The Global and Pullback Attractors for a Strongly Damped Wave Equation with Delays* ()
1. Introduction
Let 
be a bounded domain with smooth boundary
, we study the following initial boundary value problem
(1.1)
where 
is the source intensity which may depend on the history of the solution, 
are the positive constants, 
is the initial value on the interval 
where
, and 
is defined for 
as
. The assumption on 
and 
will be specified later.
It is well known that the long time behavior of many dynamical system generated by evolution equations can be described naturally in term of attractors of corresponding semigroups. Attractor is a basic concept in the study of the asymptotic behavior of solutions for the nonlinear evolution equations with various dissipation. There have been many researches on the long-time behavior of solutions to the nonlinear damped wave equations with delays. The existence of random attractors has been investigated by many authors, see, e.g., [1-4]. A new type of attractor, called a pullback attractor, was proposed and investigated for non-autonomous or these random dynamical systems. The pullback attractor describing this attractors to a component subset for a fixed parameter value is achieved by starting progressively earlier in time, that is, at parameter values that are carried forward to the fixed value. see [5-20]. However, to our knowledge, in the case of functional differential equations of second order in time, there is only partial results.
Recently, In [5], some results on pullback and forward attractor for the following strongly damped wave equation with delays
have been analyzed.
In this work, first, we apply the means in [3] to provide the existence of global attractor, for the dynamical system generated by the initial value problem (1.1). The key is to deal with the nonlinear terms and the delay term is difficult to be handled, so we aimed at showing that it is dissipative and the solution is bounded and continuous with respect to initial value. Hence we can discover the global attractor. Then, we aim to obtain the pullback attractor. The technology we use is introduced in [1], that is, we divide the semigroup into two: the one is asymptotically close to 0, while the other is uniformly compact, so we can get the pullback attractor.
Now, we state the general assumptions for problem (1.1) on 
and
.
Let
, then there exist positive constants 
such that the followings hold true
(G1).
;
(G2).
;
(G3).
;
(G4).
;
(G5).
;
(G6).
;
(G7).
.
For any
, set
, by
, there are 
and
, for any
, we have
H1. 
is continuous;
H2.
;
H3. 
such that 
H4. 
such that
H5.
, and there exists 
such that, for any
, the Frechet derivative 
satisfies
The rest of this paper is organized as follows. In Section 2, we introduce basic concepts concerning global and pullback attractor. In Section 3, we obtain the existence of the global attractor. In Section 4, we obtain the existence of the pullback attractor.
2. Preliminaries
In this section,firstly, we recall some basic concepts about the global attractor.
Definition 2.1 ([3]) Let 
be a Banach space and
be a family of operators on
. We say that 
is norm-to-weak continuous semigroup on
, if 
satisfies:
[1)]
;
[2)]
;
[3)] 
if 
and 
in
.
The strong continuous semigroup and the weak semigroup are both the norm-to-weak continuous
Definition 2.2 ([3]) The semigroup 
is called satisfying Condition (C) in 
if and only if for any bounded set 
of 
and for any
, there exist a positive constant 
and a finite dimensional subspace 
of X, such that 
is bounded and
where 
is the canonical projector.
Lemma 2.1 ([3]) Let 
be a Banach space and
be a norm-to-weak continuous semigroup on
. Then 
has a global attractor in 
provided that the following conditions hold:
1) 
has a bounded absorbing set 
in
;
2) 
satisfies Condition (C) in
.
Then, we state the concepts and some result about the process and the pullback attractor.
Instead of a family of the one-parameter map
, we need to use a two-parameter semigroup or process 
on the complete metric space
, 
denotes the value of the solution at time 
which was equal to the initial value 
at time
.
The semigroup property is replaced by the process composition property
and, obviously, the initial condition implies
.
Definition 2.3 Let 
be the two-parameter semigroup or process on the complete metric space
. A family of compact set 
is said to be a pullback attractor for 
if, for all
, it satisfies
[1)] 
for all
, and
[2)]
, for all bounded
, and all
.
Definition 2.4 The family 
is said to be
1) pullback absorbing with respect to the process
, if for all 
and all bounded
, there exists 
such that 
for all
;
2) pullback attracting with respect to the process
, if for all
, all bounded
, and all
, there exists 
such that for all 
3) pullback uniformly absorbing (respectively uniformly attracting) if 
in pact (a) (respectively 
in part (b)) does not depend on the time
.
