Dynamics of Plate Equations with Memory Driven by Multiplicative Noise on Bounded Domains ()
1. Introduction
In this paper, we investigate the existence of a random attractor for the following stochastic plate equations with linear memory and multiplicative noise on bounded domain:
(1.1)
where
and
are positive constants and
for every
, U is an open bounded set of
with smooth boundary
,
is a real function on
,
is a given external force and
is an independent two-sided real-valued wiener process on probability space
, where
is endowed with compact-open topology,
is the corresponding wiener measure, and
is the
-completion of Borel σ-algebra on Ω. We identify
with
, i.e.
Then, define the time shift
on Ω by:
The following conditions are necessary to obtain our main results.
(h1) The memory kernel
is assumed to satisfy the following conditions:
and
(h2) The nonlinear term
with
and satisfies the following conditions:
(1.2)
(1.3)
and
(1.4)
where
are constants.
Following Dafermos [1] , we introduce a new variable
defined by:
(1.5)
and let
be a Hilbert space of
-valued function on
with the inner product:
(1.6)
Set
,
. Then, the system (1.1) is equivalent to the following initial value problem in the Hilbert space E:
(1.7)
where
(1.8)
(1.9)
(1.10)
(1.11)
The stochastic plate equation is one of the fundamental stochastic partial differential equations (SPDEs) of hyperbolic type, which have been explored in [2] [3] [4] [5] . The behavior of its solutions is significantly different from those of solutions to other SPDEs.
Problem (1.1) models transversal vibration of the extensible elastic plate in a historical space, which is established based on the framework of elastic vibration by Woinowsky-Krieger [6] and Berger [7] . It can also be regarded as an elastoplastic flow equation with some kind of memory effect [1] . When
, then (1.1) reduces to determined autonomous damped plate equation.
In recent years, there have many results on the dynamics of a variety of systems related to Equation (1.1). The hyperbolic equations with memory have been studied in [8] - [15] and references therein. For instance, Khanmamedov [16] and Yue and Zhong [2] proved the existence of global attractors for plate equations with critical exponent, [17] - [22] obtained the nonlinear damped, and Ma et al. [23] [24] [25] [26] [27] obtained the strongly damped. The existence of random attractors for such system in a bounded domain has been studied in [28] . Furthermore, long-time dynamics of a plate equation with memory and time delay is considered by Feng in [29] , under suitable assumptions on real numbers
and
, the quasi-stability property of the system is established and obtained the existence of global attractor, which has finite fractal dimension, and proved the existence of exponential attractors, defined in bounded domain
with a sufficiently smooth boundary
. Shen and Ma in [30] obtained the existence of random attractors for weakly dissipative plate equations with memory and additive noise by defining the energy functionals and using the compactness translation theorem.
Crauel et al. [31] [32] [33] studied the random attractors for stochastic dynamical system. Recently, many authors have established the existence of random attractors for other equations (see [34] - [45] ). In Equation (1.1), there are fewer results and most previous authors have concentrated on the deterministic case, but there is no result of random attractors for Equation (1.1).
To prove the existence of random dynamical system (RDS) for short, the key step is to establish the compactness of the system. For our system (1.7), there are two essential difficulties in proving the compactness. Firstly, the critical growth condition (1.2) of f can be overcome by using the decomposition of solution and more accurate calculation. Secondly, the memory kernel itself, because there is no compact embedding in the history space, we introduce a new variable and define an extended Hilbert space, as well as combine with the compactness transform theorem.
The rest of the paper is organized as follows. In Section 2, we give the existence and uniqueness of the solutions. In Section 3, we devote to uniform estimates and the existence of bounded absorbing sets for the solutions and pullback compactness. In Section 4, the compactness of the random dynamical system is established by the decomposition of solution of the random differential equation into two parts. In Section 5, we prove the asymptotic compactness of the solutions, existence and uniqueness of a random attractor in E.
