The limiting dynamics in Lp(Rn) and Hs(Rn) for any p > 2, s ∈ (0, 1) are discussed for a class of fractional stochastic reaction-diffusion equations driven by a Wong–Zakai approximation process on Rn. Firstly, we prove some priori estimates and the continuity of the difference of the solution operator from L2(Rn) to Lp(Rn) and Hs(Rn) near the initial time, respectively. Finally, we show the upper semi-continuity of attractors of the approximate random system in Lp(Rn) and Hs(Rn) as the size of approximation approaches zero.

1.
L.
Arnold
,
Random Dynamical Systems
(
Springer-Verlag
,
Berlin
,
1998
).
2.
P. W.
Bates
,
K.
Lu
, and
B.
Wang
, “
Random attractors for stochastic reaction-diffusion equations on unbounded domains
,”
J. Differ. Equ.
246
,
845
869
(
2009
).
3.
T.
Caraballo
,
I. D.
Chueshov
, and
P. E.
Kloeden
, “
Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain
,”
SIAM J. Math. Anal.
38
,
1489
1507
(
2007
).
4.
T.
Caraballo
,
J. A.
Langa
, and
J. C.
Robinson
, “
Upper semicontinuity of attractors for small random perturbations of dynamical systems
,”
Commun. Partial Differ. Equ.
23
,
1557
1581
(
1998
).
5.
T.
Caraballo
,
J.
A Langa
, and
J.
C Robinson
, “
Stability and random attractors for a reaction-diffusion equation with multiplicative noise
,”
Discrete Contin. Dyn. Syst. A
6
,
875
892
(
2000
).
6.
T.
Caraballo
,
M.
J Garrido-Atienza
,
B.
Schmalfuß
, and
J.
Valero
, “
Non-autonomous and random attractors for delay random semilinear equations without uniqueness
,”
Discrete Contin. Dyn. Syst. A
21
,
415
443
(
2008
).
7.
H.
Cui
,
A. N.
Carvalho
,
A. C.
Cunha
, and
J. A.
Langa
, “
Smoothing and finite-dimensionality of uniform attractors in Banach spaces
,”
J. Differ. Equ.
285
,
383
428
(
2021
).
8.
H.
Cui
,
A. C.
Cunha
, and
J. A.
Langa
, “
Finite-dimensionality of tempered random uniform attractors
,”
J. Nonlinear Sci.
32
,
13
(
2022
).
9.
H.
Cui
,
P. E.
Kloeden
, and
F.
Wu
, “
Pathwise upper semi-continuity of random pullback attractors along the time axis
,”
Physica D
374–375
,
21
34
(
2018
).
10.
B.
Wang
, “
Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems
,”
J. Differ. Equ.
253
,
1544
1583
(
2012
).
11.
B.
Wang
, “
Attractors for reaction-diffusion equations in unbounded domains
,”
Physica D
128
,
41
52
(
1999
).
12.
P. Y.
Chen
,
R. H.
Wang
, and
X. P.
Zhang
, “
Long-time dynamics of fractional nonclassical diffusion equations with nonlinear colored noise and delay on unbounded domains
,”
Bull. Sci. Math.
173
,
103071
(
2021
).
13.
H.
Gao
,
M. J.
Garrido-Atienza
, and
B.
Schmalfuß
, “
Random attractors for stochastic evolution equations driven by fractional Brownian motion
,”
SIAM J. Math. Anal.
46
,
2281
2309
(
2014
).
14.
M.
Garrido-Atienza
,
K.
Lu
, and
B.
Schmalfuß
, “
Random dynamical systems for stochastic evolution equations driven by multiplicative fractional Brownian noise with Hurst parameters H(13,12]
,”
SIAM J. Appl. Dyn. Syst.
15
,
625
654
(
2016
).
15.
A.
Gu
,
D.
Li
,
B.
Wang
, and
H.
Yang
, “
Regularity of random attractors for fractional stochastic reaction-diffusion equations on Rn
,”
J. Differ. Equ.
264
,
7094
7137
(
2018
).
16.
H.
Lu
,
P. W.
Bates
,
S.
Lu
, and
M.
Zhang
, “
Dynamics of the 3-D fractional complex Ginzburg–Landau equation
,”
J. Differ. Equ.
259
,
5276
5301
(
2015
).
17.
H.
Lu
,
P. W.
Bates
,
J.
Xin
, and
M.
Zhang
, “
Asymptotic behavior of stochastic fractional power dissipative equations on Rn
,”
Nonlinear Anal.
128
,
176
198
(
2015
).
18.
H.
Lu
,
P. W.
Bates
,
S.
Lu
, and
M.
Zhang
, “
Dynamics of the 3D fractional Ginzburg-Landau equation with multiplicative noise on an unbounded domain
,”
Commun. Math. Sci.
14
,
273
295
(
2016
).
19.
H.
Lu
,
J.
Qi
,
B.
Wang
, and
M.
Zhang
, “
Random attractors for non-autonomous fractional stochastic parabolic equations on unbounded domains
,”
Discrete Contin. Dyn. Syst. A
39
,
683
706
(
2019
).
20.
Y.
Sun
and
H.
Gao
, “
Wong–Zakai approximations and attractors for fractional stochastic reaction–diffusion equations on unbounded domains
,”
J. Appl. Anal. Comput.
10
,
2338
2361
(
2020
).
21.
B.
Wang
, “
Asymptotic behavior of non-autonomous fractional stochastic reaction-diffusion equations
,”
Nonlinear Anal.
158
,
60
82
(
2017
).
22.
B.
Wang
, “
Dynamics of fractional stochastic reaction-diffusion equations on unbounded domains driven by nonlinear noise
,”
J. Differ. Equ.
268
,
1
59
(
2019
).
