American Institute of Mathematical Sciences

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May  2018, 23(3): 1133-1154. doi: 10.3934/dcdsb.2018145

Homogenization of trajectory attractors of Ginzburg-Landau equations with randomly oscillating terms

Gregory A. Chechkin 1,2,, , Vladimir V. Chepyzhov 3,4, and  Leonid S. Pankratov 5,

1. 

Baku branch of M.V. Lomonosov Moscow State University, Universitetskaya st., 1, Xocasan, Binagadi district, Baku, AZ 1144, Azerbaijan
2. 

M.V. Lomonosov Moscow State University, Moscow, 119991, Russian Federation
3. 

Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoy Karetniy 19, Moscow 127051, Russian Federation
4. 

Voronezh State University, Universitetskaya sq. 1, Voronezh 394018, Russian Federation
5. 

Laboratory of Fluid Dynamics and Seismic (RAEP 5top100), Moscow Institute of Physics and Technology, Institutskiy 9, Dolgoprudny, Moscow Region 141700, Russian Federation

* Corresponding author: G. A. Chechkin

To the blessed memory of I. D. Chueshov

Received  March 2017 Revised  September 2017 Published  February 2018

Fund Project: Work of GAC was supported in part by the Committee of Science of the Ministry of Education and Science of the Republic of Kazakhstan (grant no. AP05131707) and by the Russian Foundation for Basic Research (projects 18-01-00046). This research of VVC was supported by the Ministry of Education and Science of the Russian Federation (grant 14.Z50.31.0037). The work of LSP was partially supported by Russian Science Foundation (grant no. 18-11-00148).

We consider complex Ginzburg-Landau (GL) type equations of the form:
${\partial _t}u = (1 + \alpha i)\Delta u + R{\mkern 1mu} u + (1 + \beta i)|u{|^2}u + g,$
where
$R$
,
$β$
, and
$g$
 are random rapidly oscillating real functions. Assuming that the random functions are ergodic and statistically homogeneous in space variables, we prove that the trajectory attractors of these systems tend to the trajectory attractors of the homogenized equations whose terms are the average of the corresponding terms of the initial systems.
Bibliography: 52 titles.
Keywords: Trajectory attractors, random homogenization, Ginzburg-Landau equations, nonlinear equations.
Mathematics Subject Classification: Primary: 35B40, 35B41, 35B45, 35Q30.
Citation: Gregory A. Chechkin, Vladimir V. Chepyzhov, Leonid S. Pankratov. Homogenization of trajectory attractors of Ginzburg-Landau equations with randomly oscillating terms. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1133-1154. doi: 10.3934/dcdsb.2018145
References:
[1]

Y. Amirat, O. Bodart, G. A. Chechkin and A. L. Piatnitski, Boundary homogenization in domains with randomly oscillating boundary, Stochastic Processes and their Applications, 121 (2011), 1-23.   Google Scholar

[2]

V. I. Arnol'd and A. Avez, Ergodic Problems of Classical Mechanics, W. A. Benjamin Inc., New York - Amsterdam, 1968.  Google Scholar

[3]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar

[4]

N. S. Bakhvalov and G. P. Panasenko, Averaging Processes in Periodic Media, Mathematics and its Applications (Soviet Series), 36. Kluwer Academic Publishers Group, Dordrecht, 1989.  Google Scholar

[5]

K. A. Bekmaganbetov, G. A. Chechkin, V. V. Chepyzhov and A. Yu. Goritsky, Homogenization of Trajectory Attractors of 3D Navier-Stokes system with Randomly Oscillating Force, Discrete and Continuous Dynamical Systems. Series A (DCDS-A), 37 (2017), 2375-2393.   Google Scholar

[6]

K. A. Bekmaganbetov, G. A. Chechkin and V. V. Chepyzhov, Homogenization of Random Attractors for Reaction-Diffusion Systems, C R Mécanique, 344 (2016), 753-758.  doi: 10.1016/j.crme.2016.10.015.  Google Scholar

[7]