As mentioned in the introduction, our main purpose is to prove the dynamics of stochastic partial differential equations with multiplicative noise. For that matter, first, we recall some basic concepts related to random attractors for stochastic dynamical systems (see [9] [31] [32] [46] [47] [48] [49] ), which are important for getting our main results. Let
be a probability space and
be a polish space with the Borel σ-algebra
. The distance between
and
is denoted by
. If
and
, the Hausdorff semi-distance from B to C is denoted by
.
Definition 2.1.
is called a metric dynamical system if
is
-measurable,
is the identity on Ω,
for all
and
for all
.
Definition 2.2. A mapping
is called continuous cocycle on X over
and
, if for all
and
, the following conditions are satisfied:
1)
is a
measurable mapping.
2)
is identity on X.
3)
.
4)
is continuous.
Definition 2.3. Let
be the collection of all subsets of X, a set valued mapping
is called measurable with respect to
in Ω if
is a (usually closed) nonempty subset of X and the mapping
is
-measurable for every fixed
and
. Let
is called a random set.
Definition 2.4. A random bounded set
of X is called tempered with respect to
, if for p-a.e
,
where
Definition 2.5. Let
be a collection of random subset of X and
, then K is called an absorbing set of
if for all
and
, there exists,
such that:
Definition 2.6. Let
be a collection of random subset of X, the Φ is said to be
-pullback asymptotically compact in X if for p-a.e
,
has a convergent subsequence in X when
and
with
.
Definition 2.7. Let
be a collection of random subset of X and
, then
is called a
-random attractor (or
-pullback attractor) for Φ, if the following conditions are satisfied: for all
and
,
1)
is compact, and
is measurable for every
.
2)
is invariant, that is:
3)
attracts every set in
, that is for every
,
where
is the Hausdorff semi-distance given by:
Remark 2.8. Let
be a probability space with wiener measure
, the wiener shift
is defined by:
then
is an ergodic metric dynamical system.
Lemma 2.9. [31] [32] Let
be a neighborhood-closed collection of
-parameterized families of nonempty subsets of X and Φ be a continuous cocycle on X over
and
. Then, Φ has a pullback
-attractor
in
if and only if Φ is pullback
-asymptotically compact in X and Φ has a closed,
-measurable pullback
-absorbing set
, the unique pullback
-attractor
is given:
In this article, we will take
as the collection of all tempered random subsets.
Lemma 2.10. [50] For any
and any
, the following equality holds:
Lemma 2.11. [9] Let
be three Banach spaces such that
↪
↪
, the first injection being compact. Let
satisfy the following hypotheses:
1) Y is bounded in
.
2)
,
for some
.
Then, Y relatively compact in
.
3. Existence and Uniqueness of Solutions
From now on, assume that conditions (h1) - (h2) hold, the space E and the probability space
are defined in Section 1. Let
with Neumann boundary condition on U,
. We can define the powers
of A for
. The space
is the Hilbert space with the following inner product and norm, respectively:
The injection
↪
is compact if
. Then, by the generalized Poincaré inequality, there holds:
where
is the first eigenvalue of A. In particular,
,
,
, and
,
. The inner product and norm in
is denoted by
, and in
is denoted by
, respectively. By (h1), the space
is a Hilbert space of
-valued function on
with the inner product and norm, respectively:
(3.1)
(3.2)
and on
, the linear operator
has domain:
which generates a right-translation semigroup (see [1] [9] [13] [15] [51] ).
Then, Equation (1.1) can be transformed into the following system:
: (3.3)
with the initial-boundary conditions:
(3.4)
The symbol C and
are positive constants, which may change from line to line.
In this section, we show the existence, uniqueness and continuous dependence of (mild) solution of initial problem (1.7) in E, which generates a continuous RDS on E over
and
. For our purpose, we convert the problem (1.7) into a deterministic system with random parameters but without noise terms.