23.
R.
Wang
,
Y.
Li
, and
B.
Wang
, “
Random dynamics of fractional nonclassical diffusion equations driven by colored noise
,”
Discrete Contin. Dyn. Syst. A
39
,
4091
4126
(
2019
).
24.
R.
Wang
,
L.
Shi
, and
B.
Wang
, “
Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear colored noise on RN
,”
Nonlinearity
32
,
4524
4556
(
2019
).
25.
R.
Wang
,
Y.
Li
, and
B.
Wang
, “
Bi-spatial pullback attractors of fractional nonclassical diffusion equations on unbounded domains with (p, q)-growth nonlinearities
,”
Appl. Math. Optim.
84
,
425
461
(
2021
).
26.
K.
Lu
and
B.
Wang
, “
Wong–Zakai approximations and long term behavior of stochastic partial differential equations
,”
J. Dyn. Differ. Equ.
31
,
1341
1371
(
2019
).
27.
X.
Wang
,
K.
Lu
, and
B.
Wang
, “
Wong–Zakai approximations and attractors for stochastic reaction–diffusion equations on unbounded domains
,”
J. Differ. Equ.
264
,
378
424
(
2018
).
28.
D.
Cao
,
C.
Sun
, and
M.
Yang
, “
Dynamics for a stochastic reaction–diffusion equation with additive noise
,”
J. Differ. Equ.
259
,
838
872
(
2015
).
29.
F.
Li
,
Y.
Li
, and
R.
Wang
, “
Limiting dynamics for stochastic reaction-diffusion equations on the Sobolev space with thin domains
,”
Comput. Math. Appl.
79
,
457
475
(
2020
).
30.
W.
Zhao
, “
Random dynamics of stochistic p-Laplacian equations on RN with an unbounded additive noise
,”
J. Math. Anal. Appl.
455
,
1178
1203
(
2017
).
31.
W.
Zhao
, “
Long-time random dynamics of stochastic parabolic p-Laplacian equations on RN
,”
Nonlinear Anal.
152
,
196
219
(
2017
).
32.
K.
Zhu
and
F.
Zhou
, “
Continuity and pullback attractors for a non-autonomous reaction-diffusion equation in RN
,”
Comput. Math. Appl.
71
,
2089
2105
(
2016
).
33.
W.
Eugene
and
Z.
Moshe
, “
On the relation between ordinary and stochastic differential equations
,”
Int. J. Eng. Sci.
3
,
213
229
(
1965
).
34.
E.
Wong
and
M.
Zakai
, “
On the convergence of ordinary integrals to stochastic integrals
,”
Ann. Math. Stat.
36
,
1560
1564
(
1965
).
35.
Z.
Brzezniak
and
F.
Flandoli
, “
Almost sure approximations of Wong-Zakai type for stochastic partial differential equations
,”
Stochastic Process. Appl.
55
,
329
358
(
1995
).
36.
I.
Gyongy
and
A.
Shmatkov
, “
Rate of convergence of Wong-Zakai approximations for stochastic partial differential equations
,”
Appl. Math. Optim.
54
,
341
(
2006
).
37.
M.
Hairer
and
E.
Pardoux
, “
A Wong-Zakai theorem for stochastic PDEs
,”
J. Math. Soc. Jpn.
67
,
1551
1604
(
2015
).
38.
G.
Tessitore
and
J.
Zabczyk
, “
Wong-Zakai approximations of stochastic evolution equations
,”
J. Evol. Equ.
6
,
621
655
(
2006
).
39.
A.
Gu
, “
Asymptotic behavior of random lattice dynamical systems and their Wong-Zakai approximations
,”
Discrete Contin. Dyn. Syst. Ser. B
24
,
5737
5767
(
2019
).
40.
F.
Miao
,
H.
Liu
, and
J.
Xin
, “
Wong-Zakai approximations and attractors for stochastic degenerate parabolic equations on unbounded domains
,”
Stochastics Dyn.
21
,
2150033
(
2021
).
41.
J.
Shen
and
K.
Lu
, “
Wong-Zakai approximations and center manifolds of stochastic differential equations
,”
J. Differ. Equ.
263
,
4929
4977
(
2017
).
42.
X.
Wang
,
D.
Li
, and
J.
Shen
, “
Wong-Zakai approximations and attractors for stochastic wave equations driven by additive noise
,”
Discrete Contin. Dyn. Syst. B
26
,
2829
2855
(
2021
).
43.
X.
Wang
,
J.
Shen
,
K.
Lu
, and
B.
Wang
, “
Wong-Zakai approximations and random attractors for non-autonomous stochastic lattice systems
,”
J. Differ. Equ.
280
,
477
516
(
2021
).
44.
W.
Zhao
,
Wong–Zakai approximations of the non-autonomous stochastic FitzHugh–Nagumo system on RN in higher regular spaces RN in higher regular spaces
,
J. Math. Phys.
62
,
081501
(
2021
).
45.
W.
Zhao
,
Y.
Zhang
, and
S.
Chen
, “
Higher-order Wong–Zakai approximations of stochastic reaction–diffusion equations on RN
,”
Physica D
401
,
132147
(
2020
).
46.
E.
Di Nezza
,
G.
Palatucci
, and
E.
Valdinoci
, “
Hitchhiker’s guide to the fractional Sobolev spaces
,”
Bull. Sci. Math.
136
,
521
573
(
2012
).
47.
W.
Zhao
, “
Random dynamics of non-autonomous semi-linear degenerate parabolic equations on RN driven by an unbounded additive noise
,”
Discrete Contin. Dyn. Syst. B
23
,
2499
2526
(
2018
).
You do not currently have access to this content.