Random Attractors for Stochastic Reaction-Diffusion Equations with Distribution Derivatives on Unbounded Domains ()
1. Introduction
The understanding of the asymptotic behavior of dynamical system is one of the most important problems of modern mathematical physics; one way to attack the problem for dissipative deterministic dynamical systems is to consider its global attractors. This is an invariant set that attracts all the trajectories of the system. Its geometry can be very complicated and reflects the complexity of the long-time dynamical of the systems. In this paper we investigate the asymptotic behavior of solutions to the following stochastic reaction-diffusion equations with distribution derivatives and additive noise defined in the space
:
(1.1)
with initial data
(1.2)
where
is a positive constant;
is distribution derivatives;
; f is a
nonlinear function satisfying certain dissipative conditions; hj is given functions defined on
; and
is independent two sided real-valued wiener processes on probability space which will be specified later.
Stochastic differential equations of this type arise from many physical systems when random spatio-temporal forcing is taken into account. In order to capture the essential dynamics of random systems with wide fluctuations, the concept of pullback random attractors was introduced in [1] , being an extension to stochastic systems of the theory of attractors for deterministic equations found in [2] - [5] , for instance. The existence of such random attractors has been studied for stochastic PDE on bounded domains; see, e.g. [6] [7] , and for stochastic PDE on unbounded domains, see, e.g. [8] [9] , and the references therein. In the present paper, we prove the existence of such a random attractor for stochastic reaction-diffusion Equation (1.1) defined in
which is not founded.
Notice that the unboundedness of domain introduces a major difficulty for proving the existence of an attractor because Sobolev embedding theorem is no longer compact and so the asymptotic compactness of solutions cannot be obtained by the standard method. In the case of deterministic equations, this difficulty can be overcome by the energy equation approach, introduced by Ball in [10] and then employed by several authors to prove the asymptotic compactness of deterministic equations in unbounded domains. This idea was developed in [5] to prove asymptotic compactness for the deterministic version of (1.1) on
. In this paper, we provide uniform estimates on the far-field values of solutions to circumvent the difficulty caused by the unboundedness of the domains. The main contribution of this paper is to extend the method of using tail estimates of the case stochastic dissipative PDEs and prove the existence of random attractor for the stochastic reaction-diffusion equation with distribution derivatives on the unbounded domain
.
The paper is organized as follows. In Section 2, we recall some preliminaries and abstract results on the existence of a pullback random attractor for random dynamical systems. In Section 3, we transform (1.1) into a continuous random dynamical system. Section 4 is devoted to obtain the uniform estimates of solution as
. These estimates are necessary for proving the existence of bounded absorbing sets and the asymptotic compactness of the equation. In Section 5, we first establish the asymptotic compactness of the solution operator by giving uniform estimates on the tails of solutions, and then prove the estimates of a random attractor.
We denote by
and
the norm and the inner product in
and use
to denote the norm in
. Otherwise, the norm of a general Banach space X is written as
. The letters C and
are generic positive constants which may change their values form line to line or even in the same line.
As mentioned in the introduction, our main purpose is to prove the existence of a random attractor. For that matter, first, we will recapitulate basic concepts related to random attractors for stochastic dynamical systems. The reader is referred to [6] [11] -[13] for more details. Let
be a separable Hilbert space with Borel s-algebra B(X), and let
be a probability space.
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 P-a.e.
, (1)
is the iden- tity on X; (2)
for all
(3)
is continuous for all
. Hereafter, we always assume that
is a continuous RDS on X over
.
Definition 2.3. A random bounded set
of X is called tempered with respect to
if for P-a.e
,
![]()
where
.
Definition 2.4. Let D be a collection of random subsets of X and
. Then
is
called a random absorbing set for
in D if for every
and P-a.e
, there exists
such that
![]()
Definition 2.5. Let D be a collection of random subsets of X. Then
is said to be D-pullback asymptotical-
ly compact in X if for P-a.e
,
has a convergent subsequence in X whenever
,
and
with
.
Definition 2.6. Let D be a collection of random sunsets of X. Then a random set
of X is called a D-random attractor (or D-pullback 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 D, that is, for every
,
![]()
where d is the Hausdorff semi-metric given by
for any
and
. The following existence result for a random attractor for a continuous RDS can be found in [8] [13] . First, recall that a collection D of random subsets is called inclusion closed if whenever
is an arbitrary random set, and
is in D with
for all
, then
must belong to D.
