Existence and Upper Semi-Continuity of Random Attractors for Nonclassical Diffusion Equation with Multiplicative Noise on Rn ()
1. Introduction
In this paper, we investigate the asymptotic behavior of solution to the following stochastic nonclassical diffusion equations with arbitrary polynomial growth nonlinearity and multiplicative noise defined in the entire space
:
(1.1)
with the initial value condition
(1.2)
where
is the Laplacian operator with respect to the variable
,
is a real function of
and
;
are proper positive constants;
; g is a nonlinear function satisfying certain conditions;
is a two-sided real-valued Wiener process on a probability space
, where
,
is the Borel
-algebra induced by the compact-open topology of
, and
is the corresponding Wiener measure on
;
denotes the Stratonovich sense in the stochastic term. We identify
with
, i.e.,
The nonclassical diffusion equation is an important mathematical model which depicts such physical phenomena as non-Newtonian flows, solid mechanics, and heat conduction, where the viscidity, the elasticity and the pressure of medium are taken into account. Equations (1.1) is known as the nonclassical diffusion equation when (
) and the reaction-diffusion equation when (
), Equations (1.1) this kind of equation has been studied by many researchers and several excellent results have been obtained in the recent twenty years, see Refs. [1] [2] [3] [4].
Since Equations (1.1) contains the term
, it’s different from the usual reaction-diffusion equation essentially. For example, the reaction-diffusion equation has some smoothing effect, e.g., although the initial data only belongs to a weaker topology space, the solution with initial conditions will belong to a stronger topology space with higher regularity. The existence, long-time behavior and regularity of solutions of Equations (1.1) have been considered by some recent related works [5] - [16] and the references therein. However, for Equations (1.1), if the initial data
belongs to
, then the solution
is always in
and has no higher regularity because of
. There are a great number of results concerning the existence of random attractor involving stochastic partial differential equations, we refer the readers to [17] - [32]. In [20], the author has proved the existence of random attractor for the nonclassical diffusion equation with memory in
on bounded domain.
In the case of unbounded domains established the existence of pullback attractor for the stochastic nonclassical diffusion equation in [17], and existence of random attractor with additive noise in [18]. For the upper semicontinuity of corresponding attractors between autonomous and perturb non-autonomous systems, we can refer to [16] [22] [23]. However, there are fewer results on the existence and upper semi-continuity of pullback attractors for stochastic nonclassical diffusion equation with multiplicative noise on unbounded domain also gives some difficulties since the embedding is no longer compact. Consequently, for Equations (1.1), we cannot use the compact Sobolev embedding to verify the asymptotic compactness of the solutions. Most recently, by using the tail-estimates method, and some omega-limit compactness argument and useful estimates of nonlinearity of the random dynamical system, which shows that the solutions are uniformly asymptotically small when space and time variables approach infinity, the reader can refer to [23] [24] [25] [26].
This paper is organized as follows. In Section 2, we recall some basic concepts and properties for general random dynamics system. In Section 3, we provide some basic settings about Equations (1.1) and show that it generates a random dynamical system on
. In Section 4, we prove the uniform estimates of solutions, which include the uniform estimates on the tails of solutions. In Section 5, we first establish the asymptotic compactness of the solution operator by given uniform estimates on the tails of solutions, and then prove the existence of a random attractor. The existence and upper semicontinuity (in
) of random attractors are given in the last section.
2. Preliminaries
As mentioned in the introduction, our main purpose is to prove the existence of the random attractor. For that matter, first, we will recapitulate basic concepts related to random attractors for stochastic dynamical systems. The reader is referred to [19] [22] [26] [29] for more details. Let
be separable Hilbert space with the Borel
-algebra
, and
be a probability space, in the sequel, we use
and
to denote the norm and inner product of
, respectively.
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 satisfies, for
-a.e.
,
1)
is the identity on X,
2)
for all
,
3)
is continuous for all
.
Hereafter, we always assume that
is continuous RDS on X over
.
Definition 2.3 A set-valued mapping
, is said to be a random set if the mapping
is measurable for every
. If
is also closed (compact) for each
,
is called a random closed (compact) set. A random set
is said to be bounded if there exist
and a random variable
such that
for all
.
Definition 2.4 A random bounded set
is called tempered provided for
-a.e,
,
for all
,
where
.
Definition 2.5 Let
be a collection of random subset of X and
. Then
is called a random absorbing set for
in
for every
and
-a.e,
, there exist
such that
for all
.
