Random Attractors of Stochastic Non-Autonomous Nonclassical Diffusion Equations with Linear Memory on a Bounded Domain ()
1. Introduction
In this article, we investigate the asymptotic behavior of solutions to the following stochastic nonclassical diffusion equations driven by additive noise and linear memory:
(1.1)
where
is a bounded domain in
, the initial data
,
is a real valued function of
,
,
,
and
is the generalized time derivative of an infinite dimensional wiener process
defined on a probability space
, where
,
is the σ-algebra of Borel sets induced by the compact topology of
,
is a corresponding wiener measure on
for which the canonical wiener process
satisfies that both
and
are usual one dimensional Brownian motions. We may identify
with
, that is,
for all
.
To consider system (1.1), we assume that the memory kernel satisfies
(1.2)
and there exists a positive constant
such that the function
satisfies
(1.3)
And suppose that the nonlinearity satisfies as follows:
,
and for every fixed
,
satisfying
(1.4)
(1.5)
and
satisfying
(1.6)
(1.7)
where
and l are positive constants, and q is a conjugate exponent of p.
In addition, we assume that for
and
, for
;
, for
.
We assume that the time-dependent external force term
satisfies a condition
(1.8)
and for some constant
to be specified later.
Equation (1.1) has its physical background in the mathematical description of viscoelastic materials. It’s well known that the viscoelastic material exhibit natural damping, which according to the special property of these materials to retain a memory of their past history. And from the materials point of view, the property of memory comes from the memory kernel
, which decays to zero with exponential rate. Many authors have constructed the mathematical model by some concrete examples, see [1] - [7] . In [8] the authors considered the nonclassical diffusion equation with hereditary memory on a 3D bounded domains for a very general class of memory kernels
; setting the problem both in the classical past history framework and in the more recent minimal state one, the related solution semigroups are shown to possess finite-dimensional regular exponential attractors. Equation (1.1) is a special case of the nonclassical diffusion equation used in fluid mechanics, solid mechanics, and heat conduction theory (see [1] [4] [5] ). In [1] Aifantis, Urbana and Illinois discussed some basic mathematical results concerning certain new types of some equations, and in particular results showing how solutions of some equations can be expressed at in terms of solutions of the heat equation, also discussed diffusion in general viscoelastic and plastic solids. In [4] Kuttler and Aifantis presented a class of diffusion models that arise in certain nonclassical physical situations and discuss existence and uniqueness of the resulting evolution equations.
The long-time behavior of Equation (1.1) without white additive noise and
has been considered by many researchers; on a bounded domain see, e.g. [9] [10] [11] [12] [13] and the references therein. In [10] the authors proved the existence and the regularity of time-dependent global attractors for a class of nonclassical reaction-diffusion equations when the forcing term
and the nonlinear function satisfies the critical exponent growth. In [11] Sun and Yang proved the existence of a global attractor for the autonomous case provided that the nonlinearity is critical and
. The researchers in [12] obtained the Pullback attractors for the nonclassical diffusion equations with the variable delay on a bounded domain, where the nonlinearity is at most two orders growth. As far as the unbounded case for the system (1.1) the long-time behavior of solutions is concerned, most recently, by the tail estimate technique and some omega-limit compactness argument, for more details (see [14] [15] [16] [17] [18] ). In [14] Ma studied the existence of global attractors for nonclassical diffusion equations with the arbitrary order polynomial growth conditions. By a similar technique, Zhang in [16] obtained the Pullback attractors for the non-autonomous case in
, where the growth order of the nonlinearity is assumed to be controlled by the space dimension N, such that the Sobolev embedding
is continuous. However, it is regretted that some terms in the proof of [16] Lemma 3.4 are lost. Anh et al. [17] established the existence of pullback attractor in the space
, where the nonlinearity satisfied an arbitrary polynomial growth, but some additional assumptions on the primitive function of the nonlinearity were required. And the case of
with additive noise on a bounded domain, Cheng used the decomposition method of the solution operator to consider the stochastic nonclassical diffusion equation with fading memory. For the case of
, Zhao studied the dynamics of stochastic nonclassical diffusion equations on unbounded domains perturbed by a ò-random term “intension of noise”. (For more details see [2] [19] [20] [21] [22] ).
To our best knowledge, Equation (1.1) on a bounded domain in the weak topological space and the time-dependent forcing term has not been considered by any predecessors.
The article is organized as follows. In Section two, we recall the fundamental results related to some basic function spaces and the existence of random attractors. In Section three, firstly, we define a continuous random dynamical system to proving the existence and uniqueness of the solution, then prove the existence of a closed random absorbing set and establish the asymptotic compactness of the random dynamical system finally prove the existence of D-random attractor.
