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Time fractional and space nonlocal stochastic boussinesq equations driven by gaussian white noise
Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China |
This paper is concerned with the robustness of a pullback attractor as the time tends to infinity. A pullback attractor is called forward (resp. backward) compact if the union over the future (resp. the past) is pre-compact. We prove that the forward (resp. backward) compactness is a necessary and sufficient condition such that a pullback attractor is upper semi-continuous to a compact set at positive (resp. negative) infinity, and also obtain the minimal limit-set. We further prove the lower semi-continuity of the pullback attractor and get the maximal limit-set at infinity. Some criteria for such robustness are established when the evolution process is forward or backward omega-limit compact. Those theoretical criteria are applied to prove semi-uniform compactness and robustness at infinity in pullback dynamics for a Ginzburg-Landau equation with variable coefficients and a forward or backward tempered nonlinearity.
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Asymptotic behaviour of the nonautonomous SIR equations with diffusion, Commun. Pure Appl. Anal., 13 (2014), 157-173.
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T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero,
Existence of pullback attractors for pullback asymptotically compact processes, Nonlinear Anal., 72 (2010), 1967-1976.
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T. Caraballo, G. Lukaszewicz and J. Real,
Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.
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A. N. Carvalho, J. A. Langa and J. C. Robinson,
Attractors for Infinite-dimensional Nonautonomous Dynamical Systems, Appl. Math. Sciences, Springer, New York, 2013. |
[7] |
M. Carvalho and P. Varandas,
(Semi)continuity of the entropy of Sinai probability measures for partially hyperbolic diffeomorphisms, J. Math. Anal. Appl., 434 (2016), 1123-1137.
doi: 10.1016/j.jmaa.2015.09.042. |
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V. V. Chepyzhov and M. I. Vishik,
Attractors of nonautonomous dynamical systems and their dimensions, J. Math. Pures Appl., 73 (1994), 279-333.
|
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V. V. Chepyzhov and M. I. Vishik,
Attractors for Equations of Mathematical Physics, American Mathematical Society, volume 49, Providence, Rhode Island, 2002.
doi: 10.1090/coll/049. |
[10] |
H. Cui, J. A. Langa and Y. Li,
Regularity and structure of pullback attractors for reaction-diffusion type systems without uniqueness, Nonlinear Anal., 140 (2016), 208-235.
doi: 10.1016/j.na.2016.03.012. |
[11] |
H. Cui and Y. Li,
Existence and upper semicontinuity of random attractors for stochastic degenerate parabolic equations with multiplicative noises, Appl. Math. Comput., 271 (2015), 777-789.
doi: 10.1016/j.amc.2015.09.031. |
[12] |
H. Cui, Y. Li and J. Yin,
Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles, Nonlinear Anal., 128 (2015), 304-324.
doi: 10.1016/j.na.2015.08.009. |
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H. Cui, Y. Li and J. Yin,
Long time behavior of stochastic MHD equations perturbed by multiplicative noises, J. Appl. Anal. Comput., 6 (2016), 1081-1104.
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P. E. Kloeden and J. Simsen,
Pullback attractors for non-autonomous evolution equations with spatially variable exponents, Commun. Pure Appl. Anal., 13 (2014), 2543-2557.
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P. E. Kloeden,
Upper semi continuity of attractors of delay differential equations in the delay, Bull. Austra. Math. Soc., 73 (2006), 299-306.
doi: 10.1017/S0004972700038880. |
[16] |
P. E. Kloeden and J. A. Langa,
Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond, 463 (2007), 163-181.
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Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differ. Equ., 258 (2015), 504-534.
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[20] |
Y. Li and B. Guo,
Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations, J. Differ. Equ., 245 (2008), 1775-1800.
doi: 10.1016/j.jde.2008.06.031. |
[21] |
Y. Li, R. Wang and J. Yin,
Backward compact attractors for non-autonomous Benjamin-Bona-Mahony equations on unbounded channels, Discrete Contin. Dyn. Syst. B, 22 (2017), 2569-2586.
doi: 10.3934/dcdsb.2017092. |
[22] |
Y. Li and J. Yin,
A modified proof of pullback attractors in a Sobolev space for stochastic Fitzhugh-Nagumo equations, Discrete Contin. Dyn. Syst. B, 21 (2016), 1203-1223.
doi: 10.3934/dcdsb.2016.21.1203. |
[23] |
Y. Li and J. Yin,
Existence, regularity and approximation of global attractors for weakly dissipative p-Laplace equations, Discrete Contin. Dyn.Syst. S, 9 (2016), 1939-1957.
doi: 10.3934/dcdss.2016079. |
[24] |
L. Liu and X. Fu,
Existence and upper semicontinuity of (L-2, L-q) pullback attractors for a stochastic p-Laplacian equation, Commun. Pure Appl. Anal., 16 (2017), 443-473.