Definition 2.6 A random set
is said to be a random attracting set if for every tempered random set
, and
-a.e,
, we have
where
is the Hausdorff semi-distance given by
for every
.
Definition 2.7 Let
be the set of all random tempered sets in X. Then
is said to be asymptotically compact in X if for
-a.e.
,
has a convergent subsequence in X whenever
, and
with
.
Definition 2.8 A random compact set
is said to be a random attractor if it is a random attracting set and
for
-a.e.
and all
.
Theorem 2.9 Let
be a continuous random dynamical system on X over
. If there is a closed random tempered absorbing set
of
and
is asymptotically compact in X, then
is a random attractor of
, where
Moreover,
is the unique attractor of
.
Lemma 2.10 ( [21]) Let
be a Banach space and
be an autonomous dynamical system with the global attractor
in X. Given
, suppose that
is the perturbed random dynamical system with a random attractor
and a random absorbing set
. Then for
-a.e.
,
if the following conditions are satisfied:
1) For
-a.e.
,
,
, and
with
, it hold that
2) There exists some deterministic constant c such that, for
-a.e.
,
where
3) There exists a
such that, for
-a.e.
,
is precompact in X.
3. Random Dynamical System
In this section, we show that there is a continuous random dynamical system generated by the stochastic nonclassical diffusion equation defined on
with arbitrary polynomial growth nonlinearity and multiplicative noise:
(3.1)
with the initial value condition
(3.2)
where
are proper positive constants,
and
is a nonlinear function satisfying the following conditions are the same as those in [24]:
(3.3)
(3.4)
(3.5)
where
are a non-negative constant.
To model the random noise in Equation (3.1), we need to define a shift operator
on
(where
is defined in the introduction) by
(3.6)
then
is an ergodic metric dynamical system, see [20] [24].
For our purpose, it is convenient to convert Equation (3.1) into a deterministic system with a random parameter, and then show that it generates a random dynamical system.
We now introduce an Ornstein-Uhlenbeck process given by the Brownian motion. Put
(3.7)
which is called the Ornstein-Uhlenbeck process and solves the Itô equation
(3.8)
From [19] [25] [27] [28], it is known that the random variable
is tempered, and there is a
-invariant set
of full
measure such that for
every
,
is continuous in t;
; and
.
To show that Equation (3.1) generates a random dynamical system, we let
(3.9)
where u is a solution of Equation (3.1). Then we can consider the following evolution equation with random coefficients but without white noise:
(3.10)
with the initial value condition
(3.11)
Definition 3.1. A function
is called a weak solution of Equations (3.10) and (3.11) on the interval
if
and
for all test function
.
Theorem 3.2. Under the assumptions (3.3)-(3.5),
for P-a.e.
and any
, there is a unique solution
satisfying
From Theorem 3.2 above, we now define a mapping
by
for all
Then
satisfies conditions (1) and (2) in Definition 2.2. Therefore,
is a continuous random dynamical system associated with Equation (3.1) on
.
4. Uniform Estimates of Solutions
In this section, we derive uniform estimates on the solutions of (3.10) and (3.11) defined on
when
with the purpose of proving the existence of a bounded random absorbing set and the asymptotic compactness of the random dynamical system associated with the equation. In particular, we will show that the tails of the solutions for large space variable are uniformly small when time is sufficiently large. Some techniques about the unbounded case can be founded in [14] [24] [25] [26]. Here we always assume that
is the collection of all tempered random subsets of
with respect to
. The next Lemma shows that
has a random absorbing set in
.
Lemma 4.1 Assume that
, and (3.3)-(3.5) hold. Then there exists a random ball
centered at 0 with random radius
such that
is a random absorbing set for
in
, that is, for any
and
-a.e.
, there is
such that
(4.1)
Proof We first derive uniform estimates on
from which the uniform estimates on
. Multiplying Equation (3.10) with v and then integrating over
, we have
(4.2)
By the Hölder inequality and the Young inequality, we conclude
(4.3)
By condition (3.3), we get
(4.4)
Then inserting (4.3) and (4.4) into (4.2), it yields
(4.5)
where
.
Noticing that, from (4.5), let
, it follows that
(4.6)
Hence, we can rewrite the above equation as
(4.7)
By applying Gronwall’s lemma to (4.7), we find that
(4.8)
By replacing
by
in (4.8), we get
(4.9)
By the properties of Ornstein-Uhlenbeck process,
(4.10)
Notice that
is tempered, then for any
,
(4.11)
We can choose
(4.12)
And let
(4.13)
Then
, and
is a random absorbing set for
in
, which completes the proof.
