Pullback Random Attractors for Non-Autonomous Stochastic Fractional FitzHugh-Nagumo System ()
1. Introduction
In this paper, we investigate the random attractor of the non-autonomous stochastic fractional FitzHugh-Nagumo equations with additive white noise in bounded domains. Let
,
and
be a smooth bounded domain of
. We consider the following stochastic system in U:
(1)
(2)
with initial conditions
(3)
and boundary conditions
(4)
where
,
are real-valued functions on
,
and
are positive constants,
,
,
(
and the details of these spaces will be given later),
,
and
are independent two-sided real-valued Wiener processes with the standard process
on a probability space which will be specified below. The nonlinear function
is a differential function about two variables satisfying: for all
and
,
(5)
(6)
(7)
(8)
where
are constant,
,
with
, and
,
.
The FitzHugh-Nagumo system is a model for describing the signal transmission across axons in neurobiology, see [1] [2] [3]. The long term dynamics and inertial manifolds for the deterministic FitzHugh-Nagumo system have been extensively studied by many authors, see [4] [5] [6]. The existence of random attractor for the stochastic or lattice FitzHugh-Nagumo system has been investigated in [7] [8] [9] [10]. Recently, the fractional FitzHugh-Nagumo monodomain model is presented, the model consists of a coupled fractional nonlinear reaction-diffusion model and a system of ordinary differential equations. The stability and convergence are discussed using numerical method in [11]. To the best of our knowledge, there are some results on numerical calculation of deterministic fractional FitzHugh-Nagumo equation, but few results for theory study
of stochastic fractional FitzHugh-Nagumo equation, especially for
.
As far as the author is aware, the attractors of the fractional stochastic equations are not well studied, it seems that the only publications [12] [13] in this respect, where the authors researched the existence of random attractors for the fractional
stochastic equation with
. In [14] [15], the authors discussed the asymptotic behavior of fractional reaction-diffusion equation with
.
In this paper, we explore the long time behavior of the solutions when the Equations (1)-(4) are perturbed by an additive white noise. There are several difficulties in this paper. Firstly, since the FitzHugh-Nagumo equation is a coupling equations, thus the uniform estimates of solutions are slightly different from the reaction-diffusion equation [15]. On the other hand, comparing with [15], we are concerned with the existence of random attractors of the fractional FitzHugh-Nagumo equation on U driven by additive noise rather than multiplicative noise, so new difficulties arise from the estimates for some terms, especially the nonlinearity f. Finally, the lack of higher regularity of the solution
for the problem (2) which makes the difficulty to construct a compact attracting set of the random dynamical system. To achieve our goals, we must overcome these difficulties and establish the pullback asymptotic compactness of solutions in
. We decompose the second component
into a sum of two parts to overcome the lack of higher regularity like dealing with the wave equation in [16] [17].
This paper is organized as follows. In Section 2, we recall some basic concepts and define a continuous random dynamical system based on the solutions of the stochastic fractional FitzHugh-Nagumo Equations (1)-(4) in
. We derive some uniform estimates for solutions and prove the existence of a pullback random attractor by pullback asymptotic compactness of solutions in Section 3.
2. Cocycles of the Stochastic Fractional FitzHugh-Nagumo System
In this section, we first collect some well-known results from the theory of random attractors and non-autonomous random dynamical systems. For further details, readers are also referred to [18] [19] [20] [21].
Let
be a parametric dynamical system on the probability space
, in which
,
is the Borel
-algebra induced by the compact-open topology of
and P is the Wiener measure on
, the group
defined by
for
. We usually write the norm of
as
and the scalar product of
as
. We also use
to denote the norm of a Banach space X.
In the following, “property holds for a.e.
with respect to
” means that there is
with
and
for all
such that property holds for all
.
Definition 2.1. Let
be a collection of some families of nonempty subset of X and
a continuous cocycle on X,
. Then
is called a
-pullback random attractor for
if the following conditions are satisfied, for every
and a.e.
,
1)
is compact in X, and
is measurable;
2)
is strictly invariant, i.e.,
, for all
;
3)
attracts every member of
in X, i.e., for all
, we have
where
is the Hausdorff semi-distance in X.
