A Random Attractor Family of the High Order Beam Equations with White Noise ()
1. Introduction
We studied the random high order Beam equation with strong damping and white noise in this paper.
(1)
(2)
(3)
where
are given functions,
,
describe a addable white noise,
denotes a area which bounded with smooth homogeneous Dirichlet boundary,
denotes the boundary of
.
are constants,
are strong damping terms,
denotes a one-dimensional two-sided Wiener process on a probability space
,
, F denotes a Borel
-algebra generated by compact-open topology on
, P denotes a probability measure.
Random attractor is a random collection, which is measurable, compact, constant, and attract all the orbits. If it exists, it will be the smallest absorbing sets of system solution concentration, it’s also the biggest invariant set. In a sense, random attractor is reminded as a reasonable extension of the global attractor to the classical dynamical system. Recently, more and more scholars have focused on the random dynamic system.
Guo [1] wrote a book about the random infinite dimensional dynamical system, it’s the first book at home. It includes any experience of the author with random dynamic study and some research results, the latest development and results also be introduced.
Lin [2] studied the existence of stochastic attractors of high order nonlinear Beam equation.
Qin [3] proved the random attractor for stochastic Beam equations with addable white noise, Xu [4] studied the non-autonomous stochastic wave equation with dispersion and dissipation terms.
Crauel and Flandoli [5] studied the random attractor of the infinite dimensional equation. Cai and Fan [6] considered the dissipative KDV equation with multiplicative noise.
For more relevant studies, it can be referred to references in [7] - [12].
2. Preliminaries
In this section, some symbols and assumptions are introduced for convenience.
Let the operator
with Dirichlet boundary condition be selfadjoint, positive definite and linear. Set the eigenvalue of A is
, and satisfies
, when
,
. (4)
Among them
Set
and define a weighted inner product and norm in
, (5)
(6)
and
(7)
Definition 2.1 [2] Set
is a metric dynamic system, if
, measurable mapping
(8)
satisfy
1) For all
and
, mapping
satisfy
; (9)
2) For every
, mapping
continuous.
It is said that S is a continuous random dynamic system on
.
Definition 2.2 [2] It is said that the random set
is slowly increasing, if
,
, there is
and
, for all
.
Definition 2.3 [2]
denotes a collection of all random sets on X, random set
denotes a absorption set on
, if for every
and
, there is
make
. (10)
Definition 2.4 [2] The random set
becomes the random attractor of the continuous random dynamic system
on X. If the random set
satisfies:
1)
is a random compact set;
2)
is a invariant set, for every
,
;
3)
attracts all sets on
, for any
and
, we have the limit formula:
(11)
denotes the Hausdorff half distance. (
).
Definition 2.5 [2] Random set
is the random absorption set of the random dynamic system
, and the random set
satisfies
1) Random set
is a closed set on Hilbert space X;
2) For
, random set
meet the following progressive compactness conditions: For any sequence
in
, there is a convergent subsequence in space X. Then the stochastic dynamic system
has a unique global attractor
(12)
3. Existence of Random Attractors
3.1. Existence and Uniqueness of Solution
For convenience, Equations (1)-(3) can be reduced to
(13)
Set
,
, then Equation (1) is equivalent to the following stochastic differential equation
(14)
and
,
,
,
denotes a Ornstein-Uhlenbeck process, it is a stationary solution of Itô equation
(15)
, then Equation (14) can be reduced to
(16)
And
,
,
3.2. The Existence of Random Attractors
This section mainly considers existence of the random attractor of problem (1). First, we can prove that the random dynamic system
has a bounded random absorption set. For this reason, all slowly increasing subsets in the space E are denoted as
.
Lemma 1
, for every
,
, When
,
.
Then
(17)
Proof
(18)
According to the Formula (7), we can get
So set
(19)
Lemma 2
denotes a solution of problem (14), then there is a bounded random compact set
, so that there is a random variable
for any slowly increasing random set
, such that
(20)
Proof
denotes a solution of problem (16), use
to take the inner product with the Equation (16), we obtain
(21)
From Lemma 1
(22)
and
(23)
From Equations (22) and (23), Equation (21) can be written as
(24)
Set
then
(25)
According to the Formula (7), we can get
(26)
From the Gronwall’s inequality,
, we have
(27)
because
is slowly increasing, and
is continuous with respect to t, according to the literature [3], a slowly increasing random variable
can be obtained, so for
there is
(28)
Substituting
for
in (27), we get
(29)
and
(30)
Because
is slowly increasing, and
is also slowly increasing, so let
(31)
Then
also slowly increasing,
denotes a random absorption set, because
so
(32)
Then
is the random absorption set of
, and
.
So the lemma is proved.
Lemma 3 When
, for any
,
is the solution of Equation (14) under the initial value condition
. It can be decomposed into
, where
and
satisfy
(33)
(34)
Then
(35)
and there is a slowly increasing random radius
, so that for every
, satisfy
(36)
Proof
is a solution of Equation (16), from Equations (33) and (34), it can be seen that
and
satisfy
(37)
(38)
Using
and Equation (37) to take the inner product, we get
(39)
according to lemma 2 and the Gronwall’s inequality, we obtain
(40)
replace
in (40) with
, and because
is slowly increasing, then
(41)
Using
and Equation (38) to take the inner product, we get
(42)
and
.
According to lemma 1, lemma 2, Equation (29), the Gronwall’s inequality, and replace
with
, we obtain
(43)
set
(44)
for every
,
(45)
and
is slowly increasing. The lemma is proved.
Lemma 4 The stochastic dynamic system
determined by Equation (17) has a compact absorption set
under condition
,
Proof Suppose
is a closed sphere with
as the radius in space
. According to the embedding relationship
,
is a compact set in
. For any slowly increasing random set
in E, for
, according to Lemma 3.1, there is
, so for every
,
(46)
Therefore, for any slowly increasing random set
in
, there is
(47)
According to Lemma 1 to Lemma 4, there is the following theorem.
Theorem 1 Random dynamic system
has a random attractor
,
, and there is a slowly increasing random set
,
,
(48)
and
(49)
4. Conclusion
We studied a class of damped high order Beam equation stochastic dynamical systems with white noise, by using the Ornstein-Uhlenbeck process, estimating the solution of the equation and the isomorphism mapping method, then we can get the existence of the random attractor family, I wish there will be some more convenient methods can be shown off. Further we can make the inertial manifolds of the model.