doi: 10.3934/cpaa.2017023. |
[25] |
G. Lukaszewicz,
On pullback attractors in $H^1_0$ for nonautonomous reaction-diffusion equations, Internat. J. Bifur. Chaos, 20 (2010), 2637-2644.
doi: 10.1142/S0218127410027258. |
[26] |
S. H. Park and J. Y. Park,
Pullback attractor for a non-autonomous modified Swift-Hohenberg equation, Comput. Math. Appl., 67 (2014), 542-548.
doi: 10.1016/j.camwa.2013.11.011. |
[27] |
J. C. Robinson,
Stability of random attractors under perturbation and approximation, J. Differ. Equ., 186 (2002), 652-669.
doi: 10.1016/S0022-0396(02)00038-4. |
[28] |
E. O. Roxin,
Stability in general control systems, J. Differ. Equ., 1 (1965), 115-150.
doi: 10.1016/0022-0396(65)90015-X. |
[29] |
J. Simsen, M. J. D. Nascimento and M. S. Simsen,
Existence and upper semicontinuity of pullback attractors for non-autonomous p-Laplacian parabolic problems, J. Math. Anal. Appl., 413 (2014), 685-699.
doi: 10.1016/j.jmaa.2013.12.019. |
[30] |
C. Sun, D. Cao and J. Duan,
Uniform attractors for nonautonomous wave equations with Nonlinear damping, SIAM J. Appl. Dyn. Syst., 6 (2007), 293-318.
doi: 10.1137/060663805. |
[31] |
R. Temam,
Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. |
[32] |
D. Vorotnikov,
Asymptotic behavior of the non-autonomous 3D Navier-Stokes problem with coercive force, J. Differ. Equ., 251 (2011), 2209-2225.
doi: 10.1016/j.jde.2011.07.008. |
[33] |
B. Wang,
Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differ. Equ., 253 (2012), 1544-1583.
doi: 10.1016/j.jde.2012.05.015. |
[34] |
B. Wang,
Upper semicontinuity of random attractors for non-compact random dynamical systems, Electric J. Differ. Equ., 139 (2009), 1-18.
|
[35] |
B. Wang,
Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Disrete Continu. Dyn. Syst. B, 34 (2014), 269-300.
|
[36] |
G. Wang, B. Guo and Y. Li,
The asymptotic behavior of the stochastic Ginzburg-Landau equation with additive noise, Appl. Math. Comput., 198 (2008), 849-857.
doi: 10.1016/j.amc.2007.09.029. |
[37] |
Y. Wang,
On the upper semicontinuity of pullback attractors for multi-valued noncompact random dynamical systems, Discrete Contin. Dyn. Syst. B, 21 (2016), 3669-3708.
doi: 10.3934/dcdsb.2016116. |
[38] |
Y. Wang and S. Zhou,
Kernel sections and uniform attractors of multi-valued semiprocesses, J. Differ. Equ., 232 (2007), 573-622.
doi: 10.1016/j.jde.2006.07.005. |
[39] |
J. Yin, A. Gu and Y. Li,
Backwards compact attractors for non-autonomous damped 3D Navier-Stokes equations, Dynamics of PDE, 14 (2017), 201-218.
doi: 10.4310/DPDE.2017.v14.n2.a4. |
[40] |
J. Yin, Y. Li and H. Cui,
Box-counting dimensions and upper semicontinuities of bi-spatial attractors for stochastic degenerate parabolic equations on an unbounded domain, J. Math. Anal. Appl., 450 (2017), 1180-1207.
doi: 10.1016/j.jmaa.2017.01.064. |
[41] |
J. Yin, Y. Li and A. Gu,
Backwards compact attractors and periodic attractors for non-autonomous damped wave equations on an unbounded domain, Comput. Math. Appl., 74 (2017), 744-758.
doi: 10.1016/j.camwa.2017.05.015. |
[42] |
S. Zhou and Y. Bai,
Random attractor and upper semi-continuity for Zakharov lattice system with multiplicative white noise, J. Differ. Equ. Appl., 20 (2014), 312-338.
doi: 10.1080/10236198.2013.845663. |
[43] |
V. Zvyagin and S. Kondratyev,
Pullback attractors of the Jeffreys-Oldroyd equations, J. Differ. Equ., 260 (2016), 5026-5042.
doi: 10.1016/j.jde.2015.11.038. |
show all references
References:
[1] |
M. Anguiano and P. E. Kloeden,
Asymptotic behaviour of the nonautonomous SIR equations with diffusion, Commun. Pure Appl. Anal., 13 (2014), 157-173.