Proposition 2.1. Suppose X is a separable Banach spaces. Let
be an inclusion-closed collection of some families of nonempty subsets of X and
be a continuous cocycle on X over
. For all
,
, and
,
has a unique
-pullback random attractor
in
given by,
if 1)
has a compact measurable
-pullback absorbing set K in
.
2)
is
-pullback asymptotically compact in X.
To describe the main results of this raper, we review some concepts of the fractional Laplace operator on the bounded domain
(see [22] for details). Let
be the Schwartz space of rapidly decaying
functions on
, then for
, the fractional Laplace operator
is given by, for
,
,
(9)
where
is a positive constant.
Let
be the fractional Sobolev space defined by
which is equipped with the norm
Note that
is a Hilbert space with inner product given by
For convenience, we will also use the notation:
By proposition 3.6 in [22], we find that:
(10)
and hence
is an equivalent norm of
. Similar to
, we can define
and
for
(see [22]).
Since the fractional operator
is non-local, we here interpret the boundary (4) as
instead of
. From [23] and the references therein, we know such a interpretation is consistent with the non-local nature of the fractional operator. Thus, let
and
.
Then, we give the essential assumptions and define a continuous random dynamical system for the stochastic fractional FitzHugh-Nagumo equations in H. More precisely, we consider:
(11)
(12)
with initial conditions
(13)
and boundary conditions
(14)
Define
by
It is well know that
is the unique stationary solution of the following stochastic equations
(15)
(16)
In addition, assume that
, from [18], we have
(17)
and
(18)
It follows from [10] that there exists a
-invariant set of full measure (still denoted by
) such that
are both continuous in t for each
.
We now transform the stochastic Equations (11)-(14) into a pathwise deterministic one by using the random variable
. Given
and
, let
is a solution of (11)-(14), introduce variables transformation:
(19)
(20)
By (11)-(14) and (15)-(16) we get
(21)
(22)
with initial conditions
(23)
and boundary conditions
(24)
Recall that H and V are Hilbert space. Note that the definition of H is consistent with conditions (23) and (24), since
can be considered as an element of H by setting
for
. It follows from the standard arguments as in [24] that under conditions (1)-(7), problem (21)-(24) is well-posed in H. The unique solution defines two cocycles
as follows: for
and
(25)
(26)
where
. Both cocycles
and
are equivalent and so we will discuss only the cocycle
induced by Equations (21)-(24) in this paper. Let
be a family of bounded nonempty subsets of H. Such a family B is called tempered if for every
and
,
(27)
where the norm
of set B in H is given by
. From now on, we will use
to denote the collection of all tempered families of bounded non-empty subsets of H:
The functions g and h in (1)-(2) are satisfy, for every
,
(28)
(29)
and for every
,
(30)
Throughout this paper, C denotes a constant which may be different from the context.
3. Existence of Random Attractor
This section is devoted to uniform estimates of solutions for the problem (21)-(24), which are useful for constructing random pullback absorbing sets. We begin with the uniform estimates of solutions in H.
Lemma 3.1. Under conditions (5)-(8) and (28), for every
and
, there exists
such that for all
, the solution
of problem (21)-(24) satisfies
where
.
Proof. It follows from (21) and (22) that
(31)
We now estimate each term on the right-hand side of (31). For the first term, by (5) and (6) we obtain
(32)
where
satisfies the assumed conditions.
By the Young inequality, we have the following estimates on the remaining terms on the right-hand side of (31)
(33)
(34)
(35)
(36)
(37)
Since
, it follows from (31)-(37) that
(38)
Multiplying (38) by
and then integrating the above inequality on
with
, we get
(39)
Replacing
by
in the above, after changes of variables, we obtain
(40)
We now estimates the first term on the right-hand side of (40). Due to
and
is tempered, we find
(41)
Therefore, there exists
such that for all
(42)
For the remaining terms on the right-hand side of (40), we have
(43)
(44)
(45)
By (40)-(45) and (17) we obtain, for all
,
(46)
From (28), the desired estimates follow immediately.☐
We now derive uniform estimates of u in
.
