May  2020, 19(5): 2419-2443. doi: 10.3934/cpaa.2020106

Strong convergence of trajectory attractors for reaction–diffusion systems with random rapidly oscillating terms

1. 

M.V. Lomonosov Moscow State University, Kazakhstan Branch, Kazhymukan st. 11, Astana, 010010, Kazakhstan

2. 

Institute of Mathematics and Mathematical Modeling, Pushkin st. 125, Almaty, 050010, Kazakhstan

3. 

M.V. Lomonosov Moscow State University, Moscow, 119991, Russia

4. 

Institute of Mathematics with Computing Center. Subdivision of the Ufa, Federal Research Center of Russian Academy of Science, Chernyshevskogo st. 112, Ufa, 450008, Russia

5. 

Institute for Information Transmission Problems - Russian Academy of Sciences, Bolshoy Karetniy 19, Moscow 127051, Russia

6. 

National Research University Higher School of Economics, Myasnitskaya Street 20, Moscow 101000, Russia

*Corresponding author

Received  May 2019 Revised  November 2019 Published  March 2020

Fund Project: The work is partially supported by the Russian Foundation of Basic Researches (GAC project 18-01-00046 and VVC project 17-01-00515) and Russian Science Foundation (project 20-11-20272). Work of KAB is supported in part by the Committee of Science of the Ministry of Education and Science of the Republic of Kazakhstan AP05132071

We consider reaction–diffusion systems with random terms that oscillate rapidly in space variables. Under the assumption that the random functions are ergodic and statistically homogeneous we prove that the random trajectory attractors of these systems tend to the deterministic trajectory attractors of the averaged reaction-diffusion system whose terms are the average of the corresponding terms of the original system. Special attention is given to the case when the convergence of random trajectory attractors holds in the strong topology.

Citation: Kuanysh A. Bekmaganbetov, Gregory A. Chechkin, Vladimir V. Chepyzhov. Strong convergence of trajectory attractors for reaction–diffusion systems with random rapidly oscillating terms. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2419-2443. doi: 10.3934/cpaa.2020106
References:
[1]

Y. AmiratO. BodartG. A. Chechkin and A. L. Piatnitski, Boundary homogenization in domains with randomly oscillating boundary, Stoch. Process. Their Appl., 121 (2011), 1-23.  doi: 10.1016/j.spa.2010.08.011.

[2]

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

[3]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North–Holland Publishing Co., Amsterdam, 1992; Nauka, Moscow, 1989.

[4]

N. S. Bakhvalov and G. P. Panasenko, Averaging Processes in Periodic Media, Mathematics and Its Applications (Soviet Series), Vol. 36, Kluwer Academic Publishers Group, Dordrecht, 1989. doi: 10.1007/978-94-009-2247-1.

[5]

J. Ball, Global attractors for damped semilinear wave equations. Partial differential equations and applications, Discrete Contin. Dyn. Syst., 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.

[6]

K. A. BekmaganbetovG. A. ChechkinV. V. Chepyzhov and A. Yu. Goritsky, Homogenization of trajectory attractors of 3D Navier–Stokes system with Randomly Oscillating Force, Discrete Contin. Dyn. Syst., 37 (2017), 2375-2393.  doi: 10.3934/dcds.2017103.

[7]

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

[8]

A. Bensoussan, J. L. Lions and G. Papanicolau, Asymptotic Analysis for Periodic Structures, North–Holland, Amsterdam, 1978.

[9]

G. D. Birkhoff, Proof of the ergodic theorem, Proc. Natl. Acad. Sci. U. S. A., 17 (1931), 656-660.  doi: 10.1073/pnas.17.12.656.

[10]

F. Boyer F. and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, Vol. 183, Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.

[11]

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.

[12]

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

[13]

V. V. ChepyzhovM. Conti and V. Pata, A minimal approach to the theory of global attractors, Discrete Contin. Dyn. Syst., 32 (2012), 2079-2088.  doi: 10.3934/dcds.2012.32.2079.

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G. A. Chechkin, A. L. Piatnitski and A. S. Shamaev, Homogenization. Methods and Applications, American Mathematical Society, Providence, RI, 2007.

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G. A. ChechkinT. P. ChechkinaC. D'Apice and U. De Maio, Homogenization in domains randomly perforated along the boundary, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 713-730.  doi: 10.3934/dcdsb.2009.12.713.

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G. A. ChechkinT. P. ChechkinaC. D'ApiceU. De Maio and T. A. Mel'nyk, Asymptotic analysis of a boundary value problem in a cascade thick junction with a random transmission Zone, Appl. Anal., 88 (2009), 1543-1562.  doi: 10.1080/00036810902994268.

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G. A. ChechkinT. P. ChechkinaC. D'ApiceU. 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.  doi: 10.1134/S1061920810010048.