Definition 2.7. Let D be an inclusion-closed collection of random subsets of X and
a continuous RDS on X over
. Suppose that
is a closed random absorbing set for
in D and
is D-pullback asymptotically compact in X. Then
has a unique D-random attractor
which is given by
![]()
In this paper, we will take D as the collection of all tempered random subsets of
and prove the stochastic reaction-diffusion equation in
has a D-random attractor.
3. The Reaction-Diffusion Equation on Rn with Distribution Derivatives and Additive Noise
(3.1)
with initial condition
(3.2)
where
is a positive constant,
,
, for some
, ![]()
are distribution derivative,
are independent two-side real-valued wiener processes on a probability
space which will be specified below, and
with the following assumptions:
(3.3)
(3.4)
(3.5)
for
and
, where
,
,
are positive constants and
.
In the sequel, we consider the probability space
where
.
is the Borel s-algebra induced by the compact-open topology of
, and P the corresponding wiener measure on
. Then we identify
with
![]()
Define the time shift by
![]()
Then
is a metric dynamical system.
We now associate a continuous random dynamical system with the stochastic reaction-diffusion equation over
. To this end, we need to convert the stochastic equation with a random additive term in to a deterministic equation with a random parameter. Given
consider the One-dimensional Ornstein- uhlenbeck equation
(3.6)
The solution of (3.6) is given by
![]()
Note that the random variable
is tempered and
is P-a.e continuous, therefore, it follows form proposition 4.3.3 in [11] that there exists a tempered function
such that
(3.7)
where
satisfies for P-a.e ![]()
(3.8)
Then it follows form (3.7), (3.8) that, for P-a.e. ![]()
(3.9)
Putting
by (3.6) we have
![]()
The existence and uniqueness of solutions to the stochastic partial differential Equation (3.1) with initial condition (3.2) which can be obtained by standard Fatou-Galerkin methods. To show that problem (3.1), (3.2) generates a random system, we let
where u is a solution of problem (3.1), (3.2), then
satisfies
(3.10)
By a Galerkin method, one can show that if f satisfies (3.3)-(3.5), then in the case of a bounded domain with Dirichlet boundary conditions, for P-a.e.
, and for all
, (3.10) has a unique solution
![]()
with
for every T > 0, one may take the domain to be a sequence of Balls with radius approaching
to deduce the existence of a weak solution to (3.10) on
, further, one may show that
is unique and continuous with respect to
in
for all
. Let
.
Then the process u is the solution of problem (3.1), (3.2), we now define a mapping
by
(3.11)
Then
is satisfies conditions (1)-(3) in Definition 2.2 therefore
is a continuous random dynamical system associated with the stochastic reaction-diffusion equation on
. In the next two sections, we establish uniform estimates for the solutions of problem (3.1), (3.2) and prove the existence of a random attractor for
.
4. Uniform Estimates of Solutions
In this section, we drive uniform estimates on the solutions of (3.1), (3.2) 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, i.e. solutions evaluated at large values of
, are uniformly small when the time is sufficiently large.
We always assume that D is the collection of all tempered subsets of
with respect to
the next lemma shows that
has a random absorbing set in D.
Lemma 4.1. Assume that gj,
, and (3.3)-(3.5) hold. Then there exists
such that
is a random absorbing set for
in D, that is, for any
and P-a.e
,
there is
such that
![]()
Proof. We first derive uniform estimates on
from which the uniform estimates on
. Multipling (3.10) by
and then integrating over
, we have
(4.1)
For the nonlinear term, by (3.3)-(3.5) we obtain
(4.2)
on the other hand, the next two terms on the right-hand side of (4.1) are bounded by
(4.3)
the last term on the right-hand side of (4.1) is bounded by
(4.4)
where
and
.
Then it follows from (4.1)-(4.4) that
(4.5)
Note that
and
, therefore, the right-hand side of (4.5) is bounded as following
(4.6)
By (3.9), we find that for P-a.e, ![]()
(4.7)
it follows from (4.5), (4.6) that, all
,
(4.8)
which implies that for all
,
(4.9)
Let
. Applying Gronwall’s lemma, we find that, for all
,
(4.10)
By replacing
by
, we get from (4.10) and (4.7) that for all ![]()
(4.11)
Note that
.
So from (4.11) we get that, for all
,
(4.12)
By assumption
is tempered. On the other hand, by definition,
is also tempered,
therefore, if
. Then there is
such that for all ![]()
![]()
which along with (4.12) shows that, for all
,
(4.13)
Given ![]()
![]()
Then
, further, (4.13) indicates that
is a random absorbing set for
in D.
