The Global and Pullback Attractors for a Strongly Damped Wave Equation with Delays *

In this paper, we study the global and pullback attractors for a strongly damped wave equation with delays when the force term belongs to different space. The results following from the solution generate a compact set.


Introduction
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 . The assumption on   g u and   f x 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][2][3][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][6][7][8][9][10][11][12][13][14][15][16][17][18][19][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 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 con- stants such that the followings hold true , by 1

7
G G  , there are and 6 0, , . ; , and there exists such that, for any   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.

Preliminaries
In this section,firstly, we recall some basic concepts about the global attractor.
Definition 2.1 ([3]) Let X be a Banach space and     0 t S t  be a family of operators on X .We say that  , and , where , .
We set , where

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 .
with the initial value conditions we deal with the terms in (3.2) one by one as follows Hence, we can get the following inequality As our assumptions ensure that , then we can choose small enough such that . For this choice, we have By integrating over the interval   ,t  , we deduce In the Bounded set , V H , for any , there exists a constant such that So we can get by (3.16) we take the inner product of the above equation with w and we obtain , , Combining the Gronwall Lemma, we get If ,   stand for the same initial value, there has  be the eigenvalues of and u  j w be the corresponding eigenvectors, , without loss of generality, we can assume that 1, 2,

  
, for all 0, 2 m R where m is orthogonal projection and is the radius of the absorbing set.For any By Gronwall lemma, we can obtain

Existence of the Pullback Attractor
In this subsection, we assume that f H  , we aim to study the pullback attractor for the initial value problem the corresponding solutions to (1.1).Then, there exists a constant 1 0   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.
, and be the initial value for the problem (1.1) and the corresponding solutions to (1.1).Then, writing again we obtain the following.If Thus, we have

Assume that the hypotheses on g and hold with
, h 0 0 m  ,   are the positive constants.
Suppose in addition that Then exists a family

 
0 for all , where is the bounded set in and, in particular, , then inequality (4.3) holds true for or, in other words, , , , uniformly pullback absorbing for the process   , U   t .: Remark On the one hand, observe that if 0   and , then or ,we have . 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     , 0, : Noticing that for Let , V H be a bounded set, i.e. there exists such that for any 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 where Then, for , and for , we have Then, we deduce from the assumption on that h Arguing as we did in order to obtain (4.8) and (4.9), we have but, as (4.4) and (4.7) ensure if we denote by On the other hand, if , we deduce that 13) Hence, by (3.12)-(3.14)and the choice of (3.10)-(3.13)means that

. 18 )
which means that the initial boundary value problem (3solutions of the initial boundary value problem(3.1),,   are the corresponding initial value,we denote 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  , 25) This is the same as in the proof of the Theorem 3.1, except for a replacement of 1  with 1 m   .Combined with (3.23) , (3.24) and (3.4), then we have shows that Condition C is satisfied, and the proof is completed.
, and the proof has been completed.