The Family of Exponential Attractors and Random Attractors for a Class of Kirchhoff Equations ()
1. Introduction
In 1991, Eden A. et al. [1] proposed the concept of inertial fractal set and how inertial fractal set is constructed. Meanwhile, they provided some applications for people to study how to prove the existence of exponential attractors. The authors’ relevant research results can be referred to [2] [3] [4] [5].
With the advent of Kirchhoff [6] equation and the existence of its solution, scholars began to study the existence of exponential attractors of Kirchhoff equation. Recently, Jia Lan, Ma Qiaozhen [7] studied the Kirchhoff-type suspension bridge equations:
They proved the asymptotic compactness of the semigroup and showed the existence of exponential attractors by a new method of enhanced flattening property under a weaker condition of nonlinearity.
Lin Guoguang, Wang Wei [8] discussed a class of higher-order Kirchhoff-type equation with nonlinear damped term:
They obtained the exponential attractors via proving the Lipschitz continuity and discrete squeezing property of dynamical system.
In paper [9], we studied Kirchhoff equation:
(1.1)
where
,
,
is a bounded region with a smooth boundary
,
stands for the cylinder in
, the rigid term
is a general function,
is the strong dissipative term,
is the nonlinear term and
,
denotes the external force. We have proved the existence and uniqueness of solution, a family of global attractors and its dimension estimation. In this paper, we will discuss the family of the exponential attractors and its dimension estimation. Meanwhile, we will discuss the family of random attractors of stochastic Kirchhoff equation with product white noise:
(1.2)
where
is independent of time denotes a two-side process in probability space
,
,
denotes a Borel
-algebra generated by compact-open topology on
, P denotes a probability measure.
Random attractor plays an important role in stochastic dynamic systems because of its property. Lu D. proposed the concept of stochastic process and its application in the literature [10]. Then Guo Boling, Pu Xueke introduced the knowledge of random infinite dimensional dynamical system in the literature [11]. Lin G.G., Qin C.L. [12] discussed the existence of the random attractors of weekly damped Kirchhoff equation:
Following, Lin, G.G., et al. [13] proved the exponential attractor of Kirchhoff-type equations with strongly damped terms and source terms:
More relevant results can be referred to [14] [15] [16] [17].
2. Preliminaries
Combine paper [9] with some new definitions and assumptions, we have:
,
,
,
where
,
,
,
are constants.
denotes the jth eigenvalue of
with the homogeneous Dirichlet boundary on
. Define the inner of
as following:
and p satisfy the following conditions:
(A1) For
, we have
, where
are constants, and
;
(A2)
;
(A3)
;
(A4)
.
3. Exponential Attractors
Definition 3.1. Compact set
is called a family of exponential attractors of
, if a family of compact attractors
satisfies:
1)
;
2)
has finite fractal dimension
;
3) There exist constants
such that
Definition 3.2. [1] Solution semigroup
is Lipschitz continuity, if there is a bounded function
, such that
Definition 3.3. [1] Assume that solution semigroup
is a map satisfies Lipschitz continuity, then we say
is squeezing in
, if
,
,
, there is
or
where
is an orthogonal projection in
.
Theorem 3.1. [15] Assuming that
1)
possesses a family of
-compact attractors
;
2)
exists a family of positive, invariant compact sets in
;
3)
is Lipschitz continuous and squeezing on
;
then
exists a family of
-type exponential attractors
and
moreover, the fractal dimension of
satisfies
, where
is the least of N which makes squeezing found.
Theorem 3.2. [9] Suppose that (A1)-(A4) are valid. Let
,
,
, then the initial boundary value problem (1.1) has a global solution
that satisfies
,
, and there exists a nonnegative real number
and
so that
According to paper [9], the solution semigroup
of the initial boundary value problem (1.1) exists a family of
-compact attractors
, and we can define a family of positive, invariant compact sets
, where
.
Next, we prove the problem (1.1) exists a family of exponential attractors.
Let Equation (1.1) transform into a first-order evolution equation:
(3.1)
where
Lemma 3.1. (Lipschitz property)
, there is
where
.
Proof. Let
, where
,
,
, then we have
(3.2)
Taking the inner product of Equation (3.2) with W in
, and we get that
(3.3)
By using Young’s inequality, Holder’s inequality, Poincare’s inequality and differential mean value theorem, we obtain
(3.4)
(3.5)
Substitute (3.4) and (3.5) into (3.3), we have
Let
,
, then we have
(3.6)
By using Gronwall’s inequality, we get
(3.7)
Meanwhile, we obtain
.
Lemma 3.1 is proved.
