1) is
unbounded, so Sobolev embeddings are no longer compact in this case. This
introduces a major obstacle for examining the asymptotic compactness of
solutions.
To try to overcome these difficulties, we combine the method of tailestimates [20] and the asymptotic a priori estimate method [11] to prove the
asymptotic compactness of the corresponding process. We first use these
methods to prove the existence of an (H 1 (Rn ) ∩ Lp (Rn ), Lp (Rn ))-pullback
attractor. Then by verifying the condition (PDC) introduced in [7], we
obtain the existence of a pullback attractor Aˆε in H 1 (Rn ) ∩ Lp (Rn ). Next,
we study the continuous dependence on ε of solutions to problem (0.1)
as ε → 0. Hence using an abstract result derived recently by Carvalho
et. al. [5] and techniques similar to ones used in [2], we prove the upper
semicontinuity of pullback attractors Aˆε in L2 (Rn ) at ε = 0.
The rest of the thesis is organized as follows. In Chapter 1, we prove the
well-posedness of equation (0.1), that is, the existence and uniqueness of
solutions. In Chapter 2, the existence and upper semi-continuity of pullback
attractors are investigated. In conclusion, we give some ways to extend the
results.

5

Chapter 1
Existence and uniqueness of solutions
The aims of this Chapter is to prove the existence and uniqueness of solutions. In Section 2.1, we give some hypothesis and a definition of weak
solutions. The existence of weak solutions is investigated in Section 2.2.

1.1

Setting of the problem

This thesis is concerned with the non-autonomous nonclassical diffusion
equation
ut − ε∆ut − ∆u + f (x, u) + λu = g(t, x),
(1.1)
u|t=τ = uτ ,
where ε ∈ [0, 1] and λ > 0.
To study problem (1.1), we assume the following conditions:
(H1) The initial datum uτ ∈ H 1 (Rn ) ∩ Lp (Rn ) is given;
(H2) The nonlinearity f satisfies
f (x, u)u ≥ α|u|p − φ1 (x),

(1.2)

|f (x, u)| ≤ β|u|p−1 + φ2 (x),

(1.3)

fu (x, u) ≥ −γ,

(1.4)

p
where p ≥ 2, φ1 ∈ L1 (Rn ), φ2 ∈ Lq (Rn ), q = p−1
, are nonnegative
functions, α, β, γ are positive constants, |φ1 (x)| ≤ C0 for all x ∈ Rn .

6


For F (x, s) =

s
0 f (x, r)dr,

we assume

γ1 |u|p − φ3 (x) ≤ F (x, u) ≤ γ2 |u|p + φ4 (x),
where γ1 , γ2 > 0, φ3 , φ4 ∈ L1 (Rn ) are nonnegative functions. This
implies there exist positive constants ζ1 , ζ2 , C3 , C4 such that
ζ1 u

p
Lp (Rn )

F (x, u)dx ≤ ζ2 u

− C3 ≤
Rn

p
Lp (Rn )

+ C4 .

(1.5)

< +∞, ∀t ∈ R,

(1.6)

1,2
(R; L2 (Rn )) satisfies
(H3) The external force g ∈ Wloc
t

eσs
−∞

g(s)

2
L2 (Rn )

and

+ g (s)

2
L2 (Rn )

t

eσs |g(s)|2 = 0,

lim sup
k→+∞

−∞

(1.7)

|x|≥k

where σ < min{λ, 2/ε}.
Definition 1.1. A function u(t, x) is called a weak solution of (1.1) on
(τ, T ) iff
u ∈ L∞ (τ, T ; H 1 (Rn )) ∩ Lp (τ, T ; Lp (Rn )),

∂u
∈ L2 (τ, T ; H 1 (Rn )),
∂t

u|t=τ = uτ a.e. in Rn ,
and
T
τ

Rn

∂u
v + ε∇ut ∇v + ∇u∇v + f (x, u)v + λuv =

∂t

T

gv,
τ

Rn

for all test functions v ∈ C0∞ ([0, T ] × Rn ).