A. Bensoussan, J. -L. Lions and G. Papanicolau, Asymptotic Analysis for Periodic Structures, Corrected reprint of the 1978 original [MR0503330]. AMS Chelsea Publishing, Providence, RI, 2011.  Google Scholar

[8]

G. D. Birkhoff, Proof of the ergodic theorem, Proc Natl Acad Sci USA, 17 (1931), 656-660.   Google Scholar

[9]

N. N. Bogolyubov and Ya. A. Mitropolski, Asymptotic Methods in the Theory of Non-Linear Oscillations, International Monographs on Advanced Mathematics and Physics, Hindustan Publishing Corp., Delhi, Gordon & Breach Science Publishers, New York, 1961.  Google Scholar

[10]

A. Bourgeat, I. D. Chueshov and L. Pankratov, Homogenization of attractors for semilinear parabolic equations in domains with spherical traps, Comptes rendues de l'Académie des Sciences, série I, 329 (1999), 581-586.   Google Scholar

[11]

A. Bourgeat and L. Pankratov, Homogenization of reaction-diffusion equations in domains with "traps". In : Proceedings of the International Conference "Porous Media: Physics, Modelling, Simulation", Ed. A. Dmitrievsky, M. Panfilov, World Scientific, Singapore-New Jersey-London-Hong Kong, 2000,267-278. Google Scholar

[12]

L. Boutet de Monvel, I.D. Chueshov and E. Ya. Khruslov, Homogenization of attractors for semilinear parabolic equations on manifolds with complicated microstructure, Ann. Mat. Pura Appl., 172 (1997), 297-322.   Google Scholar

[13]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, 183. Springer, New York, 2013.  Google Scholar

[14]

G. A. Chechkin, A. L. Piatnitski and A. S. Shamaev, Homogenization. Methods and Applications, Translated from the 2007 Russian original by Tamara Rozhkovskaya. Translations of Mathematical Monographs, 234. American Mathematical Society, Providence, RI, 2007.  Google Scholar

[15]

G. A. Chechkin, T. P. Chechkina, C. D'Apice and U. De Maio, Homogenization in Domains Randomly Perforated Along the Boundary, Discrete Continuous Dynam. Systems -B, 12 (2009), 713-730.   Google Scholar

[16]

G. A. Chechkin, T. P. Chechkina, C. D'Apice, U. De Maio and T. A. Mel'nyk, Asymptotic Analysis of a Boundary Value Problem in a Cascade Thick Junction with a Random Transmission Zone, Applicable Analysis, 88 (2009), 1543-1562.   Google Scholar

[17]

G. A. Chechkin, T. P. Chechkina, C. D'Apice, U. De Maio and T. A. Mel'nyk, Homogenization of 3D Thick Cascade Junction with the Random Transmission Zone Periodic in One direction, Russ. J. Math. Phys., 17 (2010), 35-55.   Google Scholar

[18]

G. A. Chechkin, C. D'Apice, U. De Maio and A. L. Piatnitski, On the Rate of Convergence of Solutions in Domain with Random Multilevel Oscillating Boundary, Asymptotic Analysis, 87 (2014), 1-28.   Google Scholar

[19]

G. A. Chechkin, T. P. Chechkina, T. S. Ratiu and M. S. Romanov, Nematodynamics and Random Homogenization, Applicable Analysis, 95 (2016), 2243-2253.  doi: 10.1080/00036811.2015.1036241.  Google Scholar

[20]

V. V. Chepyzhov, A. Yu. Goritski and M. I. Vishik, Integral manifolds and attractors with exponential rate for nonautonomous hyperbolic equations with dissipation, Russ. J. Math. Phys., 12 (2005), 17-39.   Google Scholar

[21]

V. V. Chepyzhov, V. Pata and M. I. Vishik, Averaging of nonautonomous damped wave equations with singularly oscillating external forces, J. Math. Pures Appl., 90 (2008), 469--491.   Google Scholar

[22]

V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for reaction-diffusion systems, Topol. Methods Nonlinear Anal. J.Julius Schauder Center, 7 (1996), 49-76.   Google Scholar