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G. A. ChechkinC. D'ApiceU. De Maio and A. L. Piatnitski, On the rate of convergence of solutions in domain with random multilevel oscillating boundary, Asymptotic Anal., 87 (2014), 1-28.  doi: 10.3233/ASY-131194.

[19]

G. A. ChechkinT. P. ChechkinaT. S. Ratiu and M. S. Romanov, Nematodynamics and random homogenization, Appl. Anal., 95 (2016), 2243-2253.  doi: 10.1080/00036811.2015.1036241.

[20]

G. A. ChechkinV. V. Chepyzhov and L. S. Pankratov, Homogenization of trajectory attractors of Ginzburg–Landau equations with randomly oscillating terms, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1133-1154.  doi: 10.3934/dcdsb.2018145.

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[22]

V. V. ChepyzhovV. Pata and M. I. Vishik, Averaging of nonautonomous damped wave equations with singularly oscillating external forces, J. Math. Pures Appl., 90 (2008), 469-491.  doi: 10.1016/j.matpur.2008.07.001.

[23]

V. V. ChepyzhovV. Pata and M. I. Vishik, Averaging of 2D Navier-Stokes equations with singularly oscillating forces, Nonlinearity, 22 (2009), 351-370.  doi: 10.1088/0951-7715/22/2/006.

[24]

V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for reaction-diffusion systems, Topol. Methods Nonlinear Anal., 7 (1996), 49-76.  doi: 10.12775/TMNA.1996.002.

[25]

V. V. Chepyzhov and M. I. Vishik, Evolution equations and their trajectory attractors, J. Math. Pures Appl., 76 (1997), 913-964.  doi: 10.1016/S0021-7824(97)89978-3.

[26]

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

[27]

V. V. Chepyzhov and M. I. Vishik, Non-autonomous 2D Navier-Stokes system with a simple global attractor and some averaging problems, ESAIM Control Optim. Calc. Var., 8 (2002), 467-487.  doi: 10.1051/cocv:2002056.

[28]

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. 

[29]

V. V. Chepyzhov and M. I. Vishik, Non-autonomous 2D Navier-Stokes system with singularly oscillating external force and its global attractor, J. Dyn. Differ. Equ., 19 (2007), 655-684.  doi: 10.1007/s10884-007-9077-y.

[30]

V. V. ChepyzhovM. I. Vishik and W. L. Wendland, On non-autonomous sine-Gordon type equations with a simple global attractor and some averaging, Discrete Contin. Dyn. Syst., 12 (2005), 27-38.  doi: 10.3934/dcds.2005.12.27.

[31]

I. D. Chueshov and B. Schmalfuß, Averaging of attractors and inertial manifolds for parabolic PDE With random coefficients, Adv. Nonlinear Stud., 5 (2005), 461-492.  doi: 10.1515/ans-2005-0402.

[32]

M. Efendiev and S. Zelik, Attractors of the reaction-diffusion systems with rapidly oscillating coefficients and their homogenization, Ann. Inst. Henri Poincare - Anal. Non Lineaire, 19 (2002), 961-989.  doi: 10.1016/S0294-1449(02)00115-4.

[33]

M. Efendiev and S. Zelik, The regular attractor for the reaction-diffusion system with a nonlinearity rapidly oscillating in time and its averaging, Adv. Differ. Equ., 8 (2003), 673-732. 

[34]

B. Fiedler and M. I. Vishik, Quantitative homogenization of analytic semigroups and reaction-diffusion equations with Diophantine spatial sequences, Adv. Differ. Equ., 6 (2001), 1377-1408. 

[35]

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. 

[36]

J. M. Ghidaglia, A note on the strong convergence towards attractors of damped forced KdV equations, J. Differ. Equ., 110 (1994), 356-359.  doi: 10.1006/jdeq.1994.1071.

[37]

J. K. Hale and S. M. Verduyn Lunel, Averaging in infinite dimensions, J. Integral Equ. Appl., 2 (1990), 463-494.  doi: 10.1216/jiea/1181075583.

[38]

V. V. Jikov, S. M. Kozlov adnd O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer–Verlag, Berlin, 1994. doi: 10.1007/978-3-642-84659-5.

[39]

A. A. Ilyin, Averaging principle for dissipative dynamical systems with rapidly oscillatingr ight-hand sides, Sb. Math., 187 (1996), 635-677.  doi: 10.1070/SM1996v187n05ABEH000126.

[40]

A. A. Ilyin, Global averaging of dissipative dynamical systems, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 22 (1998), 165-191. 

[41] M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, GITTL, Moscow, 1956 [in Russia]. Pergamon Press, London, 1964. 
[42]

J. L. Lions and E. Magenes, Problemes oux Limites non Homogénes et Applications, Dunod, Gauthier-Villars, Paris, 1968.

[43]

J. L. Lions, Quelques Méthodes de Résolutions des Problèmes aux Limites non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.