1.2

Existence and uniqueness of solutions

In this Section, we prove that under assumptions (H1) - (H3), the problem
(1.1) has a unique weak solution. Denote by · , (·, ·) the norm and scalar
7


product of L2 (Rn ) and C an arbitrary constant, which may be different
from line to line (and even in the same line).
In order to prove the existence of a weak solution, we consider the Dirichlet problem in a bounded domain


ut − ε∆ut − ∆u + f (x, u) + λu = g(x, t), x ∈ ΩR , t > τ,
(1.8)
u|∂ΩR = 0,



u(τ, x) = uτ,R (x), x ∈ ΩR ,
where ΩR is the open ball of radius R ≥ 1 centered at 0, u0,R = u0 ψR (|x|),
and ψR is a smooth function verifying

if 0 ≤ r ≤ R − 1,

1,
ψR (r) = 0 ≤ ψR (r) ≤ 1, if R − 1 ≤ r ≤ R,


0,
if r > R.
By the Galerkin method, one can easily show that problem (1.8) has a
unique weak solution uR for any initial datum uτ,R ∈ H01 (ΩR ) ∩ Lp (ΩR ).
Theorem 1.1. Let conditions (H2) − (H3) hold. Then, problem (1.1) has
a unique weak solution for any uτ ∈ H 1 (Rn ) ∩ Lp (Rn ).
Proof. Let urj , rj → +∞, be a sequence of solutions of (1.8). We easily
conclude from Dirichlet problem that {uτ,rj } is bounded in L2 (Rn )∩H 1 (Rn ).
We have
∂t urj − ε∆(∂t urj ) − ∆urj + f (x, urj ) + λurj = g(t).

(1.9)

Multiplying (1.9) with urj in L2 (Rn ), we get
1d
2 dt

|urj |2 + ε
Ω rj

+

|∇urj |2
Ω rj

|∇urj |2

+
Ω rj

|urj |2 =

f (x, urj )urj + λ
Rn

Ωr j

8

(1.10)
g(t)urj .
Ωr j


Using (1.2) and the inequality |
we get
d
dt

urj

2

Ω rj

|∇urj |2

+ε

g(t)urj | ≤

2

urj

+λ
Ωr j

1
≤
λ

Ω rj

|g(t)|2 +

Ω rj

|urj |2 ,

Ωr j

(1.11)

|g(t)|2 + 2
Ω rj

λ
2

p
Lp (Ωrj )

|∇urj |2 + 2α urj

+2

Ωr j

Ωr j

1
2λ

φ1 (x).
Ωr j

Integrating from τ to t, t ∈ [τ, T ], in particular we obtain
t

2

t

2

2

|∇urj (t)| + 2

urj (t) + ε
Ωr j

|∇urj (s)| + 2α
τ

Ωr j

1
≤
|uτ,rj | + ε
|∇uτ,rj | +
λ
Ω rj
Ωr j
2

urj (s)
τ

Ωr j

p
Lp (Ωrj )

T

|g(s, x)|2 + 2(T − τ ) φ1

2

τ

L1 (Rn ) .

Ωr j

(1.12)
This inequality implies that
{urj } is bounded in L∞ (τ, T ; H 1 (Ωrj )) ∩ Lp (τ, T ; Lp (Ωrj )).

(1.13)

We extend these solutions to be defined on Rn in the following way
uˆrj (x) =

urj (x)ψrj (|x|),
0,

in Ωrj ,

otherwise .

(1.14)

Since (1.13), {ˆ
urj } is a bounded sequence in L∞ (τ, T ; H 1 (Rn ))∩Lp (τ, T ; Lp (Rn )).
Hence, there exists a subsequence of uˆrj (denoted again by urj ) such that
urj → u∞ weakly-* in L∞ (τ, T ; H 1 (Rn )),
urj → u∞ weakly in Lp (τ, T ; Lp (Rn )).

(1.15)

We will prove that u∞ is a weak solution of problem (1.1). Let rk be fixed.
Since rj → +∞, we can assume rk ≤ rj − 1. We define the projections in
Ωrk of urj and denote them by ukj = Lk urj . It is clear from (1.13) that {ukj }
is bounded in L∞ (τ, T ; H 1 (Ωrk )) ∩ Lp (τ, T ; Lp (Ωrk )). It follows that there
exists a subsequence (denoted again by ukj ) such that ukj = Lk urj → uk∞
weakly in Lp (τ, T ; Lp (Ωrk )) and weakly-* in L∞ (τ, T ; H 1 (Ωrk )).
9


We now check that Lk u∞ = uk∞ . Indeed, let v ∈ C0∞ ([τ, T ] × Ωrk ). The
weak-* convergence in L∞ (τ, T ; H 1 (Ωrk )) gives
T

T

Lk urj v →
τ

uk∞ v.