[23]

V. V. Chepyzhov and M. I. Vishik, Evolution equations and their trajectory attractors, J. Math. Pures Appl., 76 (1997), 913-964.   Google Scholar

[24]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49. American Mathematical Society, Providence, RI, 2002.  Google Scholar

[25]

V. V. Chepyzhov and M. I. Vishik, Global attractors for non-autonomous Ginzburg-Landau equation with singularly oscillating terms, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 29 (2005), 123-148.   Google Scholar

[26]

I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Kharkov, AKTA, 1999.  Google Scholar

[27]

I. D. Chueshov and L. S. Pankratov, Upper semicontinuity of attractors of semilinear parabolic equations with asymptotically degenerating coefficients, Mat. Fiz., Analiz, Geom., 6 (1999), 158-181.   Google Scholar

[28]

I. D. Chueshov and L. S. Pankratov, Averaging of attractors of nonlinear hyperbolic equations with asymptotically degenerate coefficients, Sb. Math., 190 (1999), 1325-1352.   Google Scholar

[29]

I. D. Chueshov and B. Schmalfuß, Averaging of attractors and inertial manifolds for parabolic PDE with random coefficients, Advanced Nonlinear Studies, 5 (2005), 461-492.   Google Scholar

[30]

Yu. A. Dubinskiĭ, Weak convergence in nonlinear elliptic and parabolic equations, Sb. Math., 67(109) (1965), 609-642.   Google Scholar

[31]

N. Dunford and J. T. Schwartz, Linear Operators. Part I: General Theory, John Wiley & Sons, Inc., New-York, 1988.  Google Scholar

[32]

M. Efendiev and S. Zelik, Attractors of the reaction-diffusion systems with rapidly oscillating coefficients and their homogenization, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 961-989.   Google Scholar

[33]

B. Fiedler and M. I. Vishik, Quantitative homogenization of global attractors for reaction-diffusion systems with rapidly oscillating terms, Asymptotic Anal., 34 (2003), 159-185.   Google Scholar

[34]

J. M. Ghidaglia and B. Héron, Dimension of the attractors associated to the Ginzburg-Landau partial differential equation, Physica D, 28 (1987), 282-304.   Google Scholar

[35]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988.  Google Scholar

[36]

J. K. Hale and S. M. Verduyn Lunel, Averaging in infinite dimensions, J. Int. Eq. Appl., 2 (1990), 463-494.   Google Scholar

[37]

A. A. Ilyin, Averaging principle for dissipative dynamical systems with rapidly oscillating right-hand sides, Sb. Math., 187 (1996), 635-677.   Google Scholar

[38]

V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994.  Google Scholar

[39]

E. Ya. Khruslov and L. S. Pankratov, Homogenization of boundary problems for GinzburgLandau equation in weakly connected domains, In: Spectral Operator Theory and Related Topics, edited by V. A. Marchenko, AMS Providence, 19 (1994), 233-268.  Google Scholar

[40]

J. -L. Lions, Quelques Méthodes de Résolutions des Problémes aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[41]

J. -L. Lions and E. Magenes, Problémes Aux Limites Non Homogénes at Applications, volume 1, Dunod, Gauthier-Villars, Paris, 1968.  Google Scholar

[42]

V. A. Marchenko and E. Ya. Khruslov, Homogenization of Partial Differential Equations, Progress in Mathematical Physics, 46. Birkhäuser Boston, Inc., Boston, MA, 2006.  Google Scholar

[43]

A. Mielke, The complex Ginzburg-Landau equation on large and unbounded domains: sharper bounds and attractors, Nonlinearity, 10 (1997), 199-222.   Google Scholar

[44]

L. Pankratov, Homogenization of Ginzburg-Landau heat flow equation in a porous medium, Applicable Analysis, 69 (1998), 31-45.   Google Scholar

[45]

E. Sánchez-Palencia, Homogenization Techniques for Composite Media, Lecture Notes in Physics, 272. Springer-Verlag, Berlin, 1987.  Google Scholar