Theorem 2.1 Let 
be a two-parameter process, and suppose 
is continuous for all
. If there exists a family of compact pullback attracting sets
, then there exists a pullback attractor
, such that 
for all
, and which is given by
We set
, where
, which are Hilbert spaces for the usual inner product and associated norms. we denote by 
the first eigenvalue of 
in
.
Our problem can be written as a second-order differential equation in
:
(2.1)
3. Existence of the Global Attractor
In this section, our objection is to show that the well-posed of the solution and the existence of global attractor for the initial boundary value problem (1.1), we assume that
.
Let 
and
, then by the transformation
. The initial boundary value problem (2.1) is equivalent to
(3.1)
with the initial value conditions
Theorem 3.1 Assume that the hypotheses on 
and 
hold for all 
and
, 
are the positive constants. Then the initial boundary value problem (3.1) has the unique solution 
for all
.
Proof. Taking the inner product of the Equation (3.1) with 
in
, we find that
(3.2)
Since 
and
we deal with the terms in (3.2) one by one as follows
(3.3)
(3.4)
(3.5)
(3.6)
(3.7)
By (3.3)-(3.7), it follows from that
Since 
and
, this will imply
, then we have
(3.8)
Set
, then (3.8) can be written as following
As our assumptions ensure that
, then we can choose 
small enough such that
. For this choice, we have
Hence, we can get the following inequality
By integrating over the interval
, we deduce
(3.9)
Since
So we can have
(3.10)
Noticing
, we obtain
(3.11)
In the Bounded set
, for any
, there exists a constant 
such that
(3.12)
(3.13)
(3.10)-(3.13) means that
(3.14)
(3.15)
Hence, by (3.12)-(3.14) and the choice of
, (3.9) can be rewritten
(3.16)
So we can get by (3.16)
which implies,for 
(3.17)
If we denote
then (3.17) yields that
(3.18)
which means that the initial boundary value problem (3.1) has the solution
.
Now, we prove the uniqueness of the solution. Assume that 
and 
are the two solutions of the initial boundary value problem (3.1), 
are the corresponding initial value,we denote
. Therefore we have
we take the inner product of the above equation with 
and we obtain
(3.19)
Since
So (3.20) can yields that
(3.20)
(3.21)
Integrating (3.21) over the interval
, we can get
Set
, then we have
Combining the Gronwall Lemma, we get
(3.22)
If 
stand for the same initial value, there has
that shows that
that is
therefore
we get the uniqueness of the solution. So the proof of the theorem 3.1. has been completed.
By the theorem 3.1,we obtain the global smooth solution 
continuously depends on the initial value
, the initial boundary value problem (1.1) generates a continuous semigroup
.
Then
is a bounded absorbing set for the semigroup 
generated by (1.1).
Under the assumption on 
and
, we can get the nonlinear term 
is compact and continuous, 
is continuous. Next, our object is to show that the 
semigroup 
satisfies cindition C.
Theorem 3.2 Assume that the hypotheses on 
and 
hold for all
, 
are positive constants. Then the 
semigroup 
associated with initial value problem (3.1) satisfies
, that is, there exists 
and 
, for any 
such that
Proof. Let 
be the eigenvalues of 
and 
be the corresponding eigenvectors, 
, without loss of generality, we can assume that
, and
.
It is well known that 
form an orthogonal basis of
. We write
Since 
and 
is compact, for any
, there exists some 
such that
(3.23)
(3.24)
where 
is orthogonal projection and 
is the radius of the absorbing set. For any
, we write
We note that
Taking the inner product of the second equation of (3.1) with 
in
, After a computation like in the proof of Theorem 3.1, we can yield that
(3.25)
This is the same as in the proof of the Theorem 3.1, except for a replacement of 
with
. Combined with (3.23) , (3.24) and (3.4), then we have
Choose 
, we can get
By Gronwall lemma, we can obtain
for all 
and
. This shows that Condition C is satisfied, and the proof is completed.