|
[2] |
J. M. Arrieta and A. N. Carvalho,
Spectral convergence and nonlinear dynamics of reaction-diffusion equations under perturbations of the domain, J. Differ. Equ., 199 (2004), 143-178.
doi: 10.1016/j.jde.2003.09.004. |
[3] |
P. W. Bates, K. Lu and B. Wang,
Attractors for lattice dynamical systems, Intern. J. Bifur. Chaos, 11 (2001), 143-153.
doi: 10.1142/S0218127401002031. |
[4] |
T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero,
Existence of pullback attractors for pullback asymptotically compact processes, Nonlinear Anal., 72 (2010), 1967-1976.
doi: 10.1016/j.na.2009.09.037. |
[5] |
T. Caraballo, G. Lukaszewicz and J. Real,
Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.
doi: 10.1016/j.na.2005.03.111. |
[6] |
A. N. Carvalho, J. A. Langa and J. C. Robinson,
Attractors for Infinite-dimensional Nonautonomous Dynamical Systems, Appl. Math. Sciences, Springer, New York, 2013. |
[7] |
M. Carvalho and P. Varandas,
(Semi)continuity of the entropy of Sinai probability measures for partially hyperbolic diffeomorphisms, J. Math. Anal. Appl., 434 (2016), 1123-1137.
doi: 10.1016/j.jmaa.2015.09.042. |
[8] |
V. V. Chepyzhov and M. I. Vishik,
Attractors of nonautonomous dynamical systems and their dimensions, J. Math. Pures Appl., 73 (1994), 279-333.
|
[9] |
V. V. Chepyzhov and M. I. Vishik,
Attractors for Equations of Mathematical Physics, American Mathematical Society, volume 49, Providence, Rhode Island, 2002.
doi: 10.1090/coll/049. |
[10] |
H. Cui, J. A. Langa and Y. Li,
Regularity and structure of pullback attractors for reaction-diffusion type systems without uniqueness, Nonlinear Anal., 140 (2016), 208-235.
doi: 10.1016/j.na.2016.03.012. |
[11] |
H. Cui and Y. Li,
Existence and upper semicontinuity of random attractors for stochastic degenerate parabolic equations with multiplicative noises, Appl. Math. Comput., 271 (2015), 777-789.
doi: 10.1016/j.amc.2015.09.031. |
[12] |
H. Cui, Y. Li and J. Yin,
Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles, Nonlinear Anal., 128 (2015), 304-324.
doi: 10.1016/j.na.2015.08.009. |
[13] |
H. Cui, Y. Li and J. Yin,
Long time behavior of stochastic MHD equations perturbed by multiplicative noises, J. Appl. Anal. Comput., 6 (2016), 1081-1104.
|
[14] |
P. E. Kloeden and J. Simsen,
Pullback attractors for non-autonomous evolution equations with spatially variable exponents, Commun. Pure Appl. Anal., 13 (2014), 2543-2557.
doi: 10.3934/cpaa.2014.13.2543. |
[15] |
P. E. Kloeden,
Upper semi continuity of attractors of delay differential equations in the delay, Bull. Austra. Math. Soc., 73 (2006), 299-306.
doi: 10.1017/S0004972700038880. |
[16] |
P. E. Kloeden and J. A. Langa,
Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond, 463 (2007), 163-181.
doi: 10.1098/rspa.2006.1753. |
[17] |
P. E. Kloeden and M. Rasmussen,
Nonautonomous Dynamical Systems, 176, American Mathematical Society, Providence, 2011. |
[18] |
Y. Li, H. Cui and J. Li,
Upper semi-continuity and regularity of random attractors on p-times integrable spaces and applications, Nonlinear Anal., 109 (2014), 33-44.
doi: 10.1016/j.na.2014.06.013. |
[19] |
Y. Li, A. Gu and J. Li,
Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differ. Equ., 258 (2015), 504-534.
doi: 10.1016/j.jde.2014.09.021. |
[20] |
Y. Li and B. Guo,
Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations, J. Differ. Equ., 245 (2008), 1775-1800.
doi: 10.1016/j.jde.2008.06.031. |
[21] |
Y. Li, R. Wang and J. Yin,
Backward compact attractors for non-autonomous Benjamin-Bona-Mahony equations on unbounded channels, Discrete Contin. Dyn. Syst. B, 22 (2017), 2569-2586.
doi: 10.3934/dcdsb.2017092. |
[22] |
Y. Li and J. Yin,
A modified proof of pullback attractors in a Sobolev space for stochastic Fitzhugh-Nagumo equations, Discrete Contin. Dyn. Syst. B, 21 (2016), 1203-1223.
doi: 10.3934/dcdsb.2016.21.1203. |
[23] |
Y. Li and J. Yin,
Existence, regularity and approximation of global attractors for weakly dissipative p-Laplace equations, Discrete Contin. Dyn.Syst. S, 9 (2016), 1939-1957.