2. Preliminaries
In this section, we recall some basic concepts and results related to function spaces and the existence of random attractors of the RDSs. For a comprehensive exposition on this topic, there is a large volume of literature, see [2] [3] [19] [23] - [29] .
Let
, with the domain
, and fractional
power space
,
, the
,
is the inner product and norm, respectively. For convenience, we use
, the norm
, and
,
.
Similar to [3] , for the memory kernel
, we denote
the Hilbert space of function
, endowed with the inner product and norm respectively,
(2.1)
Define the space
with the inner product
and the norm
We also introduce the family of Hilbert space
, and endow norm
In the following of this article, we denote
. See [2] [3] [23] for more details.
Let
, f is the Borel σ-algebra on
, and
is the corresponding Wiener measure. Define
Then
is the measurable map and
is the identity on
,
for all
. That is,
is called a metric dynamical system.
Definition 2.1.
is called a metric dynamical system if
is
-measurable,
is the identity on
,
for all
and
for all
.
Definition 2.2. A continuous random dynamical system (RDS) on X over a metric dynamical system
is a mapping
which is
-measurable and satisfied, for P-a.e.
,
1)
is the identity on X;
2)
for all
;
3)
is continuous for all
.
Definition 2.3. A random bounded set
is a family of nonempty subsets of X is called tempered with respect to
if for P-a.e.
, for all
,
where
.
Definition 2.4. Let
be the collection of all tempered random sets in X. A set
is called a random absorbing set for RDS
in
, if for every
and P-a.e.
, there exists
such that for all
,
Definition 2.5. Let
be the collection of all tempered random subsets of X. Then
is said to be asymptotically compact in X if for P-a.e.
, the sequence
has a convergent subsequence in X whenever
, and
with
.
Definition 2.6. (See [30] , [31] , [32] ) Let
be the collection of all tempered random subsets of X and
. Then
is called a D-random attractor for
if the following conditions are satisfied, for P-a.e.
,
1)
is compact, and
is measurable for every
;
2)
is invariant, that is,
,
;
3)
attracts every set in
, that is, for every
where d is the Hausdorff semi-metric given by
for any
and
.
Theorem 2.1. Let
be a continuous random dynamical system with state space X over
. If there is a closed random absorbing set
of
and
is asymptotically compact in X, then
is a random attractor of
, where
Moreover,
is the unique random attractor of
.
As mentioned in [23] , we can define a new variable to reflect the memory kernel of (1.1)
(2.2)
Hence,
(2.3)
Therefore, we can rewrite (1.1) as follows.
(2.4)
where
satisfies that there exist two positive constant C and k, such that
(2.5)
Lemma 2.1. ( [3] [33] ) Assume that
is a nonnegative function, and there exists
, such that
, then
holds for all
. Moreover, for three Banach space
and
,
and
are reflexive and
where, the embedding
is compact. Let
satisfy
1) K in
;
2)
, a.s. For some
.
Then K is relatively compact in
.
3. The Random Attractor
In this section, we prove that the stochastic nonclassical diffusion problem (2.4) has a D-random attractor. First, We convert system (2.4) with a random perturbation term and linear memory into a deterministic one with a random parameter
. For this purpose, we introduce the Ornstein-Uhlenbeck process taking the form
where
is one dimensional Wiener process defined in the introduction. Furthermore,
satisfies the stochastic differential equations
(3.1)
It is known that there exists a
-invariant set
of full P measure such that
is continuous for every
, and the random variable
is tempered, see, e.g., [2] It is easy to show that
(3.2)
where
is the Laplacian with domain
. Using the change of variable
,
satisfies the equation (which depends on the random parameter
)
(3.3)
By the Galerkin method as in [34] , under assumptions (1.2)-(1.8), for P-a.e.
, and for all
, problem (3.3) has a unique solution
in
, satisfying
.
Throughout this article, we always write
(3.4)
If u is the solution of problem (1.1) in some sense, we can define a continuous dynamical system
(3.5)
In order to prove the asymptotic compactness and the existence of global attractor, we give the following results.
Lemma 3.1. ( [23] ) Set
,
. Let the memory kernel
satisfy (1.3), then for any
,
, there exists a constant
, such that
(3.6)
We first show that the random dynamical system
has a closed random absorbing set in
, and then prove that
is asymptotically compact.
Lemma 3.2. Assume that
and (1.2)-(1.8) hold. Let
. Then for P-a.e.