Due to Lemma 2.1, Theorem 3.1 and Theorem 3.2, we obtain the following Theorem
Theorem 3.3 Assume that the hypotheses on 
and 
hold for all
, 
are positive constants. Then the 
semigroup 
associated with initial value problem (3.1) has a global attractor in E.
4. Existence of the Pullback Attractor
In this subsection, we assume that
, we aim to study the pullback attractor for the initial value problem (1.1).
From Theorem 3.1, the initial value problem (1.1) generates a family two-parameter semigroup 
in
, which can be defined by
Lemma 4.1 Let 
be the two initial values for the problem (1.1), 
is the initial time, Denote by 
and 
the corresponding solutions to (1.1). Then, there exists a constant 
which is independent of initial value value and time, such that the following estimates hold:
(4.1)
(4.2)
Proof. We denote
, by (3.22), we can get (4.1) easily.
If we consider
, then 
for any
, and
Thus,
.
Theorem 4.1 The mapping 
is continuous for any
.
Proof. Let 
be the initial value for the problem (1.1) and
. Denote by 
and 
the corresponding solutions to (1.1). Then, writing again 
we obtain the following. If
, then 
and
Thus, we have
whence
which implies the continuity of
.
Theorem 4.2 Assume that the hypotheses on 
and 
hold with
, 
are the positive constants.
Suppose in addition that
. Then exists a family 
of bounded sets in 
which is uniformly pullback absorbing fir the process
. Moreover, 
for all
, where 
is the bounded set in
.
Proof. By (3.18), we can have
and, in particular,
(4.3)
Moreover, as 
and 
for
, then inequality (4.3) holds true for
.
If we take now
, then for all 
we have 
and so
(4.4)
or, in other words,
Therefore, there exists 
such that
which means that the ball 
is uniformly pullback absorbing for the process
.
Remark: On the one hand, observe that if 
and
, then
and
with
. As a sequence of (4.4) we have
or ,we have 
On the other hand, (4.3) implies,
,
Theorem 4.3 Under the assumption in Theorem 4.1. Then there exists a compact set 
which is uniformly pullback attracting for the process
, and consequently, there exits the pullback attractor.
. Moreover, 
for all
.
Proof. For each
, the norm
is equivalent to
. This allows us to obtain absorbing ball for the original norm by proving the existence of absorbing balls for this new norm for some suitable value of
.
Indeed, let us denote
. Noticing that for 
it follows that
we then have 
.
Let 
be a bounded set, i.e. there exists 
such that for any 
it holds
Denote by 
the solution of the problem (2.1), and consider the problems:
(4.5)
(4.6)
From the uniqueness of the solution of problems (2.1), (4.5) and (4.6) it follows that
Consequently, 
can be written as
where 
and 
are the solutions of (4.5) and (4.6) respectively.
First, thanks to (4.4), but with
, it follows that
(4.7)
Furthermore, for 
and
,
with
. Thus, Equation (4.7) implies in particular
Then we can obtain that
whence,
Next, fix 
and denote
Then, for
,
(4.8)
and for
, we have
(4.9)
Then, we deduce from the assumption on 
that
and
. Arguing as we did in order to obtain (4.8) and (4.9), we have
(4.10)
and
(4.11)
Let us denote
and make use of the estimates in Theorem 4.2. On the one hand, for all
,
but, as (4.4) and (4.7) ensure
if we denote by
then, in particular,
.
Noticing that
, the Gronwall lemma leads us to
On the other hand, if
, we deduce that
and, from (4.8) and (4.10),
Once again, the Gronwall lemma implies that
Then, there exists 
such that, if
,
Recalling that
, if we fix
, take 
and denote 
we have, provided 
is large enough, that
In conclusion, there exists 
such that for all
, and all
,
Denoting
, we have for all 
where
.
But as for all 
and
, we get 
and
, and, consequently, for all 
and
,
which shows that
for all 
and
. This means that the all
is the bounded set in 
which , in addition, is uniformly absorbing for the family of operators
. As 
is the bounded set in
, then there exists 
such that
and, therefore, the bounded set 
given
is uniformly pullback absorbing for 
in
.
By Ascoli-Arzelà theorem, we can prove that 
is compact, so 
is a family of compact subsets in
, which is also uniformly pullback attracting for
, and the proof has been completed.
NOTES
#Corresponding author.