Let be a bounded domain with smooth boundary, we study the following initial boundary value problem

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.

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:

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

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

Since andwe deal with the terms in (3.2) one by one as follows

By (3.3)-(3.7), it follows from that

Since and, this will imply, then we have

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

Since

So we can have

Noticing, we obtain

In the Bounded set, for any, there exists a constant such that

(3.10)-(3.13) means that

Hence, by (3.12)-(3.14) and the choice of

, (3.9) can be rewritten

 So we can get by (3.16)

which implies,for

If we denote

then (3.17) yields that

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

Since

So (3.20) can yields that

Integrating (3.21) over the interval, we can get

Set, then we have

Combining the Gronwall Lemma, we get

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

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

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.  

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:

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,

Moreover, as and for, then inequality (4.3) holds true for.

If we take now, then for all we have and so

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:

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

Furthermore, for and,

with. Thus, Equation (4.7) implies in particular

Then we can obtain that

whence,

Next, fix and denote

Then, for,

and for, we have

Then, we deduce from the assumption on that

and

. Arguing as we did in order to obtain (4.8) and (4.9), we have

and

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.