1. Introduction
In this paper, we study the asymptotically autonomous dynamics of the following fractional nonclassical diffusion equation driven by white noise on
:
where
is positive constant,
,
,
f and
h are nonlinear functions that satisfy certain dissipative conditions.
is a one-dimensional two-sided Brownian motion over a Wiener Probability space
, where
equipped with the compact-open topology,
is the Borel
-algebra of
,
is the Wiener measure. The classical transformation
on
is given by
for all
, and thus
forms a metric dynamical system, see [
1]. For each
, we define a random variable
Then the process
is called a stationary stochastic process with a normal distribution, which satisfies
see Refs. [
2,
3]. As a consequence, we study the following Wong–Zakai approximations of Equation (
1) driven by a multiplicative noise of
:
where
,
,
. The nonlinear functions
satisfy the following conditions: for all
,
where
,
are constants, and
,
and
where number
is called the fractional critical Sobolev embedding exponent.
means that there exists a finite interval
I such that
.
When
, the fractional operator
becomes the standard Laplacian
. In this special case, the pullback attractor both in the deterministic and stochastic case for the nonclassical diffusion equation has been studied in Refs. [
4,
5,
6,
7,
8,
9,
10]. For the fractional stochastic equations driven by colored noise with
, the existence and the upper semicontinuity when
of random attractors were also studied by Refs. [
11,
12,
13]. As far as we know, there is no result available in the literature on the time-dependent uniform compactness and asymptotically autonomous upper semicontinuity of random attractors for the Wong–Zakai approximations of fractional stochastic nonclassical diffusion equation driven by a multiplicative noise of
.
Wong and Zakai first proposed the Wong–Zakai approximations for employing deterministic differential equations to approximate stochastic differential equations [
14,
15]. Their approach was then extended to multidimensional stochastic differential equations [
16,
17,
18] and stochastic differential equations driven by martingales and semimartingales [
19,
20], etc. Wong–Zakai approximations and random attractors for stochastic differential equations and lattice systems have lately been investigated in Refs. [
2,
3,
21,
22,
23,
24,
25,
26] and [
27,
28,
29,
30], respectively.
A random pullback attractor is a bi-parametric random set in the form
. The time-dependence character reflects the non-autonomous feature of the system, which should be the most significant characteristic distinguished from autonomous cases, so both the existence and some time-dependent properties of the pullback random attractor
for the cocycle generated by system (
3) in
with
are concerned. We are more specifically interested in the backward compactness of random attractor
, i.e., the compactness of
for each
. It is worth noting that the absorbing set is a union of some random sets over an uncountable index set
, hence the measurability of attractor
is uncertain. To solve this difficulty, we shall prove a crucial fact that the attractor does not vary between the two attracted universes, see Theorem 1. The time-dependence of a pullback attractor, such as the backward compactness of attractor, has recently been studied in the literature, see Refs. [
31,
32,
33,
34,
35].
In order to establish the existence result of a backward compact random attractor in
, we shall prove the backward asymptotic compactness of solutions, which indicates that the usual asymptotic compactness is uniform in the past. The main difficulties for proving such compactness comes from the fact that the Sobolev embedding
with
is not compact, and the Wiener process is nowhere differentiable with respect to time. We cannot utilize the technique of differentiating the equation with respect to
t. To get around these problems, we will use the methods of a cut-off technique [
36] and flattening properties [
37] of solutions to establish the desired backward asymptotic compactness in
.
Finally, we will show that the random attractor
is backward asymptotically autonomous to random attractor
in the sense of Hausdorff semi-distance. More precisely, for
P-a.s.
,
where
is a random attractor of the following limiting equation:
where
and
will be specified later, see Theorem 2.
Since system (
3) contains time-dependent nonlinear terms whereas system (
11) does not, it is more complicated to determine the limit process (
10), see Lemma 7. As a result, we will make certain assumptions about nonlinear terms and forcing terms, as shown in (
83)–(
85), which differs from the earlier works on an asymptotically autonomous problem in Refs. [
31,
34,
35,
38,
39].
The outline of the paper is as follows. In the next section, we give some necessary assumptions and define a non-autonomous co-cycle for system (
3). In
Section 3, we derive backward uniform estimates on the solutions in
including the backward uniform estimates on the tails and the bounded domains. In
Section 4, we prove the existence, uniqueness, and backward compactness of random attractors for problem (
3) in
. In the
Section 5, we establish the asymptotic upper semicontinuity of these attractors in
as
. Finally, we illustrate with one example the results of the paper for
briefly.