Which completes the Proof. ,
We next drive uniform estimates for u in
and for u in
.
Lemma 4.2. Assume that
and (3.3)-(3.5) hold, let
and
.
Then for every
and P-a.e
, the solutions
of problem (3.1), (3.2) and
of (3.11) with
satisfy, for all
.
(4.14)
(4.15)
where C is a positive deterministic constant independent of
and
is the tempered function in (3.7).
Proof. First, replacing t by
and then replacing
by
in (4.10) we find that
![]()
Multiply the above by
and then simplify to get.
(4.16)
By (4.7), the second term on the right-hand side of (4.16) satisfies
(4.17)
From (4.16), (4.17) it follows that
(4.18)
By (4.8) we find that, for ![]()
(4.19)
Dropping the first term on the left-hand side of (4.19) and replacing
by
, we obtain that, for all ![]()
(4.20)
By (4.7), the second term on the right-hand side of (4.20) satisfies, for all ![]()
(4.21)
Then, using (4.20) and (4.21), it follows from (4.20) that
![]()
This completes the proof. ,
Lemma 4.3. Assume that gj,
and (3.3)-(3.5) hold, Let
and
.
Then for P-a.e
, there exists
such that the solutions
of problem (3.1), (3.2) and
of (3.11) with
satisfy, for all
.
![]()
![]()
where C is a positive deterministic constant and
is the tempered function in (3.7).
Proof. First replacing t by t + 1 and then replacing
by t in (4.14), we find that
(4.22)
Note that
for
, hence, form (4.22) we have
(4.23)
Since
and
are tempered there is
such that for all ![]()
![]()
which along with (4.23) shows that, for all
,
(4.24)
Then from (4.10) using the same steps of last process applying on (4.15), we get that
(4.25)
The above uniform estimates is a special case lemma 4.2, then the lemma follows from (4.24)-(4.25). ,.
Lemma 4.4. Assume that gj,
and (3.3)-(3.5) hold, let
and
.
Then for P-a.e
, there exists
such that the solution
of problem (3.1), (3.2) satisfies, for all
.
![]()
where C is a positive deterministic constant and
is the tempered function in (3.9).
Proof. Let
be the positive constant in lemma 4.3, take
and
, by (3.11) we find that
(4.26)
By (3.9) we obtain
(4.27)
Now integration (4.26) with respect to s over (t, t + 1), by lemma 4.3 and inequality (4.27), we have
(4.28)
Then the lemma follows from (4.28). ,
Lemma 4.5. Assume that gj,
and (3.3)-(3.5) hold, let
and
. Then for P-a.e
, there exists
such that for all
.
![]()
where C is a positive deterministic constant and
is the tempered function in (3.9).
Proof. Taking the inner product of (3.10) with
in
, we get that
(4.29)
We estimates the first term in the right-hand side of (4.29) by (3.3), (3.4) we have
(4.30)
On the other hand, the second term on the right-hand side of (4.29) is bounded by
(4.31)
The last term on the right-hand side of (4.29) is bounded by
(4.32)
By (4.29)-(4.32) we get that
(4.33)
Let
(4.34)
Since
and
, there are positive constants
and
such that
![]()
which along with (3.9) shows that
(4.35)
By (4.33), (4.34) we have
(4.36)
Let
be the positive constant in lemma 4.3 take
and
. Then integrate 4.36 over (s, t + 1) to get that
![]()
Now integrating the above equation with respect to s over (t, t + 1), we find that
![]()
Replacing
by
we obtain that
(4.37)
By lemma 4.3 and 4.4, it follows from (4.37) and (4.35) that, for all ![]()
(4.38)
Then by 4.38 and 3.9, we have, for all ![]()
![]()
which completes the proof. ,
Lemma 4.6. Assume that gj,
and (3.3)-(3.5) hold, let
and
.
