Advances in Pure Mathematics
Vol.2 No.5(2012), Article ID:22809,5 pages DOI:10.4236/apm.2012.25052
Global Attractor for a Non-Autonomous Beam Equation*
1Department of Mathematic, Taiyuan University of Technology, Taiyuan, China
2CSIRO Sustainable Ecosystems, Highett, Australia
Email: renyonghua@tyut.edu.cn, #jianwen.z2008@163.com
Received April 10, 2012; revised May 13, 2012; accepted May 21, 2012
Keywords: global attractor; non-autonomous beam equation; absorbing set
ABSTRACT
This work studies the global attractor for the process generated by a non-autonomous beam equation Based on a time-uniform priori estimate method, we first in the space
establish a time-uniform priori estimate of the solution
to the equation, and conclude the existence of bounded absorbing set. When the external term
is time-periodic, the continuous semigroup of solution is proved to possess a global attractor.
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.
REFERENCES
- S. Woinowsky-Krieger, “The Effect of Axial Force on the Vibration of Hinged Bars,” Journal of Applied Mechanics, Vol. 17, 1950, pp. 35-36.
- O. F. Ma, S. H. Wang and C. K. Zhong, “Necessary and Sufficient Conditions for the Existence of Global Attractor for Semigroup and Application,” Indiana University Mathematics Journal, Vol. 51, 2002, pp. 529-551. doi.org/10.1512/iumj.2002.51.2255
- R. Temam, “Infinite Dimensional Dynamical System in Mechanics and Physical,” 2nd Edition, Spring-Verlag, Nork York, 1997.
- Q. Z. Ma and C. K. Zhong, “Existence of Strong Global Attractors for Hyperbolic Equation with Linear Memory,” Applied Mathematics and Computation, Vol. 157, No. 1, 2004, pp. 745-758. doi:10.1016/j.amc.2003.08.080
- Q. Z. Ma and C. K. Zhong, “Global Attractors of Strong Solutions for Nonclassical Diffusion Equation,” Journal of Lanzhou University, Vol. 40, 2004, pp. 7-9.
NOTES
*This work is supported by the School Foundation of TUT (900103- 03020715) and the Provincial Natural Science Foundation of Shanxi (2010011008) and (2011021002-2).
#Corresponding author.