Lemma 3.2. Under conditions (5)-(8) and (28), for every
and
, there exists
such that for all
and
, the solution u of problem (21) with
satisfies
Proof. Multiplying (21) by
, we obtain
(47)
For the nonlinear term in (47), by (6)-(8) we have
(48)
We now estimate the remaining terms on the right-hand side of (47). Using the Hölder inequality and Young inequality, we can get
(49)
(50)
(51)
For the last term on the left-hand side of (47), we obtain
(52)
It follows from (47)-(52) that
(53)
Given
and
, let
and
. Multiplying (53) by
, first integrating over
and then integrating with respect to r on
, replacing
by
we have
Let T be the constant in Lemma 4.1 and
. By the fact that
, from
and Lemma 4.1, for all
, we obtain
(54)
From (17)-(18) and Lemma 4.1, we immediately concludes the proof.☐
Note that Equation (22) has no any smoothing effect on the solutions. To overcome this difficulty, we must decompose the solution operator into two parts. Let
and
be the solution of the following problems, respectively,
(55)
(56)
(57)
(58)
Then
. Multiplying (55) by
, we obtain
(59)
Lemma 3.3. Under conditions (5)-(7) and (29), for every
and
, there exists
such that for all
, the solution
of problem (57)-(58) satisfies
(60)
Proof. Multiplying (57) by
, we obtain
(61)
For the first term on the right-hand side of (61), we can get
(62)
For the second term on the right-hand side of (61), we have
(63)
For the last term, we have
(64)
It follows from (61)-(64) that
(65)
Multiplying (65) by
, integrating over
, and then replacing
by
we have
(66)
which along with Lemma 4.1 and (17) conclude the proof.
In the following, we will prove the existence of
-pullback random attractor for problem (21)-(24).
Lemma 3.4. Suppose (5)-(7) and (30) hold. Then the continuous cocycle
of problem (21)-(24) has a closed measurable
-pullback absorbing set
which is given by
(67)
where
is defined by
(68)
Then for every
and
, there exists
such that the solution
of problem (21)-(24) with
satisfies, for all
,
(69)
In addition, the random variable
as in (68) is tempered, i.e., for any
,
(70)
Proof. As a special case of Lemma 4.1 with
, we obtain (69) immediately. Then we have
(71)
We now verify (70). By (68)
(72)
We have by (30)
(73)
☐
Next, we establish the
-pullback asymptotic compactness of
in H, for this purpose, we need to split
as follows. Given
and
, Let
(74)
and
(75)
where
is the solution of (55) with initial condition
at initial time
, and
is the solution of (57). By (25) we find that for every
, and
(76)
The
-pullback asymptotic compactness of
is presented below.
Lemma 3.5. Under conditions (5)-(7), (27) and (30), the continuous cocycle
associated with problem (21)-(24) is
-pullback asymptotically compact in H, that is, for every
and
, the
sequence
has a convergent subsequence in H whenever
and
.
Proof. Since
and
, by (59), (76) and Lemma 4.1 we find that
is bounded in H. Therefore, there exists
such that, up to a subsequence
(77)
On the other hand, by Lemma 4.2 and Lemma 4.3, there exist
and
such that for all
(78)
By the compactness of embedding and (77) we have, up to a subsequence,
(79)
which implies that there exists
such that for all
,
(80)
By (59) and (76) we have
(81)
which along with (80) implies
(82)
as desired.☐
Theorem 3.1. Suppose (5)-(7), (29) and (30) hold. Then the continuous cocycle
associated with problem (21)-(24) has a unique
-pullback attractor
in H.
Proof. Note that
is
-pullback asymptotically compact in H as demonstrated by Lemma 4.5 and has a closed measurable
-pullback absorbing set by Lemma 4.4. Thus by proposition 2.1, we find that
has a unique
-pullback random attractor
in H.☐
In conclusion, we prove the existence of a pullback random attractor in H for the random dynamical system associated with the non-autonomous stochastic fractional FitzHugh-Nagumo system.
Acknowledgements
The authors would like to thank the reviewers for their helpful comments. This work was partially supported by the National Natural Science Foundation of China (Nos. 11771444, 11861013), the Yue Qi Young Scholar Project, China University of Mining and Technology (Beijing), Guangxi Natural Science Foundation (No. 2017GXNSFAA198221), Promotion of the Basic Capacity of Middle and Young Teachers in Guangxi Universities (No. 2017KY0340), the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) (No. 2018061).