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
![](https://www.scirp.org/html/3-2340095\8d807637-e6bf-4d0f-a63c-edcd94879a88.jpg)
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
![](https://www.scirp.org/html/3-2340095\673f85e3-63d8-43e7-a123-7773abadf8b3.jpg)
H1.
is continuous;
H2.
;
H3.
such that ![](https://www.scirp.org/html/3-2340095\a2a0ac7d-e74d-451f-95be-3c38651a3a3d.jpg)
![](https://www.scirp.org/html/3-2340095\9ecdc19d-6c24-453e-8929-71d0b2aead28.jpg)
H4.
such that
![](https://www.scirp.org/html/3-2340095\39a5d4ba-32e0-47a2-963a-e488f5bf4aba.jpg)
![](https://www.scirp.org/html/3-2340095\2fb52b7f-01d5-4792-b9ce-4d907ffbdf69.jpg)
H5.
, and there exists
such that, for any
, the Frechet derivative
satisfies
![](https://www.scirp.org/html/3-2340095\12929b4a-3d6a-4777-bf7d-ce8db52f4735.jpg)
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
![](https://www.scirp.org/html/3-2340095\6c936388-e59a-4faa-bcdc-42793542b869.jpg)
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 ![](https://www.scirp.org/html/3-2340095\0d553e85-5b84-4acd-8995-ac038919843f.jpg)
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
![](https://www.scirp.org/html/3-2340095\3058abbc-2192-411c-99e5-dcdc2369158b.jpg)
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 ![](https://www.scirp.org/html/3-2340095\e7eede06-2b4a-44ae-a445-5df1eb204df9.jpg)
![](https://www.scirp.org/html/3-2340095\1d078068-7908-4d03-83a1-f45084e9d38b.jpg)
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
![](https://www.scirp.org/html/3-2340095\e4c6c30c-0ef8-4f44-ab4e-0d64d2ff7284.jpg)
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
![](https://www.scirp.org/html/3-2340095\fcdf6309-a98d-4f5f-b35a-3ecd50e7a47c.jpg)
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
![](https://www.scirp.org/html/3-2340095\6a1dd896-19fa-4b5f-826c-ae369bab4811.jpg)
Since
and
, this will imply
, then we have
(3.8)
Set
, then (3.8) can be written as following
![](https://www.scirp.org/html/3-2340095\d0e264b7-3241-46f4-ac13-7a39c803fd92.jpg)
As our assumptions ensure that
, then we can choose
small enough such that
. For this choice, we have
![](https://www.scirp.org/html/3-2340095\72fa8856-9fcf-45ce-8ae8-5ddf6dc83934.jpg)
Hence, we can get the following inequality
![](https://www.scirp.org/html/3-2340095\1f33871a-69c2-4169-9b3c-59cb0cee9c9f.jpg)
By integrating over the interval
, we deduce
(3.9)
Since
![](https://www.scirp.org/html/3-2340095\3154d14a-81e0-4ef8-aa32-63185a67e095.jpg)
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)
![](https://www.scirp.org/html/3-2340095\e640ac47-c632-489a-a832-26da05d39c40.jpg)
which implies,for ![](https://www.scirp.org/html/3-2340095\d5dc9f15-308a-48e1-a6e0-d7401235b7ee.jpg)
(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
![](https://www.scirp.org/html/3-2340095\00a722be-999a-4b6a-b0d8-491479eabb66.jpg)
we take the inner product of the above equation with
and we obtain
(3.19)
Since
![](https://www.scirp.org/html/3-2340095\496fc476-c26e-449d-9068-5c374980e8e9.jpg)
![](https://www.scirp.org/html/3-2340095\38d5aec6-0b07-4e3e-911e-aa3ab5088d27.jpg)
So (3.20) can yields that
(3.20)
(3.21)
![](https://www.scirp.org/html/3-2340095\be3def52-a2f1-418e-9863-ac57af3f9803.jpg)
Integrating (3.21) over the interval
, we can get