Ωr k

τ

(1.16)

Ωr k

On the other hand, noting that v(t, x) = 0 if x ∈
/ Ωrk and using (1.15), we
obtain
T

T

T

urj v →

Lk urj v =
τ

Ωr k

Rn

τ

and

T

u∞ v,

(1.17)

Rn

τ

T

u∞ v =

Lk u∞ v,

Rn

τ

τ

(1.18)

Ωr k

so that uk∞ = Lk u∞ .
We claim that Lk u∞ is a weak solution in [τ, T ]×Ωrk . Let v ∈ C0∞ ([τ, T ]×

Ωrk ). Since Ωrk ⊂ Ωrj , it follows that v ∈ C0∞ ([τ, T ] × Ωrj ), and using the
fact that urj is a weak solution in Ωrj we have
T

(∂t Lk urj v + ε∇∂t Lk urj ∇v
τ

Ωr k

+ ∇Lk urj ∇v + f (x, Lk urj )v + λLk urj v − g(t, x)v)
T

(1.19)

(∂t urj v + ε∇∂t urj ∇v

=
τ

Ωr j

+ ∇urj ∇v + f (x, urj )v + urj v − g(t, x)v) = 0.
In (1.9), replacing urj by Lk urj , then multiplying by ∂t Lk urj in L2 (Ωrk ), we
obtain
∂t Lk urj
Ωr k

+

2

|∇∂t Lk urj |2

+ε
Ω rk

1d
2 dt

|∇Lk urj |2 + 2
Ωr k

Ωr k

g(t, x)∂t Lk urj ≤

=
Ωr k

F (x, Lk urj ) + 2λ

Lk urj
Ωr k

∂t Lk urj
Ωr k

10

2

+

1
4

|g(t, x)|2 .
Ωr k

2

(1.20)


Thus, we deduce that
{∂t Lk urj } is bounded in L2 (τ, T ; L2 (Ωrk )).

(1.21)

Combining (1.15) (with Lk urj in place of urj , uk∞ in place of u∞ ) and (1.21),
using the Aubin-Lions Lemma in [9] we get
Lk urj → uk∞ strongly in L2 (τ, T ; L2 (Ωrk )), up to a subsequence,
and therefore
Lk urj → uk∞ a.e. in [τ, T ] × Ωrk .
Since f is a continuous function, we have
f (x, Lk urj ) → f (x, uk∞ ) a.e. in [τ, T ] × Ωrk .

(1.22)

On the other hand, from (1.3) we have

T

f (x, Lk urj )
τ

q
Lq (Ωrk )

≤C

Lk urj

p
Lp (τ,T ;Lp (Ωrk ))

+ φ2

q
Lq (Rn ) (T

− τ) .

Using (1.15) with Lk urj in place of urj , the above inequality implies that
{f (Lk urj )} is bounded in Lq (τ, T ; Lq (Ωrk )).

(1.23)

From (1.22), (1.23) and Lemma 1.3 in [9, Chapter 1], we have
f (x, Lk urj ) → f (x, uk∞ ) in Lq (τ, T ; Lq (Ωrk )).

(1.24)

Passing to the limit in (1.19) and using (1.24), we get that Lk u∞ is a
weak solution in [τ, T ] × Ωrk . Hence we get that u∞ is a weak solution of
problem (1.1). Indeed, for any v ∈ C0∞ ([τ, T ] × Rn ), there exists rk such
that v ∈ C0∞ ([τ, T ] × Ωrk ), using Lk u∞ solving (1.1) in [τ, T ] × Ωrk we can
conclude that u∞ is a weak solution of (1.1) in [τ, T ] × Rn .
It remains to prove the uniqueness of solution. Let u and v be two
solutions of problem (1.1). Denote w = u − v, we have
wt − ε∆wt − ∆w + f (x, u) − f (x, v) + λw = 0.

11

(1.25)


Taking the inner product of (1.25) with w in L2 (Rn ) we get
1d
2 dt

w

2

+ ε ∇w

2

+ ∇w

2

(f (x, u) − f (x, v))w + λ w

+
Rn

2

= 0.
(1.26)

Using (1.4), we have
d
dt

w

2

+ ε ∇w

2

≤ 2γ( w

2

+ ε ∇w 2 ),

thus
w(t)

2

+ ε ∇w(t)

2

≤ e2γ(t−τ ) ( w(τ )

2

+ ε ∇w(τ ) 2 )

by Gronwall’s lemma. This implies the uniqueness (if u(τ ) = v(τ )) and the
continuous dependence of the solution.