Then for every
and P-a.e
, there exists
and
such that the solution
of (3.10) with
satisfies, for all ![]()
![]()
Proof. Choose a smooth function
defined on
such that
for all
and
![]()
Then there exists a constant C such that
for any
, multiplying (3.10) by
in
, and integrating over
we find that
(4.39)
We now estimate the terms in (4.39) as follows, first we have
(4.40)
Note that the second term on the right-hand side of (4.40) is bounded by
(4.41)
By (4.40), (4.41), we find that
(4.42)
From (4.39) the first term on the right-hand side, we have
(4.43)
By (3.3), the first term on the right-hand side of (4.43) is bounded by
(4.44)
By (3.4), the second term on the right-hand side of (4.43) is bounded by
(4.45)
Then it follows from (4.43)-(4.45) we have that
(4.46)
For the second term on the right-hand side of (4.39) we have
(4.47)
For the last term on the right-hand side of (4.39), we have that
(4.48)
Finally, by (4.39), (4.42) and (4.47) (4.48), we have that
(4.49)
Note that (4.49) implies that
(4.50)
By lemma 4.1 and 4.5, there is
such that for all
,
(4.51)
Now integrating (4.50) over
we get that, for all ![]()
(4.52)
Replacing
by
, we obtain from (4.52) that, for all
,
(4.53)
In what follows, we estimate the terms in (4.53). First replacing t by
and then replacing
by
in (4.10), we have the following bounds for the first term on the right-hand side of (4.53)
(4.54)
where we have used (4.7). By (4.54), we find that, given
, there is
such that for all ![]()
(4.55)
By lemma 4.2, there is
such that the fourth term on the right-hand side of (4.53) satisfies
![]()
And hence, there is
such that for all
and
,
(4.56)
First replacing t by s and then replacing
by
in (4.10), we find that the third term on the right-hand side of (4.53) satisfies
![]()
This implies that there exist
and
such that for all
and
,
(4.57)
Then the second term on the right-hand side of (4.53), there exist
and
such that for all
and
we have that
(4.58)
Note that
,
. therefore, there is
such that for all
,
![]()
For the five term on the right-hand side of (4.53), we have
![]()
(4.59)
Note that
and
Hence there is
such
that for all
and
.
(4.60)
where
is the tempered function in (3.7) and
is the positive constant in the last term on the right-hand side of (4.60), By (4.60) and (3.7), (3.8), we have the following bounds for the last term on the right-hand side of (4.53):
(4.61)
Let
and
then it follows from (4.53), (4.55)-(4.61) that, for all
and
, we have
![]()
which shows that for all
and ![]()
![]()
This completes the proof. ,
Lemma 4.7. Assume that gj,
and (3.3)-(3.5) hold. Let
and
.
Then for every
and P-a.e
, there exists
and
such that, for all ![]()
![]()
Proof. Let
and
be the constant in lemma 4.6 By (4.60) and (3.7) we have, for all
and ![]()
(4.62)
then by (4.62) and lemma 4.6, we get that, for all
and ![]()
![]()
which completes the proof. ,
5. Random Attractors
In this section, we prove the existence of a D-random attractor for the random dynamical system
associated with the stochastic reaction-diffusion Equations (3.1), (3.2) on
. It follows from lemma 4.1 that
has a closed random absorbing set in D, which along with the D-pullback asymptotic compactness will imply the existence of a unique D-random attractor. The D-pullback asymptotic compactness of
is given below and will be proved by using the uniform estimates on the tails of solutions.
Lemma 5.1. Assume that gj,
and (3.3)-(3.5) hold. Then the random dynamical system ϕ is D-
pullback asymptotically compact in
; that is, for P-a.e
, the sequence ![]()
has a convergent subsequence in
provided
,
and ![]()
Proof. Let
and
Then by lemma 4.1 for P-a.e
, we have that
![]()
Hence, there is
such that, up to a subsequence,
(5.1)
Next, we prove the weak convergence of (5.1) is actually strong convergence. Given
, by lemma 4.7, there is
and
such that for all
,
(5.2)
Since
, there is
such that
for every
. Hence, it follows from (5.2) that for all
,
(5.3)
On the other hand, by lemma 4.1 and 4.5, there
such that for all
,
(5.4)
Let
be large enough such that tn ≥ T2 for n ≥ N2. then by (5.4) we have that, for all n ≥ N2,
(5.5)
Denote by
the set
. By the compactness of embedding
, it fol- lows from (5.5) that, up to a subsequence,
![]()
which shows that for the given
, there exists
such that for all
,
(5.6)
Note that
. Therefore there exists
such that
(5.7)
let
and
By (5.3), (5.6), and (6.7), we find that for all
,
![]()
which shows that
![]()
as wanted. ,
Now we are in a position to present our main result: the existence of a D-random attractor for
in ![]()
Theorem 5.2. Assume that
,
and (3.3)-(3.5) hold. Then the random dynamical system
has a unique D-random attractor in
.
Proof. Notice that
has a closed random absorbing set
in D by lemma 4.1, and is D-pullback
asymptotically compact in
by lemma 5.1. Hence the existence of a unique D-random attractor for
follows from proposition 2.7 immediately. ,
Foundation Term
This work was supported by the NSFC (11101334).
NOTES
*Corresponding author.