![](https://www.scirp.org/html/3-2340095\91f62e53-0d82-4317-b16c-bd5768634fd6.jpg)
Set
, then we have
![](https://www.scirp.org/html/3-2340095\9940fd33-2789-486a-a66a-e5d51a55db39.jpg)
Combining the Gronwall Lemma, we get
(3.22)
If
stand for the same initial value, there has
![](https://www.scirp.org/html/3-2340095\edff0643-fca1-4bb0-81bd-e5f25f852b51.jpg)
that shows that
![](https://www.scirp.org/html/3-2340095\59bd5ed9-8dbd-467f-8949-4a78cfddcac9.jpg)
that is
![](https://www.scirp.org/html/3-2340095\810abde9-667d-4ef6-b91d-493450d0b556.jpg)
therefore
![](https://www.scirp.org/html/3-2340095\ba9240df-d9db-4761-8015-f678544820b1.jpg)
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
![](https://www.scirp.org/html/3-2340095\5eebd522-af50-474b-8740-b62e465e5d41.jpg)
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
![](https://www.scirp.org/html/3-2340095\0d5c5c15-b4d6-4069-842e-60b7dce55c87.jpg)
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
![](https://www.scirp.org/html/3-2340095\d564cefe-c007-4b64-b274-703468cd4d13.jpg)
We note that
![](https://www.scirp.org/html/3-2340095\3325999b-c992-41e2-9eb1-b0df130b77e1.jpg)
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
![](https://www.scirp.org/html/3-2340095\296bc47d-b01d-4705-9332-7dc4912f7039.jpg)
Choose
, we can get
![](https://www.scirp.org/html/3-2340095\f46e0e0b-922c-459c-80b8-8156c2483d31.jpg)
By Gronwall lemma, we can obtain
![](https://www.scirp.org/html/3-2340095\7bf25ea6-6c28-444a-a91d-4a22c237ac46.jpg)
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
![](https://www.scirp.org/html/3-2340095\45799111-d589-4aaf-917a-25d855f301bb.jpg)
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
![](https://www.scirp.org/html/3-2340095\fcd7a969-3495-45d2-a52e-92902b8e5982.jpg)
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
![](https://www.scirp.org/html/3-2340095\53bcff2a-1252-49c8-b711-2e462f185cdb.jpg)
Thus, we have
![](https://www.scirp.org/html/3-2340095\1248e4ef-2b23-42b2-a6cb-acd689f5fa4f.jpg)
whence
![](https://www.scirp.org/html/3-2340095\c01fb2a1-90a4-453c-88aa-f0a62feb1e09.jpg)
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 ![](https://www.scirp.org/html/3-2340095\d30332b5-4925-496c-9c3f-d28092a7f0bc.jpg)
which is uniformly pullback absorbing fir the process
. Moreover,
for all
, where
is the bounded set in
.
Proof. By (3.18), we can have
![](https://www.scirp.org/html/3-2340095\8910a396-3857-46a5-8670-6db722d2d4ac.jpg)
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,
![](https://www.scirp.org/html/3-2340095\363c8bf9-a23c-461c-bc9d-5d0014ac0679.jpg)
Therefore, there exists
such that
![](https://www.scirp.org/html/3-2340095\205c80b3-f590-4ca3-9221-5ed982cef177.jpg)
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
![](https://www.scirp.org/html/3-2340095\0b0b2e4a-0b83-4d0f-91e0-d410ca535b50.jpg)
or ,we have ![](https://www.scirp.org/html/3-2340095\532cf5f4-5781-48d4-8ce2-be13b70e4b3f.jpg)
![](https://www.scirp.org/html/3-2340095\a1b90283-b104-41ab-92d2-03f5019f756a.jpg)
On the other hand, (4.3) implies,
,
![](https://www.scirp.org/html/3-2340095\27713d2a-26bb-48ff-b383-47ba4f167f14.jpg)
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
![](https://www.scirp.org/html/3-2340095\5a23486c-fd2b-4e20-b410-05436174c85f.jpg)
we then have
.