12


Chapter 2
Existence and upper semicontinuity
of pullback attractors
2.1

Existence of pullback attractors

We first recall some basic concepts related to pullback attractors for dynamical systems.
Let X, Y be two Banach spaces with the norms · X and · Y respectively, and X ⊂ Y . Denote by B(X) the set of all bounded subsets of X.

For A, B ⊂ X, the Hausdorff semi-distance between A and B is defined by
distX (A, B) = sup inf x − y
x∈A y∈B

X.

Let {U (t, τ ) : t ≥ τ, τ ∈ R} be a process in X, i.e., U (t, τ ) : X → Y such
that U (τ, τ ) = Id and U (t, s)U (s, τ ) = U (t, τ ) for all t ≥ s ≥ τ, τ ∈ R.
Definition 2.1. The process {U (t, τ )} is said to be (X, Y ) - pullback asymptotically compact if for any t ∈ R, any sequence τn → −∞, and any bounded
sequence {xn }, the sequence {U (t, τn )xn } is relatively compact in Y .
Definition 2.2. A process {U (t, τ )} is called satisfying Condition (PDC)
in Y if for any fixed t ∈ R, D ∈ B(X), and any η > 0, there exist τ0 =
τ0 (D, η, t) ≤ t and a finite dimensional subspace Y1 of Y such that:
(i) P (

τ ≤τ0

U (t, τ )D) is bounded in Y ,

(ii) (IdY − P )y

Y

≤ η, ∀y ∈

τ ≤τ0

U (t, τ )D,

13

where P : Y → Y1 is a bounded projector, IdY is the identity.
Lemma 2.1. [7] If a process {U (t, τ )} satisfies Condition (PDC) in Y ,
then {U (t, τ )} is (X, Y ) - pullback asymptotically compact.
Definition 2.3. The family Aˆ = {A(t) : t ∈ R} ⊂ B(X) is said to be an
(X, Y ) - pullback attractor for {U (t, τ )} if
(1) A(t) is compact in Y , for all t ∈ R,
(2) Aˆ is invariant, i.e.,
U (t, τ )A(τ ) = A(t), for all t ≥ τ,
(3) Aˆ is (X, Y ) - pullback attracting, i.e.,
lim distY (U (t, τ )D, A(t)) = 0, for all D ∈ B(X), and all t ∈ R,

τ →−∞

(4) if {C(t) : t ∈ R} is another family of closed (X, Y ) - pullback attracting
sets then A(t) ⊂ C(t), for all t ∈ R.
Theorem 2.2. [7] Let {U (t, τ )} be a process satisfying the following conditions:
(i) {U (t, τ )} is norm-to-weak continuous on Y , i.e., U (t, τ )xn
in Y , as xn → x in X, for all t ≥ τ, τ ∈ R;

U (t, τ )x

(ii) there exists a family of (X, Y ) - pullback absorbing sets B = {B(t) : t ∈
R} ⊂ Y , i.e., for any t ∈ R, any D ∈ B(X), there is τ0 = τ0 (D, t) ≤ t
such that
U (t, τ )D ⊂ B(t);
τ ≤τ0

(iii) {U (t, τ )} is (X, Y ) - pullback asymptotically compact.

Then {U (t, τ )} has a unique (X, Y ) - pullback attractor Aˆ = {A(t) : t ∈ R},
and
Y

U (t, τ )B(τ ) ,

A(t) =
s≤t τ ≤s
Y

where A denotes the closure of A with respect to the norm topology in Y .
14


Remark 2.1. In fact, the results of Lemma 2.1 and Theorem 2.2 were
proved in [7] for the case of usual pullback attractors (i.e. the case X = Y )
instead of bi-spaces pullback attractors. But they can be obtained almost
directly from [7] with some obvious changes, so we omit the proof here.
The following result, which is about the upper semicontinuity of pullback
attractors was proved [5].
Definition 2.4. [5] Let {Uε (t, τ ) : ε ∈ [0, 1]} be a family of evolution
processes in a Banach space X with corresponding pullback attractors Aˆε =
{Aε (t) : ε ∈ [0, 1]}. For any bounded interval I ⊂ R, we say that {Aˆε }ε∈[0,1]
is upper semicontinuous in Y at ε = 0 for t ∈ I if
lim sup distY (Aε (t), A0 (t)) = 0.

