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

Yangrong Li; Lianbing She; Jinyan Yin

doi:10.3934/dcdsb.2018058

Previous Article
Dynamics of a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity
DCDS-B Home
This Issue
Next Article
Time fractional and space nonlocal stochastic boussinesq equations driven by gaussian white noise

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 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 & 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[8]	V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimensions, J. Math. Pures Appl., 73 (1994), 279-333. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[17]	P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, 176, American Mathematical Society, Providence, 2011. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[28]	E. O. Roxin, Stability in general control systems, J. Differ. Equ., 1 (1965), 115-150. doi: 10.1016/0022-0396(65)90015-X. Google Scholar
[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. Google Scholar
[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. Google Scholar
[31]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. Google Scholar
[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. Google Scholar
[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. Google Scholar
[34]	B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electric J. Differ. Equ., 139 (2009), 1-18. Google Scholar
[35]	B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Disrete Continu. Dyn. Syst. B, 34 (2014), 269-300. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[8]	V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimensions, J. Math. Pures Appl., 73 (1994), 279-333. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[17]	P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, 176, American Mathematical Society, Providence, 2011. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[28]	E. O. Roxin, Stability in general control systems, J. Differ. Equ., 1 (1965), 115-150. doi: 10.1016/0022-0396(65)90015-X. Google Scholar
[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. Google Scholar
[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. Google Scholar
[31]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. Google Scholar
[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. Google Scholar
[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. Google Scholar
[34]	B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electric J. Differ. Equ., 139 (2009), 1-18. Google Scholar
[35]	B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Disrete Continu. Dyn. Syst. B, 34 (2014), 269-300. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

[1]	Bo You, Yanren Hou, Fang Li, Jinping Jiang. Pullback attractors for the non-autonomous quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1801-1814. doi: 10.3934/dcdsb.2014.19.1801
[2]	Xiaobin Sun, Jianliang Zhai. Averaging principle for stochastic real Ginzburg-Landau equation driven by $ \alpha $-stable process. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1291-1319. doi: 10.3934/cpaa.2020063
[3]	Lu Zhang, Aihong Zou, Tao Yan, Ji Shu. Weak pullback attractors for stochastic Ginzburg-Landau equations in Bochner spaces. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021063
[4]	N. I. Karachalios, Hector E. Nistazakis, Athanasios N. Yannacopoulos. Asymptotic behavior of solutions of complex discrete evolution equations: The discrete Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems, 2007, 19 (4) : 711-736. doi: 10.3934/dcds.2007.19.711
[5]	N. Maaroufi. Topological entropy by unit length for the Ginzburg-Landau equation on the line. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 647-662. doi: 10.3934/dcds.2014.34.647
[6]	Hans G. Kaper, Peter Takáč. Bifurcating vortex solutions of the complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems, 1999, 5 (4) : 871-880. doi: 10.3934/dcds.1999.5.871
[7]	Jingna Li, Li Xia. The Fractional Ginzburg-Landau equation with distributional initial data. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2173-2187. doi: 10.3934/cpaa.2013.12.2173
[8]	Satoshi Kosugi, Yoshihisa Morita, Shoji Yotsutani. A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions. Communications on Pure & Applied Analysis, 2005, 4 (3) : 665-682. doi: 10.3934/cpaa.2005.4.665
[9]	Jun Yang. Vortex structures for Klein-Gordon equation with Ginzburg-Landau nonlinearity. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2359-2388. doi: 10.3934/dcds.2014.34.2359
[10]	Noboru Okazawa, Tomomi Yokota. Subdifferential operator approach to strong wellposedness of the complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems, 2010, 28 (1) : 311-341. doi: 10.3934/dcds.2010.28.311
[11]	Sen-Zhong Huang, Peter Takáč. Global smooth solutions of the complex Ginzburg-Landau equation and their dynamical properties. Discrete & Continuous Dynamical Systems, 1999, 5 (4) : 825-848. doi: 10.3934/dcds.1999.5.825
[12]	Simão Correia, Mário Figueira. A generalized complex Ginzburg-Landau equation: Global existence and stability results. Communications on Pure & Applied Analysis, 2021, 20 (5) : 2021-2038. doi: 10.3934/cpaa.2021056
[13]	Yong Zhou, V. Vijayakumar, R. Murugesu. Controllability for fractional evolution inclusions without compactness. Evolution Equations & Control Theory, 2015, 4 (4) : 507-524. doi: 10.3934/eect.2015.4.507
[14]	Hans G. Kaper, Bixiang Wang, Shouhong Wang. Determining nodes for the Ginzburg-Landau equations of superconductivity. Discrete & Continuous Dynamical Systems, 1998, 4 (2) : 205-224. doi: 10.3934/dcds.1998.4.205
[15]	Mickaël Dos Santos, Oleksandr Misiats. Ginzburg-Landau model with small pinning domains. Networks & Heterogeneous Media, 2011, 6 (4) : 715-753. doi: 10.3934/nhm.2011.6.715
[16]	Fanghua Lin, Ping Zhang. On the hydrodynamic limit of Ginzburg-Landau vortices. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 121-142. doi: 10.3934/dcds.2000.6.121
[17]	Renhai Wang, Yangrong Li. Backward compactness and periodicity of random attractors for stochastic wave equations with varying coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4145-4167. doi: 10.3934/dcdsb.2019054
[18]	Yan Zheng, Jianhua Huang. Exponential convergence for the 3D stochastic cubic Ginzburg-Landau equation with degenerate noise. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5621-5632. doi: 10.3934/dcdsb.2019075
[19]	Shujuan Lü, Chunbiao Gan, Baohua Wang, Linning Qian, Meisheng Li. Traveling wave solutions and its stability for 3D Ginzburg-Landau type equation. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 507-527. doi: 10.3934/dcdsb.2011.16.507
[20]	Hongzi Cong, Jianjun Liu, Xiaoping Yuan. Quasi-periodic solutions for complex Ginzburg-Landau equation of nonlinearity $\|u\|^{2p}u$. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 579-600. doi: 10.3934/dcdss.2010.3.579

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors