Attractors for the Stochastic Lattice Selkov Equations with Additive Noises ()
1. Introduction
In this paper, we study that the stochastic lattice Selkov system with the cubic nonlinearity and additive white noises on an infinite lattice is considered in [1] and [2] :
(1.1)
with initial conditions
(1.2)
where
denotes the integer set,
,
,
are positive constants,
,
is independent Brownian motions.
The reversible Selkov model is derived from a set of the two reversible chemical reactions, which has been studied by [3] [4] and other authors:
The original Selkov model corresponds to the two irreversible reactions, where the product
is an inert product. Let
and
are respectively the concentrations of the reactants Q and P, Equation (1.1) can be regarded as a Selkov system (see [5] ) on
:
(1.3)
For the Equation (1.1), the solution mapping defines a random dynamical system, which is a parametric dynamical system, and pullback absorbing set has been proved, see [1] and [2] . Random attractors are the appropriate objects for describing asymptotic dynamics of such a parametric dynamical system. Therefore, in this paper, we would prove the existence of a random attractor for the stochastic lattice Selkov Equation (1.1).
This paper is organized as follows. In the next section, we recall basic concepts and results related to random attractors. In Section 3, using the transformation of Ornstein-Uhlenbeck process, the stochastic Selkov equation with white noise is transformed into a noiseless determined Selkov equation with random variables as parameters. In Section 4, we prove the pullback asymptotic compactness for the random dynamical system. Then the existence of a random attractor is proved.
2. Preliminaries
Firstly, we introduce the relevant definitions of random attractor, which can be taken from [6] [7] [8] .
Let
be a complete separable metric space,
be a probability space,
.
Definition 2.1.
is called a metric dynamical system if
is
measurable,
for all
, and
for all
.
Definition 2.2. A continuous random dynamical system (RDS) on H over a metric dynamical system
is a mapping
which is
-measurable and satisfies, for every
,
1)
is the identity on H;
2) Cocycle property:
for all
;
3)
is strongly continuous.
Definition 2.3. Suppose
is a random dynamical system, a random set
is called a random
attractor if the following hold:
1)
is a random compact set, i.e.,
is measurable for every
and
is compact for every
;
2)
is strictly invariant, i.e., for every
and all
one has
;
3)
attracts all sets in
, i.e. for all
and
we have
where
is the Hausdorff semi-metric (here,
).
Theorem 2.1 ( [9] , Proposition 4.1) Let
be an absorbing set for the random dynamical system
which is closed and which satisfies for
the following asymptotic compactness condition: each sequence
with
has a convergent subsequence in H. Then the random dynamical system
has a unique global random attractor
3. Ornstein-Uhlenbeck Process
Let
, with the inner product and norm as follows:
Then
is a Hilbert space. Set
be the product Hilbert space. In view of the cubic term
, we need
to make (1.1) hold in
.
Introducing an Ornstein-Uhlenbeck process (O-U process) (see [10] ) in
on
given by the Wiener process:
and y solve the Itô equations:
There exists a
-invariant set
of full P measure such that
1) the mappings
are continuous for each
;
2) the random variables
is tempered.
Let
From (1.3), we have
(3.1)
with the initial value condition
4. Pullback Asymptotic Compactness
From Theorem 2.1, to prove the existence of a random attractor for the random dynamical system generated by (1.1), it is necessary to obtain the pullback absorbing property and the pullback asymptotic compactness. The pullback absorbing property has been obtained by [2] . For the pullback asymptotic compactness, we have the following lamma.
Lemma 4.1. Assume the initial functions
, where
is the absorbing set. Then for every
, there exist
and
such that the solution
of (1.1) satisfies
for all
.
Proof. We choose a smooth function
such that
for all
and
(4.1)
and there exists a positive constant
, such that
for
.
We first consider the random Equation (3.1). Let r be a fixed positive integer which will be specified later. Taking the inner product of the Equation (3.1) with
and
in E, respectively, we get
(4.2)
By
, we have
(4.3)
And
(4.4)
From (4.2), we have
(4.5)
Then from (4.2)-(4.5), we find that
(4.6)
For the third term and forth term in the right-hand side of (4.6), we have
(4.7)
For the fifth term and sixth term in the right-hand side of (4.6), we have
(4.8)
where
are positive constants depending only on
. For the last two terms in the right-hand side of (4.6),
(4.9)
From (4.6)-(4.9), we have
(4.10)
Let
. By Gronwall’s inequality in [11] , we have that for
,
(4.11)
Replace
by
. From (4.11) in [2] , with t replaced by
and
by
, it follows that
(4.12)
Thus, there exists a
such that if
, then
(4.13)
From (4.11), we have
(4.14)
where
. Recall that
, which implies that
, and
is tempered in [2] . Thus there exists
and
such that for
and
, we have
(4.15)
Since
, there exists
such that for
,
(4.16)
Finally, we estimate the last terms on the right-hand side of (4.11). Let
to be determined later. We have for
(4.17)
Using (4.12) in [2] , we have
(4.18)
Thus, by choosing
we have for
,
(4.19)
For the fixed
, from Lebesgue’s theorem, there is
such that for
,
(4.20)
Therefore, by letting
(4.21)
for
and
, we have
(4.22)
which implies that
(4.23)
provided
is large enough. This completes the proof of the lemma.
We are now ready to show the pullback asymptotic compactness of the random set
.
Lemma 4.2. For
, the set
is pullback asymptotically compact in the sense of each sequence
with
having a convergent subsequence in
.
Proof. We follow the method of [9] . Let
for each sequence
as
, and
;
this implies that there exists
such that
Since
is a bounded absorbing set, for large n,
; thus there exists
, and a subsequence
such that
(4.24)
Next, we will show that
is also strongly convergent in the norm
in
, i.e., for each
there is
such that for
,
From Lemma 4.1, for any
, there exists
and
such that for
,
(4.25)
Since
, there exists
such that
(4.26)
Let
. From the weak convergence (4.24), we have for each
as
,
which implies that there exists
such that for
,
(4.27)
Combining (4.25)-(4.27), we obtain that for
,
(4.28)
Hence, we have completed the proof of Lemma 4.2.
Theorem 4.1. The random dynamical systems
possess a random attractor in
.
Proof. Note that random dynamical system is pullback asymptotically compact in E by Lemma 4.1 and 4.2. On the other hand, the random dynamical system has a pullback absorbing set by Lemma 4.1 in [2] . Then the existence and uniqueness of a random attractor follow from Theorem 2.1 immediately.
Acknowledgements
The paper is supported by Shanghai Natural Science Foundation (16ZR1414000), Humanities and Social Sciences Project of the Ministry of Education (16YJCZH043).