ε→0 t∈I

Theorem 2.3. [5] Let {Uε (t, τ ) : ε ∈ [0, 1]} be a family of processes with
corresponding pullback attractors {Aε (.) : ε ∈ [0, 1]}. Then for any bounded

interval I ⊂ R, {Uε (t, τ ) : ε ∈ [0, 1]} is upper semicontinuous in Y at 0 for
t ∈ I if for each t ∈ R, for each compact subset K of X and each T > 0,
the following conditions hold:
(i)

sup sup Uε (t, τ )x − U0 (t, τ )x

Y

→ 0 as ε → 0,

τ ∈[t−T,t] x∈K

Aε (t) is bounded in Y for given t0 ,

(ii)
ε∈[0,1] t≤t0

Aε (t) is compact in Y for each t ∈ R.

(iii)
0<ε≤1

Now, we want to establish the existence of pullback attractors for equation (1.1). Thanks to Theorem 1.1, problem (1.1) defines a process
U (t, τ ) : H 1 (Rn ) ∩ Lp (Rn ) → H 1 (Rn ) ∩ Lp (Rn ),
with U (t, τ )uτ is the unique weak solution of (1.1) subject to uτ as initial
datum at time τ.
15

Proposition 2.4. Under assumptions (1.2)-(1.6), for any t ∈ R, any D ∈
B(H 1 (Rn ) ∩ Lp (Rn )), there exists τ0 (t, D) such that
u(t)

2

+ ∇u(t)

2

+ u(t)

p
Lp (Rn )

t

≤C 1+e

−σt

eσs g(s)

2

,

−∞

for any τ ≤ τ0 , any uτ ∈ D, where u(t) = U (t, τ )uτ . This implies that

the process {U (t, τ )} corresponding to (1.1) has a family of (H 1 (Rn ) ∩
Lp (Rn ), H 1 (Rn ) ∩ Lp (Rn )) - pullback absorbing sets B = {B(t) : t ∈ R}.
Proof. Taking the inner product of (1.1) with u in L2 (Rn ), we have
1d
2 dt

u

2

+ ε ∇u

2

+ ∇u

2

+ (f (x, u), u) + λ u

2

= (g(t), u).

(2.1)

Hence using hypothesis (1.2), Cauchy’s inequality and the assumption σ <
min{λ, 2/ε}, we have
d
dt

u
+C

2

+ ε ∇u
u

2

2

+ σ( u

+ ∇u

2

+ u

2

+ ε ∇u 2 )

p
Lp (Rn )

1
≤

g(t)
λ

(2.2)
2

+ 2 φ1

L1 (Rn ) ,

thus,
d σt
e
u 2 + ε ∇u 2
dt
+ Ceσt u 2 + ∇u

(2.3)
2

+ u

p
Lp (Rn )

σt

≤ Ce

g(t)

2

σt

+ Ce .

Integrating (2.3) from τ to s, s ∈ [τ, t − 1], we get
s
σs

2

2

στ

e ( u(s) +ε ∇u(s) ) ≤ e ( uτ

2

2

eσz g(z) 2 +Ceσs .

+ε ∇uτ )+C
−∞

(2.4)
On the other hand, integrating (2.3) from s to s + 1, where s ∈ [τ, t − 1],

16


and using (2.4), in particular we get
s+1

eσt

2

u

2

+ ∇u

p
Lp (Rn )

+ u

s

s+1
σs

2

≤ Ce ( u(s)

2

eσr g(r)

+ ε ∇u(s) ) + C

2

+ Ceσ(s+1)

s
t

≤ Ceστ ( uτ

2

+ ε ∇uτ 2 ) + C

eσr g(r)

2

+ Ceσt .

−∞

(2.5)
Combining (1.5) and (2.5) we obtain

s+1

eσt λ u

2

+ ∇u

2

+2

F (x, u)
Rn

s
s+1

eσt

≤C

u

2

+ ∇u

2

+ u

s

p
Lp (Rn )

(2.6)

+C

t

≤ Ceστ ( uτ

2

+ ε ∇uτ 2 ) + C

eσr g(r)

2

+ Ceσt .