2. Non-Autonomous Co-Cycles for Fractional Equations
In this section, we first briefly review the concepts of fractional derivatives and fractional Sobolev spaces. Let
with
be the non-local, fractional Laplace operator defined by
where P.V. means the principal value of the integral and
is a constant given by
Let
with
be the fractional Sobolev space given by
which is a Hilbert space equipped with the inner product and the norm:
By Ref. [
40], we also have
and thus
The reader is referred to Ref. [
40] for more details on fractional operators.
It follows from Refs. [
40,
41] and (
9) that
where
p is the number in (
4) and
.
Under conditions (
4)–(
8), for each
,
, then as in [
42], the system (
3) has a unique solution
, where
. This allow us to define a continuous cocycle (see [
43])
given by
We take a universe
on
, which consists of all backward tempered set-valued mappings
satisfying
where
denotes the supremum of norms of all elements. Meanwhile, let
be the usual universe of all tempered sets on
, which satisfies
Obviously, is a subset of .
In order to obtain the
-pullback attractor
, we make the following assumption for some
,
where we use
(resp.
) to denote the norm of
(resp.
) here and after. Indeed, by ([
35], Lemma 4.2), one can show that the growth rate
can be arbitrary, which means for
, we have
Based on this, (
19) can imply the following tempered condition with arbitrary growth rates:
Since the systems defined on the entire space
, we also need the following assumptions
3. Backward Uniform Estimates of Solutions
In this section, we derive uniform estimates on the solutions of (
3) when
, with the purpose of proving the existence of a bounded
-pullback absorbing set that is uniform in the past and the backward asymptotic compactness of the random dynamical system associated with the equation.
Lemma 1. Let (
4)–(
9)
and (
19)
be satisfied. Then for every , and , there is a such that for all , the solution of (
3)
with satisfies where is a constant independent of τ, ω and , is given by which is well-defined due to (
19).
Proof. Let
be fixed, and
. Multiplying Equation (
3) by
u and integrate over
, we obtain
We now consider the right hand side of (
26) one by one. For the nonlinear term, we have
For the last term, by Young’s inequality, we have
Substituting (
27), (
28) into (
26), we see that
where
.
Multiplying (
29) by
, then integrating over
with
, and replace
by
, we find that
Taking the supremum with respect to the time over
in (
30) and using (
2) and (
19), we see that
and
Therefore, it follows from (
30) to (
32), and we have
which implies (
23).
Furthermore, taking (
)-th power of (
30) and multiplying by
, we integrate the result over
with
to get
Taking the supremum with respect to the time over
in (
34), and by
with
, we have
which together with (
34) implies the desired result (
24). This completes the whole proof. □
Lemma 2. Let (
4)–(
9)
hold. Then for every , and , the derivative of the solution of (
3)
satisfies, for all , Proof. Multiplying (
3) by
and integrate over
, we have
Using the assumption (
5) and the Sobolev embedding inequality given in (
12), we obtain
By (
12) and (
7), we obtain
Applying Young’s inequality, we have
It follows from (
36) to (
39) that the desired result (
35) follows immediately. □
We now derive the uniform estimates on the tails of the solutions. To this end, we introduce a smooth function
defined for
such that
for
and satisfies
for all
and
for all
. Letting
for
and
, by ([
44], Lemma 3.4), we have
Lemma 3. Let (
4)–(
9)
and (
19)
be satisfied. Then for every , and , there exists , such that for all and , the solution of (
3)
with satisfies where , and .
Proof. By Ref. [
12], Lemma 2 and a slight modification, we have the following energy equation:
We now multiply the above by
, then integrate the result over
with
and replace
by
in the resulting inequality to obtain
For the first term on the right-hand side of (
44), since
with
and
, we have, as
,
For the second term on the right-hand side of (
44), by Lemma 1, there exists
such that for all
, as
,
By (
2) and (
19), we have
tends to 0 as
. In addition, it follows from assumption (
21), we get that
For the last term on the right-hand side of (
44), apply (
2) and the assumption (
22), we have
which tends to 0 as
.