[44]

V. A. Marchenko and E. Ya. Khruslov, Boundary Value Problems in Domains with Fine-Grain Boundary, Naukova Dumka, Kiev, 1974 [in Russian].

[45]

V. A. Marchenko and E. Ya. Khruslov, Homogenization of Partial Differential Equations, Vol. 46, Birkhäuser Boston, Inc., Boston, MA, 2006.

[46]

I. MoiseR. Rosa and X. Wang, Attractors for non-compact semigroups via energy equations, Nonlinearity, 11 (1998), 1369-1393.  doi: 10.1088/0951-7715/11/5/012.

[47]

L. S. Pankratov and I. D. Cheushov, Averaging of attractors of nonlinear hyperbolic equations with asymptotically degenerate coefficients, Sb. Math., 190 (1999), 1325-1352.  doi: 10.1070/SM1999v190n09ABEH000427.

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R. Rosa, The global attractor for the 2D Navier–Stokes flow on some unbounded domains, Nonlinear Anal. Theory Methods Appl., 32 (1998), 71-85.  doi: 10.1016/S0362-546X(97)00453-7.

[49]

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

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R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, Vol. 68, 2$^nd$ edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[51]

M. I. Vishik and V. V. Chepyzhov, Averaging of trajectory attractors of evolution equations with rapidly oscillating terms, Sb. Math., 192 (2001), 11-47.  doi: 10.1070/SM2001v192n01ABEH000534.

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M. I. Vishik and V. V. Chepyzhov, Approximation of trajectories lying on a global attractor of a hyperbolic equation with an exterior force that oscillates rapidly over time, Sb. Math., 194 (2003), 1273-1300.  doi: 10.1070/SM2003v194n09ABEH000765.

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M. I. Vishik and V. V. Chepyzhov, Non-autonomous Ginzburg-Landau equation and its attractors, Sb. Math., 196 (2005), 791-815.  doi: 10.1070/SM2005v196n06ABEH000901.

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M. I. Vishik and V. V. Chepyzhov, Attractors of dissipative hyperbolic equations with singularly oscillating external forces, Math. Notes, 79 (2006), 483-504.  doi: 10.1007/s11006-006-0054-2.

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M. I. Vishik and B. Fiedler, Quantitative averaging of global attractors of hyperbolic wave equations with rapidly oscillating coefficients, Russ. Math. Surv., 57 (2002), 709-728.  doi: 10.1070/RM2002v057n04ABEH000534.

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show all references

References:
[1]

Y. AmiratO. BodartG. A. Chechkin and A. L. Piatnitski, Boundary homogenization in domains with randomly oscillating boundary, Stoch. Process. Their Appl., 121 (2011), 1-23.  doi: 10.1016/j.spa.2010.08.011.

[2]

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

[3]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North–Holland Publishing Co., Amsterdam, 1992; Nauka, Moscow, 1989.

[4]

N. S. Bakhvalov and G. P. Panasenko, Averaging Processes in Periodic Media, Mathematics and Its Applications (Soviet Series), Vol. 36, Kluwer Academic Publishers Group, Dordrecht, 1989. doi: 10.1007/978-94-009-2247-1.

[5]

J. Ball, Global attractors for damped semilinear wave equations. Partial differential equations and applications, Discrete Contin. Dyn. Syst., 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.

[6]

K. A. BekmaganbetovG. A. ChechkinV. V. Chepyzhov and A. Yu. Goritsky, Homogenization of trajectory attractors of 3D Navier–Stokes system with Randomly Oscillating Force, Discrete Contin. Dyn. Syst., 37 (2017), 2375-2393.  doi: 10.3934/dcds.2017103.

[7]

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

[8]

A. Bensoussan, J. L. Lions and G. Papanicolau, Asymptotic Analysis for Periodic Structures, North–Holland, Amsterdam, 1978.

[9]

G. D. Birkhoff, Proof of the ergodic theorem, Proc. Natl. Acad. Sci. U. S. A., 17 (1931), 656-660.  doi: 10.1073/pnas.17.12.656.

[10]

F. Boyer F. and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, Vol. 183, Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.

[11]

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.

[12]

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

[13]

V. V. ChepyzhovM. Conti and V. Pata, A minimal approach to the theory of global attractors, Discrete Contin. Dyn. Syst., 32 (2012), 2079-2088.  doi: 10.3934/dcds.2012.32.2079.

[14]

G. A. Chechkin, A. L. Piatnitski and A. S. Shamaev, Homogenization. Methods and Applications, American Mathematical Society, Providence, RI, 2007.