−∞

Multiplying (1.1) by ut , we obtain
2 ut

2

2

+ 2ε ∇ut

+

d
λ u
dt

Since 2|(g(t), ut )| ≤ g(t)
d
λ u
dt

2

2

2

+ ∇u

2

+2

F (x, u)

= 2(g(t), ut ).

Rn

(2.7)

+ ut 2 , we have

+ ∇u

2

+2

F (x, u)

≤ g(t) 2 .

(2.8)

Rn

Thus
d σt
e
λ u
dt

2

σt

≤ σe

+ ∇u

2

+2

F (x, u)
Rn

λ u

2

+ ∇u

2

(2.9)
σt

+2

F (u) + e
Rn

17

2

g(t) .


From (2.6) and (2.9) and using uniform Gronwall’s inequality, we get
eσt λ u

2

2

+ ∇u

F (x, u)

+2
Rn

(2.10)

t
στ

≤ C e ( uτ

2

2

σt

+ ε ∇uτ ) + e +

e

σs

g(s)

2

.

−∞

By (1.5) again
u

2

+ ∇u

2

p
Lp (Rn )

+ u

≤C λ u

2

2

+ ∇u

+2

F (x, u) + C

(2.11)

Rn
t

≤C e

−σt στ

e (|uτ

2

−σt

2

eσs g(s)

+ ε ∇uτ ) + 1 + e

2

.

−∞

Because uτ ∈ D is bounded in H 1 (Rn ) ∩ Lp (Rn ),
lim sup eστ ( uτ
τ →−∞

2

+ ε ∇uτ 2 ) = 0.

Then, we get τ0 ≤ t such that,
u

2

+ ∇u

2

+ u

p
Lp (Rn )

t
−σt

eσs g(s)

≤C 1+e

2

,

−∞

for all τ ≤ τ0 , all uτ ∈ D. This completes the proof.
Remark 2.2. The family of (H 1 (Rn )∩Lp (Rn ), H 1 (Rn )∩Lp (Rn )) - pullback
absorbing sets B = {B(t)}t∈R of {U (t, τ )} obtained in Proposition 2.4 is
also (H 1 (Rn ) ∩ Lp (Rn ), L2 (Rn ), (H 1 (Rn ) ∩ Lp (Rn ), Lp (Rn )) and (H 1 (Rn ) ∩
Lp (Rn ), H 1 (Rn )) - pullback absorbing sets of the process {U (t, τ )}.
Lemma 2.5. [26] Let X, Y be two Banach spaces, X ∗ , Y ∗ be respectively
their dual spaces. Suppose that X is dense in Y , the injection i : X → Y
is continuous and its adjoint i∗ : Y ∗ → X ∗ is dense, and {U (t, τ )} is a
continuous or weak continuous process on Y . Then {U (t, τ )} is norm-toweak continuous on X if and only if for t ≥ τ , τ ∈ R, U (t, τ ) maps a
compact set of X to be a bounded set of X.
18


By the above lemma and the fact that {U (t, τ )} is continuous in H 1 (Rn )

and L2 (Rn ), we deduce that {U (t, τ )} is norm-to-weak in Lp (Rn ).
Lemma 2.6. For any t ∈ R, any D ∈ B(H 1 (Rn ) ∩ Lp (Rn )), there exists
τ0 (t, D) ≤ t such that
t

ut (t)

2

2

+ ∇ut (t)

≤C 1+e

−σt

eσs

g(s)

2

+ g (s)

2

, (2.12)

−∞

for any τ ≤ τ0 , any uτ ∈ D, where ut (s) =

d
dt

(U (t, τ )uτ ) |t=s .

Proof. By differentiating equation (1.1) with respect to t, we have
utt − ε∆utt − ∆ut + fu (x, u)ut + λut = g (t).

(2.13)

Taking the inner product of (2.13) with ut in L2 (Rn ) and using (1.3), we
get
d
dt

ut

2

2

+ ε ∇ut

+ 2 ∇ut

2

+ 2λ ut

2

2

≤ 2γ ut

+ (g (t), ut ). (2.14)

By Young’s inequality we can obtain
d σt
e ( ut
dt

2

+ ε ∇ut 2 ) ≤ Ceσt ( ut

2

+ ε ∇ut 2 ) + Ceσt g (t) 2 . (2.15)

By (2.7) we get
Ceσt

ut

2

σt

≤ Ce

2

+ ε ∇ut
λ u

2

+

d σt
e
λ u
dt

+ ∇u

2

2

2

+ ∇u

+2
σt

+2

F (u)
Rn

F (u) + Ce

(2.16)
2

g(t) .