Due to Ornstein-Uhlenbeck process deducing by the Brownian motion, which holds the Itô differential equation:
(3.5)
and hence, the solution is given by:
(3.6)
It is known from [48] [49] , the random variable
is tempered and there is a
-invariant set
of full P measure such that for every
,
is continuous in t and:
(3.7)
Equation (3.6) has a random fixed point in the sense of random dynamical systems generating a stationary solution known as the stationary Ornstein-Uhlenbeck process (see [31] [32] [36] [52] for more details).
For convenience, in the following, we write
as Ω. Next, we need to transform the stochastic system into deterministic with a random parameter, then show that it generates a random dynamical system.
Let:
(3.8)
(3.9)
where
is the smallest eigenvalue of operator A with Neumann boundary condition on U.
In this paper, we assume that:
(3.10)
where
.
By (3.8) and (1.1), we can obtain the following random evolution equation:
(3.11)
Then, the problem (3.11) is equivalent to the following determined system with random parameter in E:
(3.12)
where
(3.13)
(3.14)
In line with [9] [53] , we know that the operator L in (1.9) is the infinitesimal generator of C0-semigroup
of contractions on E for
. Since
, and
is an isomorphism of E, the operator
also generates a C0-semigroup
of contractions on E. By the assumptions (h2) and the embedding relation
↪
, it is easy to check
is locally Lipschitz continuous with respect to
, by the classical semigroup theory concerning the (local) existence and uniqueness solution of evolution differential equation [53] , we have the following theorem.
Theorem 3.1. Assume that (h1) - (h3) hold. Then, for each
and for any
, there exists
such that (3.12) has a unique mild function
such that
satisfies the integral equation:
(3.15)
Moreover,
is jointly continuous in
and measurable in
.
From Theorem 3.1, we know that for P-a.s. each
, then the following results hold for all
:
1) If
then
.
2)
is jointly continuous into t and measurable in
.
3) The solution mapping of (3.12) satisfies the properties of Random Dynamical System.
We notice that a unique solution
of (3.12) can define a continuous random dynamical system over
and
. Hence, the solution mapping:
(3.16)
generates a random dynamical system. Moreover,
(3.17)
We also define the following transformation:
(3.18)
similar to (3.12), we get that:
(3.19)
where
(3.20)
(3.21)
and
It is easy to see that:
(3.22)
and
(3.23)
are continuous RDS over
and
associated with system (3.7) and (3.15) respectively.
We introduce the isomorphism
,
, which has inverse isomorphism
, it follows that
with mapping:
(3.24)
is a random dynamical system from above discussion, we show that the two RDS are equivalent.
4. Random Absorbing Set
In this section, we will show the existence of a random absorbing set for the RDS
in the space E.
Lemma 4.1. Suppose that (h1) - (h2) hold. Then, there exists a closed tempered absorbing ball
of E, centered at 0 with random radius
such that for any bounded non-random set
, there exists a deterministic
, such that the solution
of (3.12) with initial value
satisfies, for P-a.s.
,
(4.1)
that is,
Proof. Taking the inner product
of (3.12) with
, we have:
(4.2)
Similar to the proof of Lemma 2 in [54] , we have:
(4.3)
Then, by using (h1), we find that:
(4.4)
(4.5)
Applying (4.3)-(4.5), Hölder inequality, Young inequality and Poincaré inequality, we obtain that:
(4.6)
It follows from a simple computation that:
(4.7)
Hence, combining (4.6) and (4.7), we find that:
(4.8)
Let us estimate the right hand side of (4.2):
(4.9)
By the Cauchy-Schwartz inequality, we find that:
(4.10)
(4.11)
(4.12)
(4.13)
(4.14)
(4.15)
Then, we estimate nonlinear term (4.9), by (h2) and the Hölder inequality, we get that:
(4.16)
Applying (1.2)-(1.4), we have:
(4.17)
(4.18)
Thus, due to (4.16)-(4.18), we obtain that:
(4.19)
where
.
