In this paper, we introduce a new class of functions satisfying spacial absolutely continuous (see Definition 3.1), denoted by

$L^{2}_{sac}(\mathbb {R};\mathbb {R}^{n})$
Lsac2(R;Rn)⁠, which are translation bounded but not normal (see [S. S. Lu, H. Q. Wu, and C. K. Zhong, “
Attractors for non-autonomous 2D Navier-Stokes equations with normal external forces
,” Discrete Contin. Dyn. Syst. A13(
3
), 701−719 (2005)]
and Definition 3.1) in
$L^{2}_{loc}(\mathbb {R};\mathbb {R}^{n})$
Lloc2(R;Rn)
. Then the asymptotic a priori estimate is applied to some nonlinear reaction-diffusion equations with external forces
$g(x,s)\in L^{2}_{sac}(\mathbb {R};\mathbb {R}^{n})$
g(x,s)Lsac2(R;Rn)
. We obtain the existence of uniform attractor together with its structure in the bi-spaces
$(L^{2}(\mathbb {R}^{n}), L^{2}(\mathbb {R}^{n}))$
(L2(Rn),L2(Rn))
and
$(L^{2}(\mathbb {R}^{n}), L^{p}(\mathbb {R}^{n}))(p>2)$
(L2(Rn),Lp(Rn))(p>2)
without any restriction on the growing order of the nonlinear term.

1.
J. M.
Arrieta
,
J. W.
Cholewa
,
T.
Dlotko
, and
A.
Rodriguez-Bernal
, “
Asymptotic behavior and attractors for reaction-diffusion equations in unbounded domains
,”
Nonlinear Anal.
56
,
515
554
(
2004
).
2.
A. V.
Babin
and
M. I.
Vishik
,
Attractors of Evolution Equations
(
North-Holland
,
Amsterdam
,
1992
).
3.
J. W.
Cholewa
and
T.
Dlotko
, “
Bi-spaces global attractors in abstract parabolic equations
,”
in: Evol. Equations, in: Banach Cent Publ. Polish Acad. Sci., Warsaw
60
,
13
26
(
2003
).
4.
V. V.
Chepyzhov
and
M. I.
Vishik
, “
Attractors of non-autonomous dynamical systems and their dimension
,”
J. Math. Pures Appl.
73
,
279
333
(
1994
).
5.
V. V.
Chepyzhov
and
M. I.
Vishik
, “
Non-autonomous evolutionary equations with translation compact symbols and their attractors
,”
Acad. Sci., Paris, C. R.
321
,
153
158
(
1995
).
6.
V. V.
Chepyzhov
and
M. I.
Vishik
, “
Trajectory attractors for 2D Navier-Stokes systems and some generalizations
,”
Topol. Methods Nonlinear Anal.
8
,
217
243
(
1996
).
7.
V. V.
Chepyzhov
and
M. I.
Vishik
, “
Trajectory attractors for reaction-diffusion systems
,”
Topol. Methods Nonlinear Anal.
7
,
49
76
(
1996
).
8.
V. V.
Chepyzhov
and
M. I.
Vishik
,
Attractors for Equations of Mathematical Physics
, American Mathematical Society Colloquium Publications Vol. 49 (
American Mathematical Society
,
Providence, RI
,
2002
).
9.
G. X.
Chen
and
C. K.
Zhong
, “
Uniform attractors for non-autonomous p-Laplacian equations
,”
Nonlinear Anal.
68
,
3349
3363
(
2008
).
10.
M.
Efendiev
and
S.
Zelik
, “
The attractor for a nonlinear reaction-diffusion system in an unbounded domain
,”
Commun. Pure Appl. Math.
54
,
625
688
(
2001
).
11.
A.
Haraux
, “
Recent results on semilinear wave equations with dissipation
,”
Pitman Res. Notes Math
141
,
150
157
(
1986
).
12.
A.
Haraux
,
Systémes Dynamiques Dissipatifs et Applications
(
Masson
,
Paris
,
1991
).
13.
A.
Haraux
, “
Attractors of asymptoticlly compact process and applications to nonlinear partial differential equations
,”
Commun. Partial Differ. Equ.
13
,
1383
1414
(
1988
).
14.
J. K.
Hale
,
Asymptotic Behavior of Dissipative Systems
,
Mathematical Surveys and Monographs Vol. 25
(
American Mathematical Society
,
Providence, RI
,
1988
).
15.
S. S.
Lu
,
H. Q.
Wu
, and
C. K.
Zhong
, “
Attractors for non-autonomous 2D Navier-Stokes equations with normal external forces
,”
Discrete Contin. Dyn. Syst. A
13
(
3
),
701
719
(
2005
).
16.
Q. F.
Ma
,
S. H.
Wang
, and
C. K.
Zhong
, “
Necessary and sufficient conditions for the existence of global attractors for semigroups and applications
,”
Indiana Univ. Math. J.
51
,
1541
1559
(
2002
).
17.
S.
Ma
,
C. K.
Zhong
, and
H. T.
Song
, “
Attractors for non-autonomous 2D Navier-Stokes equations with less regular symbols
,”
Nonlinear Anal.
71
,
4215
4222
(
2009
).
18.
V.
Pata
and
C.
Santina
, “
Longtime behavior of semilinear reaction-diffusion equations on the whole space
,”
Rend. Semin. Matermatico Univ. di Padova
105
,
233
251
(
2001
).
19.
J. C.
Robinson
,
Infinite-Dimensional Dynamical Systems an Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors
(
Cambridge University Press
,
Cambridge
,
2001
).
20.
C. Y.
Sun
,
D. M.
Cao
, and
J. Q.
Duan
, “
Non-autonomous wave dynamics with memory–asymptotic regularity and uniform attractor
,”
Discrete Contin. Dyn. Syst
9
(
3
),
743
761
(
2008
).
21.
H. T.
Song
,
S.
Ma
, and
C. K.
Zhong
, “
Attractors of nonautonomous reaction-diffusion equations
,”
Nonlinearity
22
,
667
681
(
2009
).
22.
H. T.
Song
and
C. K.
Zhong
, “
Attractors of nonautonomous reaction-diffusion equations in Lp
,”
Nonlinear Anal.
68
,
1890
1897
(
2008
).
23.
C. Y.
Sun
and
C. K.
Zhong
, “
Attractors for the semilinear reaction-diffusion equation with distribution derivatives in unbounded domains
,”
Nonlinear Anal.
63
,
49
65
(
2005
).
24.
R.
Temam
,
Infinite Dimensional System in Mechanics and Physics
(
Springer
,
New York
,
1997
).
25.
B. X.
Wang
, “
Attractors for reaction-diffusion equations in unbounded domains
,”
Physica D
128
,
41
52
(
1999
).
26.
X. J.
Yan
and
C. K.
Zhong
, “
Lp-uniform attractor for nonautonomous reaction-diffusion equations in unbounded domains
,”
J. Math. Phy.
49
,
1
17
(
2008
).
27.
C. K.
Zhong
,
M. H.
Yang
, and
C. Y.
Sun
, “
The existence of global attractors for the norm-to-weak continuous semigroup and its application to the nonlinear reaction-diffusion equations
,”
J. Differ. Equations
223
,
367
399
(
2006
).
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