1. Introduction
Let 
be an open bounded connected domain in 
with smooth boundary
. In this work, we are devoted to the investigation of the following problem for a perturbed non-autonomous beam equation
(1)
with the following initial and boundary conditions
(2)
(3)
where 
denotes a real-valued unknown function, and describes the transversal motion of the non-autonomous beam. 
and 
is the time-periodic external force.
System (1)-(3) is derived from the vibrations of an non-autonomous beam equation, and its dynamical setting is presented by Woinowsky-Krieger [1] as the new idea in fields of J. Appl. Mech. For the non-autonomous wave equations, there are many interesting results were published focusing different respects (see [2,3]). The focus of this work is the study of the long-term properties of the dynamical system generated by global attractor, please refer the reader to [4,5] and references therein.
This paper is organized as follows. In Section 2, we introduce the main assumptions and discuss the existence of a family of solution operators 
to the problem (1)-(3). Section 3 is devoted to the existence of the bounded absorbing set. In Section 4, we show the continuity of the semigroup operator. Finally, in Section 5, the global attractor is obtained.
2. Assumptions and the Existence of the Solution Operators S(t, τ)
Firstly, we assume that the nonlinear functions 
satisfied the following conditions. Namely, there exist two constants 
such that
(H1)
(4)
(H2)
(5)
(H3)
(6)
(H4)
(7)
For simplicity, we define
and
with the usual notation, we write 
and the scalar products and norms on 
and
, respectively
For the linear self-adjoint and positive operator
, let us write
,
.
The space 
is dense in
. Next, we define the power 
of
, 
, which operate on the spaces
, and write
, which turns out to be a Hilbert space with the inner product and the norm
and 
is an isomorphism from 
onto
,
.
From the Poincaré inequality, there exists the constant
, such that:
where 
is the first-eigenvalue of
. For
, we consider the abstract Cauchy problem on 
in the unknown variables 
(8)
The following well-posedness result holds.
Theorem 1. Suppose that 
and the conditions (4)-(7) hold. If the nonlinear functions 
. Then, for any initial value 
is given in
, problem (8) admits a unique solution 
in the class
where
.
Furthermore, calling 
the difference of any two solutions corresponding to initial data having norm less than or equal to
, there exists 
such that
(9)
We omit the proof, based on a standard FeadoGalerkin approximation procedure together with a slight generalization of the usual Gronwall’s lemma. Theorem 1 translates into the existence of the solution operators
acting as
Remark 1. In the non-autonomous case, namely, when both 
and
are time-dependent, the two-parameter family 
fulfills the semigroup property
.
is the identity operator,
.
Thus, 
is a continuous semigroup of operators on
.
3. The Absorbing Set
In this section, we prove the existence of an absorbing set for the semigroup
. Combining with (4) and (5), there exist two constants 
such that
(10)
(11)
Lemma 1. Under the hypotheses of Theorem 1, For the ball
, 
, centered at 0 of radius M, is an absorbing set for the semigroup 
in E.
Proof. Let us begin with 
be fixed and 
is chosen such that
. We set
and rewrite (1) as follows
(12)
Taking the scalar product in H of (12) with
, and we obtain the desired form
(13)
Using the Hölder inequality and the Poincaré inequality, we have the estimate
(14)
Exploiting (11), we lead to
(15)
and,
(16)
where
. From (13)-(16), it follows that
(17)
So, in the light of condition (10), we have
Taking
, we obtain
(18)
where
.
Applying the Gronwall’s Lemma, we obtain the following absorbing inequality
or
(19)
Taking
.
Obviously,
. Furthermore, let us denote 
be a bounded closed ball of 
centered at 0 with radius 
So, 
is a bounded absorbing set of analytic semigroup 
of (1)-(3). We complete the proof.
4. Continuity of the Semigroup
In this section, we prove the continuity of the semigroup 
in
. For this reason, we assume
(20)
Lemma 2. Under the hypotheses of Theorem 1, the mapping
, for
, is continuous in
, and the semigroup 
associated with the initial-boundary value problem (1) is a 
-semigroup in
.
Proof. Assume that 
be a bounded positive invariant set for
, and initial-data 
. Let 
be the two corresponding solutions of (1).
Assume
. We claim that the proof is similar to the Lemma 1. Then, from (1) we have
(21)
By multiplying (21) by 
and integrating over
, we find
(22)
Next, we are devoted to estimate (22). By the same method that we obtained (14), it follows that
(23)
And, according to (20), for
, there exists a constant
, such that
(24)
By (24) and the Sobolev embedding theorem, we can obtain 
are uniformly bounded in
, that is, there exists a constant
, such that
(25)
From (24), (25), it follows that
(26)
Thus
Meanwhile, we know easily
(27)
Due to the continuity of
, we can note the fact that
We see that easily
.
This implies
(28)
where
.
Combining with (27) and (28), it is obtained directly
(29)
Thanks to (22)-(29), and the usual Gronwall’s Lemma, we have
So we complete the proof.
5. Existence of the Global Attractor
Theorem 2. Under the hypotheses of Theorem 1, the semigroup 
associated with the initial-boundary value problem (1) possesses a global attractor 
in 
which attracts all bounded subsets of
.
Proof. Let 
be a solution of (12) with initial value, for any
, and, it can be decomposed into 
where 
, satisfy, respectively,
(30)
and
(31)
Applying two lemmas showed below. We also define a new inner product and norm in
. Obviously, by embedding theorem, (30) are easily concluded through some simple computation
. It is sufficient to prove that 
is asymptotically smooth in
.
Taking the inner product in 
of (12) with 
, we obtain
According to the uniform boundedness of 
in space
, the Sobolev Embedding theorem and the Gromwall’s Lemma, combining with Lemma 1, we have the conclusion similar to below,
. Thus, this leads to 
possesses a global attractor 
in 
which attracts all bounded subsets of
. So we end the proof.
NOTES
#Corresponding author.