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June  2018, 23(4): 1535-1557. doi: 10.3934/dcdsb.2018058

Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE

Yangrong Li , , Lianbing She and  Jinyan Yin

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author: Yangrong Li, Email: liyr@swu.edu.cn

Received  April 2017 Revised  August 2017 Published  June 2018 Early access  February 2018

Fund Project: This work is supported by National Natural Science Foundation of China grant 11571283 and Natural Science Foundation of Guizhou Province (KY[2016]103).

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.

Keywords: Evolution process, pullback attractor, longtime robustness, forward compactness, backward compactness, Ginzburg-Landau equation.
Mathematics Subject Classification: Primary: 37L55, 37L30; Secondary: 35B41.
Citation: Yangrong Li, Lianbing She, Jinyan Yin. Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1535-1557. doi: 10.3934/dcdsb.2018058
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.

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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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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.

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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