Lemma 4.2 Assume that
, and (3.3)-(3.5) hold. Then there exists a tempered random variable
, such that for any
and
, there exists a
such that the solution
of (3.10) satisfies for
-a.e.
, for all
,
(4.14)
Proof By substituting t by
and
by
in (4.8) for any
, we find that
(4.15)
Multiplying two sides of Equation (4.15) by
, then simplifying it, we find that for all
(4.16)
By the Gronwall lemma to (4.6), we get that for all
,
(4.17)
which obviously gives
(4.18)
By replacing
by
into (4.18), we get
(4.19)
Together with (4.16) and (4.19), we have
(4.20)
Replacing
by t and t by
in (4.20), we have
(4.21)
For
, to yield that
(4.22)
By the property of
and temperedness of
, there exists
such that for all
, from (4.21) and (4.22), we find that
(4.23)
It is easy to check that
is tempered. This completes the proof.
Lemma 4.3 Assume that
, (3.3)-(3.5) hold. Then, there exists a tempered random variable
, such that for any
and
, there exists a
such that the solution
of (3.10) satisfies for
-a.e.
, for all
,
(4.24)
(4.25)
Proof Taking the inner product of Equation (3.10) with
in
, we have
(4.26)
Now, we estimate the first term on the right-hand side of (4.26) by the condition (3.5), we get
(4.27)
On the other hand, in the second term on the right-hand side of (4.26) by Hölder’ inequality and Young inequality, we conclude
(4.28)
Then inserting (4.27) and (4.28) into (4.26), it yields
(4.29)
where
.
Noticing that, from (4.29), let
, it follows that
(4.30)
Hence, we can rewrite the above equation as
(4.31)
By applying the Gronwall lemma to (4.30), we find that
(4.32)
which obviously gives
(4.33)
By replacing
by
and t by
into (4.33), we get
(4.34)
Thanks to
(4.35)
Together with (4.34) and (4.35), we have
(4.36)
For
, to yield that
(4.37)
Together with (4.36) and (4.37), we have
(4.38)
By the property of
and temperedness of
, there exists
such that for all
, from (4.38), we find that
(4.39)
It is easy to check that
is tempered.
Now, let
be the non-negative constant in Lemma 4.2 and Equation (4.39), take
and
. Then integrate (4.31) over
, we find that
(4.40)
Now integrating (4.40) with respect to s over
, we conclude that
(4.41)
Replacing
by
(4.42)
By Lemma 4.2 and Equation (4.39), it follows from (4.42) it yield that, for all
(4.43)
This proof is concluded.
Lemma 4.4 Assume that
, and (3.3)-(3.5) hold. Let
and
. Then, for any
, there exist
and
, such that the solution
of Equation (3.10) satisfies for
-a.e.
,
,
(4.44)
Proof We first need to define a smooth function
from
into
such that
on
and
on
, which evidently implies that there is a positive constant c such that the
for all
. For convenience, we write
.
Multiplying Equation (3.10) with
and integrating over
, we have
(4.45)
where
(4.46)
where
is a non-negative constant.
By condition (3.3), we get
(4.47)
For the last term on the right-hand side of (4.45), we have that
(4.48)
Then inserting (4.46)-(4.48) into (4.45) to see that
(4.49)
Hence, we can rewrite (4.49) as
(4.50)
By applying the Gronwall’s lemma to (4.50), for every
, we find that
(4.51)
Then, substituting
by
into (4.51), we have that
(4.52)
Then, we estimate every term on the right-hand side of (4.52). Firstly by (4.8), and replacing t by
and
by
, then we get
(4.53)
Then, there exists
, such that for all
, then
(4.54)
For the second term on the right-hand side of (4.52), Since
, there are
and
, such that for all
and
, then
(4.55)
For the last term on the right-hand side of (4.52). By replacing t by s and
by
in (4.8), we get
(4.56)
Then, by
, there exist
and
, such that for all
and
, we find that
(4.57)
By letting
, and
.
Then, inserting (4.54) and (4.55) and (4.57) into (4.52), for all
and
, we obtain that
(4.58)
which shows that
(4.59)
This proof is completed.
5. Random Attractors
In this section, we prove the existence of a global random attractor for the random dynamical system
associated with the stochastic reaction-diffusion Equations (3.1) and (3.2) on
. The main result of this section can now be stated as follows.
Lemma 5.1 Assume that
, and (3.3)-(3.5) hold. Then the random dynamical system
generated by (3.10) is asymptotically compact in
, that is, for
-a.e.
, the sequence
has a convergent subsequence in
provided
,
and
.