Lemma 3.2. (discrete squeezing)
, if
then we have
Proof. Applying
to Equation (3.2), we get
(3.8)
Taking the inner product of Equation (3.8) with
in
, we obtain
(3.9)
Similar to the process of Lemma 3.1, we have
(3.10)
where
,
.
By using Gronwall’s inequality, we get
Suppose
, then we have
Therefore, when
, we have
Lemma 3.2 is proved.
In fact, Theorem 3.1 has provided the theoretical basis to prove the existence of random attractors. Because of the value of
, furthermore, we can obtain the existence theorem of the family of exponential attractors via Lemma 3.1 and Lemma 3.2 as following:
Theorem 3.3. Assume (A1)-(A4) are valid,
, then
, the solution semigroup
of the initial boundary value problem (1.1) exists a family of
-type exponential attractors
, and
4. Random Attractors
Define the time-translation operator on
:
, then
constitutes an ergodic, metric dynamical system. According to paper [8], we have some definitions and theorems as following:
Definition 4.1.
is a metric dynamical system,
is measurable.
is called a continuous stochastic dynamical system, if map
satisfies
1)
;
2)
;
3)
is continuous.
Definition 4.2. [8]
is called a family of tempered random sets, if
where
,
.
Definition 4.3. [8]
is the set of all the random sets on
. Random set
is called a family of absorption sets on
, if
,
, such that
Definition 4.4. [8] Random set
is called a family of random attractors of continuous random dynamic system
on
, if
satisfies:
1)
is a family of random compact sets;
2)
is a family of invariant sets, which means
;
3)
attracts all sets on
, which means
, there is
where
denotes the Hausdorff half-distance.
Theorem 4.1. Assume the family of random sets
is a family of random absorbing sets of the stochastic dynamic system
, and it satisfies:
1) random set
is closed set on
;
2)
,
is asymptotically compact. Namely, when
, there exists a convergent subsequence in
for
.
Then there exists a unique family of global attractors
of the stochastic dynamic system
, and
Rewrite Equation (1.2) into a stochastic differential equation:
(4.1)
Equation (4.1) can be reduced to
(4.2)
where
,
,
,
Let
, where
denotes Ornstein-Uhlenbeck process and is the solution of Ito equation:
Let
, furthermore, rewrite Equation (4.1)
(4.3)
where
,
.
Lemma 4.1.
, there is
where
,
.
Proof.
Let
,
, we obtain
Lemma 4.2. Assume
is the solution of problem (4.3), then there exists a bounded random compact set
, such that for arbitrarily random set
, there is a random variable
, we have
Proof. Suppose
is the solution of problem (4.3). Taking the inner product of Equation (4.3) with
in
, we obtain
(4.4)
According to Lemma 4.1, we get
(4.5)
(4.6)
Substitute (4.5)-(4.6) into (4.4), we have
Let
,
,
, then we obtain
(4.7)
By using Gronwall’s inequality, we get
(4.8)
Because
is tempered,
is continuous about t. According to paper [11], we can obtain a temper random variable
, so that
, we have
(4.9)
Substitute
for
in inequality (4.8), and let
, then we have
(4.10)
Because
is tempered and
is tempered,
is also tempered. Then
is a random attractor set. Because of
is a random attractor set of
.
Lemma 4.3. When
,
, assume
is the solution of problem (4.2), and we decompose
, where
(4.11)
(4.12)
then
,
.
Also, there exists a temper random radius
, such that
,
satisfies
.
Proof. Suppose
is the solution of Equation (4.4), then we can know from Equation (4.11) and Equation (4.12) that
satisfy
(4.13)
(4.14)
Taking the inner product of Equation (4.13) with
in
, we obtain
(4.15)
By using Gronwall’s inequality and Lemma 4.1, we have
(4.16)
Substitute
for
in inequality (4.16), and
is tempered, thus
Taking the inner product of Equation (4.14) with
in
, with Lemma 4.1 and Lemma 4.2 we have
(4.17)
Substitute
for
in inequality (4.17), and by using Gronwall’s inequality, we obtain
(4.18)
Thus, there exists a temper random radius
, such that
,
.
Lemma 4.4. The stochastic dynamic system
determined by problem (4.3) has a family of compact absorbing sets
, while
,
.
Proof. Suppose
be a closed ball with radius
in
. According to Rellich-Kondrachov Compact Embedding Theorem,
. Then
is the compact set in
. For any tempered random set
, according to Lemma 4.3, we have
, then for all
, when
, via Lemma 4.2 we have
Lemma 4.4 is proved.
According to the lemma mentioned above, we verified two conditions in Theorem 4.1. Similarly, we can obtain the theorem as following:
Theorem 4.2. The stochastic dynamic system
has a family of random attractors
, and there exists a slowly increasing family of random sets
, such that
and