[46]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, 2nd edition. Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. Google Scholar

[47]

M. I. Vishik and V. V. Chepyzhov, Averaging of trajectory attractors of evolution equations with rapidly oscillating terms, Sb. Math., 192 (2001), 11-47.   Google Scholar

[48]

M. I. Vishik and V. V. Chepyzhov, The nonautonomous Ginzburg-Landau equation and its attractors, Sb. Math., 196 (2005), 791-815.   Google Scholar

[49]

M. I. Vishik and V. V. Chepyzhov, Trajectory attractors of equations of mathematical physics, Russian Math. Surveys, 66 (2011), 637-731.   Google Scholar

[50]

M. I. Vishik and B. Fiedler, Quantitative averaging of global attractors of hyperbolic wave equations with rapidly oscillating coefficients, Russian Math. Surveys, 57 (2002), 709-728.   Google Scholar

[51]

S. V. Zelik, The attractor for a nonlinear reaction-diffusion system with a supercritical nonlinearity and it's dimension, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 24 (2000), 1-25.   Google Scholar

[52]

V. V. Zhikov, On two-scale convergence, Journal of Mathematical Sciences, 120 (2003), 1328-1352.   Google Scholar

show all references

References:
[1]

Y. Amirat, O. Bodart, G. A. Chechkin and A. L. Piatnitski, Boundary homogenization in domains with randomly oscillating boundary, Stochastic Processes and their Applications, 121 (2011), 1-23.   Google Scholar

[2]

V. I. Arnol'd and A. Avez, Ergodic Problems of Classical Mechanics, W. A. Benjamin Inc., New York - Amsterdam, 1968.  Google Scholar

[3]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar

[4]

N. S. Bakhvalov and G. P. Panasenko, Averaging Processes in Periodic Media, Mathematics and its Applications (Soviet Series), 36. Kluwer Academic Publishers Group, Dordrecht, 1989.  Google Scholar

[5]

K. A. Bekmaganbetov, G. A. Chechkin, V. V. Chepyzhov and A. Yu. Goritsky, Homogenization of Trajectory Attractors of 3D Navier-Stokes system with Randomly Oscillating Force, Discrete and Continuous Dynamical Systems. Series A (DCDS-A), 37 (2017), 2375-2393.   Google Scholar

[6]

K. A. Bekmaganbetov, G. A. Chechkin and V. V. Chepyzhov, Homogenization of Random Attractors for Reaction-Diffusion Systems, C R Mécanique, 344 (2016), 753-758.  doi: 10.1016/j.crme.2016.10.015.  Google Scholar

[7]

A. Bensoussan, J. -L. Lions and G. Papanicolau, Asymptotic Analysis for Periodic Structures, Corrected reprint of the 1978 original [MR0503330]. AMS Chelsea Publishing, Providence, RI, 2011.  Google Scholar

[8]

G. D. Birkhoff, Proof of the ergodic theorem, Proc Natl Acad Sci USA, 17 (1931), 656-660.   Google Scholar

[9]

N. N. Bogolyubov and Ya. A. Mitropolski, Asymptotic Methods in the Theory of Non-Linear Oscillations, International Monographs on Advanced Mathematics and Physics, Hindustan Publishing Corp., Delhi, Gordon & Breach Science Publishers, New York, 1961.  Google Scholar

[10]

A. Bourgeat, I. D. Chueshov and L. Pankratov, Homogenization of attractors for semilinear parabolic equations in domains with spherical traps, Comptes rendues de l'Académie des Sciences, série I, 329 (1999), 581-586.   Google Scholar

[11]

A. Bourgeat and L. Pankratov, Homogenization of reaction-diffusion equations in domains with "traps". In : Proceedings of the International Conference "Porous Media: Physics, Modelling, Simulation", Ed. A. Dmitrievsky, M. Panfilov, World Scientific, Singapore-New Jersey-London-Hong Kong, 2000,267-278. Google Scholar

[12]