Collecting (4.2), (4.8), (4.19) and (4.9)-(4.15), we show that:
choose
. Due to
, then we have the following equivalent system:
where
(4.20)
That is,
(4.21)
So, applying Gronwall’s Lemma to (4.21), we have:
(4.22)
Substitiuting
by
, from (4.22), we have:
(4.23)
Since
is stationary and ergodic, it follows from (3.2) and the ergodic theorem that:
(4.24)
(4.25)
From (4.24) and (4.25), we know that there exists
such that for any
,
(4.26)
Next, we need to obtain that for any
,
(4.27)
Indeed, by (4.26), we have:
In order to obtain (4.27), for any
, there holds:
Solving this quadratic inequality, we find that:
Since
is tempered, it follows from (4.27) that the following integral is bounded:
(4.28)
According to (1.3)-(1.5), we have:
(4.29)
It follows from theorem 3.1 and
that:
(4.30)
Combining with (4.27)-(4.30), there exits
such that for all
,
Then, we complete the proof.
5. Decomposition of Solutions
In order to obtain regularity estimates later, as in [52] , we decompose the equations (3.3). At first, we will give the following decomposition on nonlinearity
and
satisfies the following conditions:
(5.1)
and
(5.2)
where
,
are constants.
We decompose the solution
of the system (3.12) into the two parts:
where
,
solve the following equations, respectively:
(5.3)
and
(5.4)
where
(5.5)
To prove the existence of a compact random attractor for the random dynamical system Φ, we need to get the solutions of systems (5.3) and (5.4), which one decays exponentially and another is bounded in higher regular space. In order to get the regularity estimate, we will prove some priori estimates for the solutions of systems (5.3) on
as follows.
Lemma 5.1. For any P-a.e.
, there exists
such that the solution
of (5.3) with initial data
satisfies:
(5.6)
Proof. Taking the inner product
of (5.3) with
, we show that:
(5.7)
Similar to the proof of (4.8), we obtain that:
(5.8)
Now, we estimate the third term of (5.7). According to (5.1)3, we get:
(5.9)
Thus, it follows from (5.7)-(5.10) and (5.3) that:
(5.10)
where
. By Gronwall’s Lemma to (5.10), we have:
(5.11)
According to (5.1), we have:
(5.12)
Combining (5.11)-(5.12) with
, we get:
(5.13)
So, the proof is completed.
Lemma 5.2. For any P-a.e.
, there exists
such that the solution
of (5.3) with initial data
satisfies:
(5.14)
Proof. We consider (5.1), (5.7) and similar to Lemma 5.1, we conclude that:
(5.15)
Applying interpolation inequality, we have:
(5.16)
Hence, combining (5.15)-(5.16) with Lemma 5.1, we find that there exists
, such that:
(5.17)
Due to (5.7)-(5.8), (5.1)2 and (5.17), we can obtain the following result:
that is,
(5.18)
where
.
By applying Gronwall’s inequality to (5.18), it yields:
(5.19)
Then, the proof is completed.
Next, we estimate the component
in (5.4).
Lemma 5.3. For any P-a.e.
, there exists
such that the solution
of (5.4) with initial data
satisfies:
(5.20)
where
(5.21)
and
is increasing function.
Proof. By (4.1), (5.6) and
, there exists a random variable
such that:
(5.22)
Taking the inner product of
of (5.4) with
, we find that:
(5.23)
In later calculations, we will use the following embedding relations:
↪
↪
↪
↪
(5.24)
Similar to the proof of (4.8), we deduce that:
(5.25)
Next, we will deal with the right-hand side of (5.23). Using (4.5)-(4.10) and (5.21), we get:
(5.26)
(5.27)
(5.28)
(5.29)
(5.30)
(5.31)
For the nonlinear term, we have:
Firstly, we deal with the term:
(5.32)
by (5.2)1, (5.24) and Lemma 5.1, we have:
(5.33)
and
(5.34)
Secondly, we consider the following term:
(5.35)
According to (1.2), (5.24), Lemma 4.1, 5.1 and
, we obtain that:
(5.36)
(5.37)
and
(5.38)
In addition, by (1.2), (5.2)1, (5.24) and Lemma 5.1, we find that:
(5.39)
and
(5.40)
Thus, combining with (5.23)-(5.40), we can show that:
(5.41)
that is,
(5.42)
where
(5.43)
Let
By Gronwall’s inequality to (5.42), we have:
Similar to the proof of Lemma 4.1, we have:
(5.44)
Due to (4.26) again, there exists
such that:
(5.45)
where c satisfies (4.10). Thus, it follows from (5.39)-(5.40) and (5.43)-(5.46) that:
(5.46)
where
is increasing function. Then, the proof is finished.