, there is a positive random function
and a constant
such that for all
,
the solution of (3.3) has the following uniform estimate
(3.7)
Proof. Taking the inner product of the first equation of (3.3) with
, we have
(3.8)
From (2.2) and (2.3), we obtain
(3.9)
Hence, we can rewrite (3.8) as follows
(3.10)
By Young inequality and Lemma 3.1, we get
(3.11)
From the first term on the right hand side of (3.8)
, First we estimate
. By (1.4)-(1.5) and using a similar arguments as (4.2) in [35] , we have
(3.12)
By using (1.6)-(1.7), we arrive at
(3.13)
By the young inequality, and using assumption (1.6), we see that
(3.14)
(3.15)
where
. Then, it follows from (3.13)-(3.15) that
(3.16)
On the other hand, we have
(3.17)
From the last term of (3.8), we obtain
(3.18)
Then, we substituting (3.11), (3.12) and (3.16)-(3.18) into (3.10) we conclude that
then we have
(3.19)
Furthermore, let
(3.20)
Then (3.19)-(3.20), it implies
(3.21)
According to Grnowall's Lemma, we obtain
(3.22)
Substituting
by
, then from (3.22), we have that
(3.23)
Recalling that
is tempered such that
(3.24)
Note that
is the tempered, and
,
, we can choose
(3.25)
Then
is the tempered since
has at most linear growth rate at infinity, now the proof is completed.
To prove the asymptotic compactness of the solution, we decompose the solution
of (3.3) as follows [3] [23] :
where
satisfy the following problems, respectively
(3.26)
and
(3.27)
here the nonlinearity
are satisfies (1.4)-(1.7). The drifting term
and the forcing term satisfies a condition as in (1.8),
, for any
, such that
(3.28)
Set
, we find that
(3.29)
Let
, where
satisfies (3.26), and
,
is the solution of (3.27). Then for
and
we have that
(3.30)
and
(3.31)
The same of the problem (3.3), we also have the corresponding existence and uniqueness of solutions for (3.30) and (3.31). For the convenience, we obtain the solution operators of (3.30) and (3.31) by
and
respectively. Then, for every
, we get
Next, we give some Lemmas to prove the asymptotic compactness.
Lemma 3.3. Assume that the condition on
hold. Let
. Then for P-a.e.
, there is a constant
,
, if
then for all
, the solution of (3.30) satisfies the following uniform estimate
(3.32)
where the positive random function
is defined in Lemma 3.2.
Proof. From (3.10) we substituting
by
, respectively. Similar to the proof the Lemma 3.2, we compute
(3.33)
Since
,
are tempered, we can choose
,
, such that (3.32) is satisfied.
Lemma 3.4. Assume that the condition on
hold. Let
. Then for P-a.e.
, there is a positive random function
and
such that for every given
, the solution of (3.31) has the following uniform estimates
(3.34)
where
.
Proof. Multiplying (3.31) by
and integrating over
, we can get
(3.35)
From (2.2) and (3.31), we obtain
(3.37)
hence
and
By
and the mean value theorem, we have
(3.38)
Using the embedding theorem, we have
(3.39)
where we have used inequality
, so
and the embedding theorem
Note that
(3.40)
Thanks to Lemma 3.1, the property of the solution of (3.3) and (3.26), and (3.36)-(3.40), we conclude that
(3.42)
where
.
(3.43)
Thus, for every given
, we get
(3.44)
where
is a random function.
The proof is complete.
Since
and (3.31), it follows
(3.45)
for more information on
, see [23] , we have
Lemma 3.5. Let
is a projection operator, setting
,
is a random bounded absorbing set from Lemma 3.4,
is the solution operator of (3.31), and under the assumption of Lemma 3.4, there is a positive random function
depend on T, such that
1)
is bounded in
;
2)
.
Proof. By the random translation, (3.44) and Lemma 3.4, we can prove this Lemma.
Therefore, Lemma 2.1 implies that
is relatively compact in
. And use the compact embedding
, we have that
Lemma 3.6. Let
be the corresponding solution operator of (3.31), and the assumption of Lemma 3.4 and 3.5 hold, then for any
,
is relatively compact in
.
Now we are on a position to prove the existence of a random attractor for the stochastic nonclassical diffusion equation with linear memory and additive white noise.
Theorem 3.1. Let
be the solution operator of equation (3.3), and the conditions of the lemma 3.6 hold, then the random dynamical system
has a unique random attractor in
.
Proof. Notice that
has a closed absorbing set
by Lemma 3.2, and is relatively compact in
by Lemma 3.3 and Lemma 3.6. Hence the existence of a unique D-random attractor follows from Theorem 2.1 immediately.
Foundation Term
This work was supported by the NSFC (11561064), and NWNU-LKQN-14-6.