L. Boutet de Monvel, I.D. Chueshov and E. Ya. Khruslov, Homogenization of attractors for semilinear parabolic equations on manifolds with complicated microstructure, Ann. Mat. Pura Appl., 172 (1997), 297-322.   Google Scholar

[13]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, 183. Springer, New York, 2013.  Google Scholar

[14]

G. A. Chechkin, A. L. Piatnitski and A. S. Shamaev, Homogenization. Methods and Applications, Translated from the 2007 Russian original by Tamara Rozhkovskaya. Translations of Mathematical Monographs, 234. American Mathematical Society, Providence, RI, 2007.  Google Scholar

[15]

G. A. Chechkin, T. P. Chechkina, C. D'Apice and U. De Maio, Homogenization in Domains Randomly Perforated Along the Boundary, Discrete Continuous Dynam. Systems -B, 12 (2009), 713-730.   Google Scholar

[16]

G. A. Chechkin, T. P. Chechkina, C. D'Apice, U. De Maio and T. A. Mel'nyk, Asymptotic Analysis of a Boundary Value Problem in a Cascade Thick Junction with a Random Transmission Zone, Applicable Analysis, 88 (2009), 1543-1562.   Google Scholar

[17]

G. A. Chechkin, T. P. Chechkina, C. D'Apice, U. De Maio and T. A. Mel'nyk, Homogenization of 3D Thick Cascade Junction with the Random Transmission Zone Periodic in One direction, Russ. J. Math. Phys., 17 (2010), 35-55.   Google Scholar

[18]

G. A. Chechkin, C. D'Apice, U. De Maio and A. L. Piatnitski, On the Rate of Convergence of Solutions in Domain with Random Multilevel Oscillating Boundary, Asymptotic Analysis, 87 (2014), 1-28.   Google Scholar

[19]

G. A. Chechkin, T. P. Chechkina, T. S. Ratiu and M. S. Romanov, Nematodynamics and Random Homogenization, Applicable Analysis, 95 (2016), 2243-2253.  doi: 10.1080/00036811.2015.1036241.  Google Scholar

[20]

V. V. Chepyzhov, A. Yu. Goritski and M. I. Vishik, Integral manifolds and attractors with exponential rate for nonautonomous hyperbolic equations with dissipation, Russ. J. Math. Phys., 12 (2005), 17-39.   Google Scholar

[21]

V. V. Chepyzhov, V. Pata and M. I. Vishik, Averaging of nonautonomous damped wave equations with singularly oscillating external forces, J. Math. Pures Appl., 90 (2008), 469--491.   Google Scholar

[22]

V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for reaction-diffusion systems, Topol. Methods Nonlinear Anal. J.Julius Schauder Center, 7 (1996), 49-76.   Google Scholar

[23]

V. V. Chepyzhov and M. I. Vishik, Evolution equations and their trajectory attractors, J. Math. Pures Appl., 76 (1997), 913-964.   Google Scholar

[24]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49. American Mathematical Society, Providence, RI, 2002.  Google Scholar

[25]

V. V. Chepyzhov and M. I. Vishik, Global attractors for non-autonomous Ginzburg-Landau equation with singularly oscillating terms, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 29 (2005), 123-148.   Google Scholar

[26]

I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Kharkov, AKTA, 1999.  Google Scholar

[27]

I. D. Chueshov and L. S. Pankratov, Upper semicontinuity of attractors of semilinear parabolic equations with asymptotically degenerating coefficients, Mat. Fiz., Analiz, Geom., 6 (1999), 158-181.   Google Scholar

[28]

I. D. Chueshov and L. S. Pankratov, Averaging of attractors of nonlinear hyperbolic equations with asymptotically degenerate coefficients, Sb. Math., 190 (1999), 1325-1352.   Google Scholar

[29]

I. D. Chueshov and B. Schmalfuß, Averaging of attractors and inertial manifolds for parabolic PDE with random coefficients, Advanced Nonlinear Studies, 5 (2005), 461-492.   Google Scholar

[30]