Rn

Integrating the above identity from r to r + 1, r ∈ [τ, t − 1], and using (2.6)
and (2.10) we have
r+1

eσs

ut

2

+ ε ∇ut

2

r

(2.17)

t

≤ C eστ

uτ

2

+ ε ∇uτ

2

+ eσt +

eσs g(s)
−∞

19

2

.


From (2.15) and (2.17), using uniform Gronwall’s lemma, we get τ0 ≤ t
such that
t

2

ut

+ ε ∇ut

2

≤C 1+e

−σt

eσs

g(s)

2

+ g (s)

2

,

(2.18)

−∞

for any τ ≤ τ0 and any uτ ∈ D.
Next, we estimate the tail of solutions

Lemma 2.7. For any η > 0, any t ∈ R, and any D ∈ B H 1 (Rn ) ∩ Lp (Rn ) ,
there exist K0 > 0 and τ0 ≤ t such that
(|u|2 + ε|∇u|2 ) ≤ η, for all K ≥ K0 , τ ≤ τ0 , uτ ∈ D.

(2.19)

|x|≥K

Proof. Let θ : R+ → R+ be a smooth function satisfying 0 ≤ θ(s) ≤ 1 for
s ≥ 0, and
θ(s) = 0 for 0 ≤ s ≤ 1; θ(s) = 1 for s ≥ 2.
Then there exists a constant C such that |θ (s)| ≤ C for s ≥ 0. Taking the
2
u in L2 (Rn ), we get
inner product of (1.1) with θ |x|
k2
1d
2 dt

|x|2
|x|2
2
|u|
+
ε
|∇u|2 + λ
θ
θ
2
2

k
k
n
n
n
R
R
R
2
2
|x|
|x|
|∇u|2 +
u.f (u)
+
θ
θ
2
k
k2
Rn
Rn
2x
|x|2
2x
|x|2
+
θ
.u.∇u + ε
θ

.u.∇ut
2
2
k2
k2
Rn k
Rn k
|x|2
=
g(t)θ
u.
k2
Rn
θ

|x|2
k2

|u|2

(2.20)
First, we estimate some terms on the left hand side of (2.20). By (1.2), we
have
|x|2
|x|2
|x|2
p
θ
uf (u) ≥ α
θ

|u| −
θ
φ1 (x)
k2
k2
k2
Rn
Rn
Rn
(2.21)
≥−
φ1 (x).
|x|≥k

20


Because θ (s) = 0 for all 0 ≤ s < 1 and s > 2, we obtain

Rn

|x|2
k2

2x
θ
k2

u∇u ≤

√
k≤|x|≤ 2k

2|x|
C
C|u||∇u|
≤
u . ∇u . (2.22)
k2
k

Similarly,
2x
θ
2
Rn k
Next, the right hand side
g(t)θ

C
u . ∇ut .
k

u∇ut ≤

|x|2
g(t)θ
u =
u
k2

|x|≥k
λ
1
|x|2
2
≤
|u|
+
θ
2 |x|≥k
k2
2λ

|x|2
k2

Rn

|x|2
k2

(2.23)

(2.24)
|g(t)|2 .
|x|≥k

Combining (2.20)-(2.24), we have
d
dt

|x|2
k2

θ
Rn

2

|u| + ε

θ
Rn

|∇u|2

|x|2
|∇u|2
2
k
Rn
Rn
1
C
≤
|g(t)|2 + 2
φ1 (x) + ( u 2 + ∇u
λ |x|≥k
k
|x|≥k

+σ

|x|2
k2

|x|2
k2

θ

|u|2 + ε

θ

2

+ ∇ut 2 ).
(2.25)

By Gronwall’s lemma, we have
θ
Rn

|x|2
k2

≤e

2

|u| + ε

−σ(t−τ )

θ
Rn

( uτ

2

|x|2
k2
2

+ ε ∇uτ

|∇u|2

t

eσs |g(s)|2

eσs φ1 (x) +
|x|≥k

|x|≥k

τ

t

−σt

+ 2e−σt
τ

t

e−σt
)+
λ
Ce
k

eσs ( u

2

+ ∇u

2

+ ∇ut 2 ) .