Proof Let
,
and
. Then by Lemma 4.1, for
-a.e.
, we have that
is bounded in
.
Hence, there exist
such that, up to a subsequence,
(5.1)
Next, we prove the weak convergence of (5.1) is actually strong convergence. Given
, by Lemma 4.4, there exist
,
and
, such that
for every
(5.2)
On the other hand, by Lemma 4.1 and 4.3, there exist
, such that for all
,
(5.3)
Let
be large enough such that
for
. Then by (5.3) we find that, for all
,
(5.4)
Denote by
. By the compactness of embedding
↪
. It follows from (5.4) that, up to a subsequence depending on
(5.5)
which shows that for the given
, there exist
, such that for all
,
(5.6)
Note that
. Therefore, there exist
, such that
(5.7)
By letting
, and
.
Then, by (5.2), (5.6) and (5.7), we find that for all
,
(5.8)
which shows that
(5.9)
This as desired.
We are now in a position to present our main result, the existence of a global random attractor for
in
.
Lemma 5.2 Assume that
, and (3.3)-(3.5) hold. Then the random dynamical system
generated by (3.10) has a unique global random attractor in
.
Proof Notice that the random dynamical system
has a random absorbing set
in
by Lemma 4.1. On the other hand, by Lemma 5.1, the random dynamical system
is asymptotically compact in
. Then by Theorem 2.9, the random dynamical system
generated by (3.10) has a unique global random attractor in
.
6. Upper Semi-Continuity of Random Attractor in
In this section, we investigate the existence and upper semi-continuity of random attractors for (3.1) and (3.2) by studying (3.10) and (3.11). To indicate the dependence of solutions on b, we respectively write the solutions of (3.1) and (3.2) and (3.10) and (3.11) as
and
. Let
be the solution of the following deterministic system corresponding to (3.10) and (3.11):
(6.1)
In fact, the system (6.1) is also equivalent to (3.1) and (3.2) provided
, that is,
, where
is the solution of corresponding to (3.1) and (3.2).
Remark 6.1 Correspondingly, the deterministic and autonomous system
generated by (6.1) is readily verified to admit a global attractor
in
.
The next lemma shows the convergence
provided
with
, which is important for the upper semi-continuity of random attractors.
Lemma 6.2 Assume that
and (3.3)-(3.5) hold. Then, for each
and
-a.e.
, there exist constants
and
independent of b, such that
(6.2)
Proof Let
. Then, by (3.10) and (3.11) and (6.1), W satisfies
(6.3)
where we have used the relations
. Taking the inner product of (6.3) with W in
we find that
(6.4)
By conditions (3.4) and (3.5), to yield that
(6.5)
On the other hand,
(6.6)
Then inserting (6.5) and (6.6) into (6.4) to see that
(6.7)
Since
, by (5.7) we conclude that
(6.8)
Hence, we can rewrite (6.8) as
(6.9)
where
, independent of b and
-a.s. bounded for each
;
(6.10)
By applying Gronwall’s lemma to (6.9), we find that
(6.11)
Now for each fixed
and
-a.e.
, consider the last term in (6.11). First, notice that, by (6.10),
(6.12)
According to (4.9), by replacing
with
, we conclude that
(6.13)
where
is independent of b,
-a.s. bounded for each fixed t, and given by
(6.14)
where
is the tempered random variable given by (4.12).
By taking
in (6.13) we find that
(6.15)
Similarly, from (4.8) we know that, when
,
(6.16)
Therefore, from (6.11)-(6.16), to yield that
(6.17)
where
is
-a.s. bounded for each
(since
) is pathwise continuous) and independent of b. This proof is completed.
Theorem 6.3 Assume that
and (3.3)-(3.5) hold. Then,
-a.e.
, we have
Proof To achieve the result, it suffices to verify conditions (1), (2) and (3) in Lemma 2.10.
Notice that, condition (1) is actually proved by Lemma 6.2. For condition (2), since Lemma 4.1, has proved that random dynamical system
possesses a closed random absorbing set
, which is given by
where
it is readily to obtain that,
-a.e.,
which deduces condition (2) immediately. Now consider condition (3). Given
. From Lemma 4.3 we know that
is also closed and tempered random absorbing set
in
, where
with
and
are tempered random variables in
and continuous in b. Let
Then, we know
is compact in
. From
it follows that
is precompact in
. Hence, condition (3) is clear and this proof is completed.
Acknowledgements
The authors would like to express their gratitude to the anonymous reviewers for their valuable suggestions that have improved the quality of this paper.