Finally, it follows from (
45) to (
49) that there exists
, such that for all
and
, we take the supremum over
in (
44), and we have
which implies the first result (
41). Furthermore, by (
51), it is easy to see that
Interchanging
x and
y in (
52) yields
Combining (
52) and (
53), we obtain the second result (
42). This completes the whole proof. □
In order to establish the backward pullback asymptotic compactness of solutions in
, we need to approximate
ba a family of bounded domains. For every
and
, let
Then for and , and for some constant independent of .
Multiplying (
3) by
and by [
12], we have
with initial-boundary conditions:
Let
and
which are considered as subspaces of
and
, respectively. Considering the following eigenvalue problem in
V:
It is easy to see that the above has a family of eigenvalues
such that
and the corresponding eigenfunctions
in
V form an orthonormal basis of
H. Write
and let
be the orthonomal projector defined by
where
.
Lemma 4. Let (
4)–(
9)
and (
19)
be satisfied. Then for every , and , there exists , such that for all and , the solution of (
3)
with satisfies Proof. Multiplying (
55) by
, we get
By (
5) and Gagliardo–Nirenberg inequality (
14), the first term on the right-hand side of (
58) is bounded by
where
due to
. On the other hand, by assumption (
7) and embedding inequality (
12) and (
13) as well as Hölder’s inequality, we have
Since
and by (
40), we obtain in the second line of (
58)
By [
44], the remaining terms on the right-hand side of (
58) is bounded by
Substituting (
59)–(
62) into (
58), we see from Lemma 2 that
where
as
. Applying the uniform Gronwall inequality over
with
and replacing
by
in the resulting inequality, then we take the supremum over
, and we conclude that
Due to
and
with
for all
, we have
Given
, let
, such that for all
and
. Then by (
19) and Lemma 1, we find that the remaining terms on the right-hand side of (
64) satisfies
which implies (
57) as desired. □
4. Existence of Backward Pullback Random Attractors
In this section, we show that the cocycle , generated by the Wong–Zakai approximations for fractional nonclassical diffusion equations, has unique -pullback random attractors in . We establish the existence of -pullback absorbing sets in firstly.
Lemma 5. Assume that (
4)–(
9)
and (
19)
hold. Then the co-cycle Φ for problem (
3)
has a -pullback random absorbing set given by and a -pullback absorbing set given by where M and are the same as given in Lemma 1.
Proof. By Lemma 1,
is an absorbing set and
is measurable since it is the integral of some random variables. Hence,
is a random set. Next, we show
. For this end, we first prove that
is tempered with any growth rate
. Indeed, letting
, by (
2) and (
19), we have,
which tends to 0 as
. So,
and thus
due to
.
On the other hand, it is easy to see that
is increasing, which means
if
. Given
, we have
where we have used
. Then we have
as required. □
Remark 1. In this lemma, the measurability of is unknown since it is the union of some random sets over an uncountable index set . However, we will prove the measurability of in Theorem 1.
We then show the -pullback backward asymptotic compactness of .
Lemma 6. Under conditions (
4)–(
9)
and (
19),
the continuous co-cycle Φ
associated with problem (
3)
is -pullback backward asymptotically compact in , that is, for every , and , the sequence has a convergent sub-sequence in whenever and . Proof. By Lemmas 1, 3 and 4, the proof is standard, so we omit it. □
Finally, we will establish the existence, uniqueness, and backward compactness of pullback random attractors of in .
Theorem 1. Suppose that conditions (
4)–(
9)
and (
19)
hold. Let Φ
be the continuous co-cycle for problem (
3)
and (resp. ) be the universe of backward tempered sets (resp. tempered sets). Then - (i)
Φ has a -pullback bi-parametric attractor such that is backward compact in ;
- (ii)
Φ has a -pullback random attractor ;
- (iii)
for all , that is, is a -pullback random attractor with the backward compactness.
Proof. (i) By Lemma 5, we find that
is a
-pullback absorbing set for
in
. On the other hand, by Lemma 6 we know that
is
-pullback backward asymptotically compact in
. Then it follows from the abstract result as given by [
43] we show that
has a
-pullback bi-parametric attractor
given by
where the measurability of
is unknown. However, we will prove that
is a random attractor in (ii) and (iii).
Next, we prove
is backward compact. We define
In this case, the
is compact. Indeed, if there is a
, then
We choose three sequences by
and
such that
which implies that
Let
, then there are
and
such that
By the
-backward asymptotic compactness of
given by Lemma 6, there is a
and a index subsequence
, such that
Both (
69) and (
70) imply
as
and thus
is compact as desired.
On the other hand, we have
Hence, is backward compact.