doi: 10.3934/dcdss.2016079. |
[24] |
L. Liu and X. Fu,
Existence and upper semicontinuity of (L-2, L-q) pullback attractors for a stochastic p-Laplacian equation, Commun. Pure Appl. Anal., 16 (2017), 443-473.
doi: 10.3934/cpaa.2017023. |
[25] |
G. Lukaszewicz,
On pullback attractors in $H^1_0$ for nonautonomous reaction-diffusion equations, Internat. J. Bifur. Chaos, 20 (2010), 2637-2644.
doi: 10.1142/S0218127410027258. |
[26] |
S. H. Park and J. Y. Park,
Pullback attractor for a non-autonomous modified Swift-Hohenberg equation, Comput. Math. Appl., 67 (2014), 542-548.
doi: 10.1016/j.camwa.2013.11.011. |
[27] |
J. C. Robinson,
Stability of random attractors under perturbation and approximation, J. Differ. Equ., 186 (2002), 652-669.
doi: 10.1016/S0022-0396(02)00038-4. |
[28] |
E. O. Roxin,
Stability in general control systems, J. Differ. Equ., 1 (1965), 115-150.
doi: 10.1016/0022-0396(65)90015-X. |
[29] |
J. Simsen, M. J. D. Nascimento and M. S. Simsen,
Existence and upper semicontinuity of pullback attractors for non-autonomous p-Laplacian parabolic problems, J. Math. Anal. Appl., 413 (2014), 685-699.
doi: 10.1016/j.jmaa.2013.12.019. |
[30] |
C. Sun, D. Cao and J. Duan,
Uniform attractors for nonautonomous wave equations with Nonlinear damping, SIAM J. Appl. Dyn. Syst., 6 (2007), 293-318.
doi: 10.1137/060663805. |
[31] |
R. Temam,
Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. |
[32] |
D. Vorotnikov,
Asymptotic behavior of the non-autonomous 3D Navier-Stokes problem with coercive force, J. Differ. Equ., 251 (2011), 2209-2225.
doi: 10.1016/j.jde.2011.07.008. |
[33] |
B. Wang,
Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differ. Equ., 253 (2012), 1544-1583.
doi: 10.1016/j.jde.2012.05.015. |
[34] |
B. Wang,
Upper semicontinuity of random attractors for non-compact random dynamical systems, Electric J. Differ. Equ., 139 (2009), 1-18.
|
[35] |
B. Wang,
Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Disrete Continu. Dyn. Syst. B, 34 (2014), 269-300.
|
[36] |
G. Wang, B. Guo and Y. Li,
The asymptotic behavior of the stochastic Ginzburg-Landau equation with additive noise, Appl. Math. Comput., 198 (2008), 849-857.
doi: 10.1016/j.amc.2007.09.029. |
[37] |
Y. Wang,
On the upper semicontinuity of pullback attractors for multi-valued noncompact random dynamical systems, Discrete Contin. Dyn. Syst. B, 21 (2016), 3669-3708.
doi: 10.3934/dcdsb.2016116. |
[38] |
Y. Wang and S. Zhou,
Kernel sections and uniform attractors of multi-valued semiprocesses, J. Differ. Equ., 232 (2007), 573-622.
doi: 10.1016/j.jde.2006.07.005. |
[39] |
J. Yin, A. Gu and Y. Li,
Backwards compact attractors for non-autonomous damped 3D Navier-Stokes equations, Dynamics of PDE, 14 (2017), 201-218.
doi: 10.4310/DPDE.2017.v14.n2.a4. |
[40] |
J. Yin, Y. Li and H. Cui,
Box-counting dimensions and upper semicontinuities of bi-spatial attractors for stochastic degenerate parabolic equations on an unbounded domain, J. Math. Anal. Appl., 450 (2017), 1180-1207.
doi: 10.1016/j.jmaa.2017.01.064. |
[41] |
J. Yin, Y. Li and A. Gu,
Backwards compact attractors and periodic attractors for non-autonomous damped wave equations on an unbounded domain, Comput. Math. Appl., 74 (2017), 744-758.
doi: 10.1016/j.camwa.2017.05.015. |
[42] |
S. Zhou and Y. Bai,
Random attractor and upper semi-continuity for Zakharov lattice system with multiplicative white noise, J. Differ. Equ. Appl., 20 (2014), 312-338.
doi: 10.1080/10236198.2013.845663. |
[43] |
V. Zvyagin and S. Kondratyev,
Pullback attractors of the Jeffreys-Oldroyd equations, J. Differ. Equ., 260 (2016), 5026-5042.
doi: 10.1016/j.jde.2015.11.038. |
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