Lemma 5.4. Assume that (h2) holds. For any
, there exists
and
, such that
-a.s.
,
(5.47)
(5.48)
Proof. Combining (5.14), (5.20) with the technique used in [55] , we can finish the proof.
Lemma 5.5. Assume that (h1) - (h2) and (5.1)-(5.2) hold. There exists a random radius
such that for P-a.e.
, the solution
of (5.4) satisfies:
(5.49)
where
is given in (5.21).
Proof. Taking the inner product of
of (5.4) with
, we find that:
(5.50)
First, we deal with the nonlinearity in (5.50). Applying (5.1), (5.6), (5.22) and Hölder’s inequality, we have:
(5.51)
(5.52)
(5.53)
By (5.26)-(5.31), (5.34), (5.38) and (5.50)-(5.53), we obtain that:
(5.54)
where
By Gronwall’s inequality to (5.54), we get that:
(5.55)
where
For any
such that
is so small,
Next, similar to the proof of Lemma 4.1, we know that:
(5.56)
where c satisfies (4.10).
Hence, combining (5.39)-(5.40) with tempered
, we obtain that:
Then, the proof is completed.
6. Random Attractors
In this section, we establish the existence of a
-random attractor for the random dynamical system Φ associated with system (3.12) on
, that is, by Lemma 4.1, Φ has a closed random absorbing set in
, which along with the
-pullback asymptotic compactness and then implies the existence of a unique
-random attractor. Next, due to decomposition of solutions, we shall prove the
-pullback asymptotic compactness of Φ (see [56] [57] ).
Since
, we get:
(6.1)
(6.2)
Denote
as:
(6.3)
is the solution of (3.12), where
. Next, it follows from Lemma 5.5 and (6.1)-(6.2) that:
(6.4)
which implies
is bounded in
. Also, by Lemmas 4.1, 5.5 and (6.2), there holds:
(6.5)
By (h1) and (6.5), for any
, we find that:
(6.6)
which shows that
is bounded. It follows from Lemma 2.11 that the set
is relatively compact in
. Next, we investigate the main result about the existence of a random attractor for Random Dynamical System Φ.
Lemma 6.1. Assume that (h1) - (h2) hold. Then, for any
, the RDS Φ associated with (3.12) possesses a uniformly
-attracting set
and possesses a
-random attractor
.
Proof For any
, as Lemma 5.5, let
be the closed ball in
of radius
. Set:
(6.7)
Then,
. Because
↪
is compact, and
is compact in
. At the same time,
is compact in
, then
is compact in E. Next, we prove the following attraction property of
: for every
,
(6.8)
Indeed, firstly, according to Lemma 4.1, there exists closed, tempered and measurable absorbing set
such that
, for any
,
(6.9)
Moreover, let:
Assume that
and
. Making use of the property 3) of Φ and (6.9), we deduce that:
(6.10)
For any
, choose
, where
. Due to (6.10) and Lemma 5.1, we have:
So, according to Lemma 5.2, we find that for
,
that is,
Thus, (6.8) holds. Therefore, applying Lemma 2.11 and Theorem 4.1, we obtain that the RDS Φ possesses a
-pullback random attractors
.
Then, the proof is completed.
Acknowledgement
The authors would like to thank anonymous reviewers for their helpful comments that improved the presentation of this work. Mohamed Y. A. Bakhet acknowledges and thanks the financial support from the University of Juba, South Sudan.