Let
be a bounded set, i.e. there exists
such that for any
it holds
![](https://www.scirp.org/html/3-2340095\268f11d7-2d7c-43d8-aade-edb251bc9b7c.jpg)
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
![](https://www.scirp.org/html/3-2340095\efd04ccb-2690-4722-ae09-4c45b9232e9b.jpg)
Consequently,
can be written as
![](https://www.scirp.org/html/3-2340095\afd312cd-8f3e-4bda-a55a-a2e72ed14e0e.jpg)
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
,
![](https://www.scirp.org/html/3-2340095\93feb130-860a-4061-aaf4-9a6487fc7975.jpg)
with
. Thus, Equation (4.7) implies in particular
![](https://www.scirp.org/html/3-2340095\5e3e598a-1c48-4085-bfa2-77f051d2e806.jpg)
Then we can obtain that
![](https://www.scirp.org/html/3-2340095\17ef3759-50ad-4a7d-9a19-11f60a50aa87.jpg)
whence,
![](https://www.scirp.org/html/3-2340095\e9d7399b-779a-4d7f-9ba6-e2b676ee8092.jpg)
Next, fix
and denote
![](https://www.scirp.org/html/3-2340095\e96ddaad-aa19-4112-a43d-f673227a6df3.jpg)
![](https://www.scirp.org/html/3-2340095\0fae79df-2296-4ce5-b828-a7b4423b2b9b.jpg)
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
,
![](https://www.scirp.org/html/3-2340095\4ca2982b-55f0-4fb9-85ee-7df91213f444.jpg)
but, as (4.4) and (4.7) ensure
![](https://www.scirp.org/html/3-2340095\d0e5753b-0491-4f8f-a73e-ab9a169e0890.jpg)
if we denote by
![](https://www.scirp.org/html/3-2340095\371bf5ef-2ece-4485-91f8-3783ae288cae.jpg)
then, in particular,
.
Noticing that
, the Gronwall lemma leads us to
![](https://www.scirp.org/html/3-2340095\a714f78f-2aab-4c61-9b98-138c298fff45.jpg)
On the other hand, if
, we deduce that
![](https://www.scirp.org/html/3-2340095\671c5a3c-e48b-4f0d-ab59-ce08bb82cde2.jpg)
and, from (4.8) and (4.10),
![](https://www.scirp.org/html/3-2340095\cab209eb-c192-4f02-97bb-6c35ae6820fd.jpg)
Once again, the Gronwall lemma implies that
![](https://www.scirp.org/html/3-2340095\88eddedf-ada8-470f-bce8-276f27ba884f.jpg)
Then, there exists
such that, if
,
![](https://www.scirp.org/html/3-2340095\3dda7bb9-6fb0-4911-81be-4544dc5ab2fa.jpg)
Recalling that
, if we fix
, take
and denote
we have, provided
is large enough, that
![](https://www.scirp.org/html/3-2340095\016d1f8d-9d8e-432a-af87-8bfb26036c53.jpg)
In conclusion, there exists
such that for all
, and all
,
![](https://www.scirp.org/html/3-2340095\1705fdf6-89f9-4fb1-8111-6181eb31a752.jpg)
Denoting
, we have for all ![](https://www.scirp.org/html/3-2340095\8b2cc69d-0697-4e82-a916-911801347bc8.jpg)
![](https://www.scirp.org/html/3-2340095\a6830231-2679-4c89-8acb-758969602c04.jpg)
where
.
But as for all
and
, we get
and
, and, consequently, for all
and
,
![](https://www.scirp.org/html/3-2340095\232c588c-5650-4b5d-8e8f-718325eefa23.jpg)
which shows that
![](https://www.scirp.org/html/3-2340095\827b46e7-ccc4-4493-8930-d1c9d151ba52.jpg)
for all
and
. This means that the all
is the bounded set in ![](https://www.scirp.org/html/3-2340095\702136f0-e2a0-4e06-9bd3-9d0247c163f1.jpg)
which , in addition, is uniformly absorbing for the family of operators
. As
is the bounded set in
, then there exists
such that
![](https://www.scirp.org/html/3-2340095\57e43a2e-a411-48b6-a3ac-067dc487faec.jpg)
and, therefore, the bounded set
given
![](https://www.scirp.org/html/3-2340095\84417ff3-ff7f-4d3b-bbf8-bd84ae426a07.jpg)
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.