Yu. A. Dubinskiĭ, Weak convergence in nonlinear elliptic and parabolic equations, Sb. Math., 67(109) (1965), 609-642.   Google Scholar

[31]

N. Dunford and J. T. Schwartz, Linear Operators. Part I: General Theory, John Wiley & Sons, Inc., New-York, 1988.  Google Scholar

[32]

M. Efendiev and S. Zelik, Attractors of the reaction-diffusion systems with rapidly oscillating coefficients and their homogenization, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 961-989.   Google Scholar

[33]

B. Fiedler and M. I. Vishik, Quantitative homogenization of global attractors for reaction-diffusion systems with rapidly oscillating terms, Asymptotic Anal., 34 (2003), 159-185.   Google Scholar

[34]

J. M. Ghidaglia and B. Héron, Dimension of the attractors associated to the Ginzburg-Landau partial differential equation, Physica D, 28 (1987), 282-304.   Google Scholar

[35]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988.  Google Scholar

[36]

J. K. Hale and S. M. Verduyn Lunel, Averaging in infinite dimensions, J. Int. Eq. Appl., 2 (1990), 463-494.   Google Scholar

[37]

A. A. Ilyin, Averaging principle for dissipative dynamical systems with rapidly oscillating right-hand sides, Sb. Math., 187 (1996), 635-677.   Google Scholar

[38]

V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994.  Google Scholar

[39]

E. Ya. Khruslov and L. S. Pankratov, Homogenization of boundary problems for GinzburgLandau equation in weakly connected domains, In: Spectral Operator Theory and Related Topics, edited by V. A. Marchenko, AMS Providence, 19 (1994), 233-268.  Google Scholar

[40]

J. -L. Lions, Quelques Méthodes de Résolutions des Problémes aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[41]

J. -L. Lions and E. Magenes, Problémes Aux Limites Non Homogénes at Applications, volume 1, Dunod, Gauthier-Villars, Paris, 1968.  Google Scholar

[42]

V. A. Marchenko and E. Ya. Khruslov, Homogenization of Partial Differential Equations, Progress in Mathematical Physics, 46. Birkhäuser Boston, Inc., Boston, MA, 2006.  Google Scholar

[43]

A. Mielke, The complex Ginzburg-Landau equation on large and unbounded domains: sharper bounds and attractors, Nonlinearity, 10 (1997), 199-222.   Google Scholar

[44]

L. Pankratov, Homogenization of Ginzburg-Landau heat flow equation in a porous medium, Applicable Analysis, 69 (1998), 31-45.   Google Scholar

[45]

E. Sánchez-Palencia, Homogenization Techniques for Composite Media, Lecture Notes in Physics, 272. Springer-Verlag, Berlin, 1987.  Google Scholar

[46]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, 2nd edition. Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. Google Scholar

[47]

M. I. Vishik and V. V. Chepyzhov, Averaging of trajectory attractors of evolution equations with rapidly oscillating terms, Sb. Math., 192 (2001), 11-47.   Google Scholar

[48]

M. I. Vishik and V. V. Chepyzhov, The nonautonomous Ginzburg-Landau equation and its attractors, Sb. Math., 196 (2005), 791-815.   Google Scholar

[49]

M. I. Vishik and V. V. Chepyzhov, Trajectory attractors of equations of mathematical physics, Russian Math. Surveys, 66 (2011), 637-731.   Google Scholar

[50]

M. I. Vishik and B. Fiedler, Quantitative averaging of global attractors of hyperbolic wave equations with rapidly oscillating coefficients, Russian Math. Surveys, 57 (2002), 709-728.   Google Scholar

[51]

S. V. Zelik, The attractor for a nonlinear reaction-diffusion system with a supercritical nonlinearity and it's dimension, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 24 (2000), 1-25.   Google Scholar

[52]

V. V. Zhikov, On two-scale convergence, Journal of Mathematical Sciences, 120 (2003), 1328-1352.   Google Scholar

Figure 1.  Attractors of the Ginzburg-Landau Equations
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