τ

(2.26)
Since uτ ∈ D is bounded in H (R ) ∩ L (R ), and φ1 ∈ L (R ), we have
1

n

p

lim sup e−σ(t−τ ) ( uτ
τ →−∞

21

2

n

+ ∇uτ 2 ) = 0,

1

n

(2.27)


and

t

lim sup 2e

−σt

k→+∞

−∞

eσs φ1 (x) = 0.

(2.28)

eσs |g(s)|2 = 0.

(2.29)

|x|≥k

Using (1.7), we have
e−σt
lim sup
λ
k→+∞

t
|x|≥k

τ

From (2.3), integrating from τ to t we have
t

eσs

u

2

2

+ ∇u

+ u

τ

p
Lp (Rn )
t

στ

≤ Ce

uτ

2

+ ∇uτ

2

eσs g(s)

+C

2

+ Ceσt < +∞.

τ

(2.30)

Similarly, from (2.16), we get that
t

eσs ∇ut (s)

2

τ
στ

≤C e

uτ

2

+ ∇uτ

2

+

uτ pLp (Rn )

t
σt

eσs g(s)

+e +

2

< +∞.

−∞

(2.31)
Combining (2.30) and (2.31) we get
Ce−σt
lim sup lim sup
k
k→+∞ τ →−∞

t

eσs ( u

2

+ ∇u

2

+ ∇ut 2 ) = 0.

(2.32)

τ

Applying (2.27)-(2.29) and (2.32) to (2.26), we obtain the existence of K0 >
0 and τ0 ∈ R such that, for any k ≥ K0 , any τ ≤ τ0 and any uτ ∈ D, we
have
|x|2
|x|2
2
|u|
+
ε
θ
|∇u|2 ≤ η,
θ
2
2
k
k
Rn
Rn
hence

2
2
√ |u| + ε
√ |∇u| ≤ η.
|x|≥ 2k

|x|≥ 2k

22


Theorem 2.8. The process {U (t, τ )} corresponding to problem (1.1) possesses an (H 1 (Rn ) ∩ Lp (Rn ), L2 (Rn )) - pullback attractor.
Proof. Since {U (t, τ )} has a family of (H 1 (Rn ) ∩ Lp (Rn ), L2 (Rn )) - pullback
absorbing sets, we remain to show that {U (t, τ )} is (H 1 (Rn )∩Lp (Rn ), L2 (Rn ))
- pullback asymptotically compact.
Let D ∈ B(H 1 (Rn ) ∩ Lp (Rn )), {τn } be a sequence in R such that
τn → −∞, and {uτn } ⊂ D. We will prove that for any η > 0, the set
{U (t, τn )uτn }n≥1 has a finite covering of balls of radius η in L2 (Rn ). For a
given K > 0, denote by
ΩK = {x : |x| ≤ K}

and

ΩcK = {x : |x| > K}.

By Lemma 2.7, there exists K = K(t, D, η) and τ0 = τ0 (t, D, η) such that
for τ ≤ τ0 ,
U (t, τ )uτ L2 (ΩcK ) ≤ η/4.
Since τn → −∞, there is N1 > 0 such that τn ≤ τ0 for all n ≥ N1 , and
hence we obtain that, for all n ≥ N1 ,

U (t, τn )uτn

L2 (ΩcK )

≤ η/4.

(2.33)

On the other hand, by Proposition 2.4, there exist C = C(t) > 0 and
N2 > 0 such that for all n ≥ N2 ,
U (t, τn )uτn

H 1 (ΩK )

≤ C.

(2.34)

By the compactness of the embedding H 1 (ΩK ) ⊂ L2 (ΩK ), the sequence
{U (t, τn )uτn } is precompact in L2 (ΩK ). Therefore, for the given η > 0,
{U (t, τm )uτn } has a finite covering in L2 (ΩK ) of balls of radius less than η/4,
which combine with (2.33) shows that {U (t, τn )τn } has a finite covering in
L2 (Rn ) of balls of radius less than η, and thus {U (t, τn )uτn } is precompact
in L2 (Rn ).
Lemma 2.9. For any η > 0, any t ∈ R, and any D ∈ B(H 1 (Rn ) ∩ Lp (Rn )),
there exist M > 0, τ0 ≤ t such that
mes(Ω(|U (t, τ )uτ | ≥ M )) ≤ η, ∀τ ≤ τ0 , ∀uτ ∈ D,
where Ω(|u(t)| ≥ M ) = {x ∈ Rn : |u(t, x)| ≥ M } and mes is the Lebesgue
measure.
23