(ii) By Lemma 5, we know that co-cycle
has a
-pullback absorbing set
as given in Lemma 5 and
is a random set. Using a similar process in
Section 3, one can show that
is
-pullback asymptotically compact. Then, by the abstract result in [
43],
has a
-pullback random attractor
given by
(iii) By the definition of
and
in Lemma 5, we have the inclusion
. Therefore, by (
67) and (
71), we see that
On the other hand, since
is a subset of
and
, we have
and thus the
-pullback attractor
attracts
. Since
is invariant, we have
By the compactness of
, we have
, which together with (
72) implies that
for all
. These finish the proof. □
5. Asymptotic Upper Semicontinuity
In this section, we will prove the pullback random attractor
of original non-autonomous stochastic Equation (
3) backward converges to the autonomous random attractor
of the following problem:
with the initial value
The nonlinear functions
satisfy the following conditions: for all
,
where
,
are constants, and
,
.
Let
be a universe of all tempered parametric sets
in
, satisfying that
In fact, similar to
Section 3 and
Section 4, we can also prove that problem (
74) and (
75) under (
76)–(
80) and
generates an autonomous co-cycle
given by
Hence, we can prove that has a -pullback random attractor in .
In order to consider the backward asymptotically autonomous problem, we further assume the functions
f and
h satisfy
and
where the assumption (
85) is weaker than the convergence condition used in [
34,
45] that
.
Lemma 7. Suppose that assumption (
4)–(
9)
, (
19)
, (
76)–(
80)
, and (
83)–(
85)
hold. Then the solution u of (
3)
is backward asymptotically autonomous to the solution of (
74)
. More precisely, whenever as .
Proof. Given
for
. Let
, then minus (
3) by (
74), and take the inner product of result with
, we obtain that
For the nonlinear term, let
, there exists
between
u and
, by (
78), (
83), and (
85), we have
For the external force term, we use the Young inequality, and have
For the last term on the right hand side of (
87), let
, there exists
between
u and
, by (
80) and (
84), we obtain
Thus by (
87)–(
90), we have
where
Applying the Gronwall inequality to (
91) over
, we get
where we have used the fact that the continuity of
on
. By assumption (
85), for a positive integer positive
On the other hand, by using the energy inequality (
29) on
for
, we obtain
where
are positive constants independent of
and
t. The Gronwall inequality implies that for all
,
which is bounded as
due to (
85). Hence,
is bounded as desired. Since
as
, the desired result follows immediately. □
Finally, we summarize the main results as follows.
Theorem 2. Suppose that assumption (
4)–(
9)
, (
19)
, (
76)–(
80)
, and (
83)–(
85)
hold. Then the stochastic non-autonomous system (
3)
has a -pullback backward compact random attractor such that it is backward asymptotically autonomous to the random attractor in . More precisely, Proof. Then it suffices to prove
. Let
. If
, then
, and hence there exists
. This implies
. Suppose that the backward upper semi-continuity (
93) is not true, then there is a
and
such that
for all
. We can take a
such that
Similar to the Lemma 5, we also find that
and the attraction of
, then there exists a
such that
and
Furthermore, by the continuity of
, we see that
By the invariance of random attractor
, we have
Hence, there exists a
such that
If
, then
and thus
By Theorem 1,
is backward compact, then
is a pre-compact set, which means that there is a subsequence
of
such that
Note that
due to the
-invariance of
. Applying the backward asymptotically autonomous (
86) at sample
for
and
as
, we have
if
k is large enough. This together with (
95), we obtain that
which contradicts with (
94). Hence, the backward upper semicontinuity (
93) holds true. □
6. Conclusions
In view of Theorem 1, we can see that fractional nonclassical diffusion equations driven by Wong–Zakai approximations in
with
exists in backward time-dependent uniform compact random attractors when the growth rate of nonlinearities have a subcritical range. Theorem 2 implies that the components of the random attractors of a non-autonomous dynamical system in time can converge to those of the random attractor of the limiting autonomous dynamical system (time-independent) in
. In this paper, system (
3) contains time-dependent nonlinear terms, so it is more complicated to determine the limit process (
10), which differs from the earlier works.
We give an example for and the nonlinearity f and forcing term g, which satisfy all the assumptions in the paper.
Given
and
, let
where
and
,
for some
and
. Then, we can verify that
So we find that
f and
g satisfy all the assumptions with
In this case, Theorems 1 and 2 are clearly true.