[15]

G. A. ChechkinT. P. ChechkinaC. D'Apice and U. De Maio, Homogenization in domains randomly perforated along the boundary, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 713-730.  doi: 10.3934/dcdsb.2009.12.713.

[16]

G. A. ChechkinT. P. ChechkinaC. D'ApiceU. De Maio and T. A. Mel'nyk, Asymptotic analysis of a boundary value problem in a cascade thick junction with a random transmission Zone, Appl. Anal., 88 (2009), 1543-1562.  doi: 10.1080/00036810902994268.

[17]

G. A. ChechkinT. P. ChechkinaC. D'ApiceU. 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.  doi: 10.1134/S1061920810010048.

[18]

G. A. ChechkinC. D'ApiceU. De Maio and A. L. Piatnitski, On the rate of convergence of solutions in domain with random multilevel oscillating boundary, Asymptotic Anal., 87 (2014), 1-28.  doi: 10.3233/ASY-131194.

[19]

G. A. ChechkinT. P. ChechkinaT. S. Ratiu and M. S. Romanov, Nematodynamics and random homogenization, Appl. Anal., 95 (2016), 2243-2253.  doi: 10.1080/00036811.2015.1036241.

[20]

G. A. ChechkinV. V. Chepyzhov and L. S. Pankratov, Homogenization of trajectory attractors of Ginzburg–Landau equations with randomly oscillating terms, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1133-1154.  doi: 10.3934/dcdsb.2018145.

[21]

V. V. ChepyzhovA. 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. 

[22]

V. V. ChepyzhovV. Pata and M. I. Vishik, Averaging of nonautonomous damped wave equations with singularly oscillating external forces, J. Math. Pures Appl., 90 (2008), 469-491.  doi: 10.1016/j.matpur.2008.07.001.

[23]

V. V. ChepyzhovV. Pata and M. I. Vishik, Averaging of 2D Navier-Stokes equations with singularly oscillating forces, Nonlinearity, 22 (2009), 351-370.  doi: 10.1088/0951-7715/22/2/006.

[24]

V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for reaction-diffusion systems, Topol. Methods Nonlinear Anal., 7 (1996), 49-76.  doi: 10.12775/TMNA.1996.002.

[25]

V. V. Chepyzhov and M. I. Vishik, Evolution equations and their trajectory attractors, J. Math. Pures Appl., 76 (1997), 913-964.  doi: 10.1016/S0021-7824(97)89978-3.

[26]

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

[27]

V. V. Chepyzhov and M. I. Vishik, Non-autonomous 2D Navier-Stokes system with a simple global attractor and some averaging problems, ESAIM Control Optim. Calc. Var., 8 (2002), 467-487.  doi: 10.1051/cocv:2002056.

[28]

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. 

[29]

V. V. Chepyzhov and M. I. Vishik, Non-autonomous 2D Navier-Stokes system with singularly oscillating external force and its global attractor, J. Dyn. Differ. Equ., 19 (2007), 655-684.  doi: 10.1007/s10884-007-9077-y.

[30]

V. V. ChepyzhovM. I. Vishik and W. L. Wendland, On non-autonomous sine-Gordon type equations with a simple global attractor and some averaging, Discrete Contin. Dyn. Syst., 12 (2005), 27-38.  doi: 10.3934/dcds.2005.12.27.

[31]

I. D. Chueshov and B. Schmalfuß, Averaging of attractors and inertial manifolds for parabolic PDE With random coefficients, Adv. Nonlinear Stud., 5 (2005), 461-492.  doi: 10.1515/ans-2005-0402.

[32]

M. Efendiev and S. Zelik, Attractors of the reaction-diffusion systems with rapidly oscillating coefficients and their homogenization, Ann. Inst. Henri Poincare - Anal. Non Lineaire, 19 (2002), 961-989.  doi: 10.1016/S0294-1449(02)00115-4.

[33]

M. Efendiev and S. Zelik, The regular attractor for the reaction-diffusion system with a nonlinearity rapidly oscillating in time and its averaging, Adv. Differ. Equ., 8 (2003), 673-732. 

[34]

B. Fiedler and M. I. Vishik, Quantitative homogenization of analytic semigroups and reaction-diffusion equations with Diophantine spatial sequences, Adv. Differ. Equ., 6 (2001), 1377-1408. 

[35]

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. 

[36]

J. M. Ghidaglia, A note on the strong convergence towards attractors of damped forced KdV equations, J. Differ. Equ., 110 (1994), 356-359.  doi: 10.1006/jdeq.1994.1071.

[37]

J. K. Hale and S. M. Verduyn Lunel, Averaging in infinite dimensions, J. Integral Equ. Appl., 2 (1990), 463-494.  doi: 10.1216/jiea/1181075583.

[38]

V. V. Jikov, S. M. Kozlov adnd O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer–Verlag, Berlin, 1994. doi: 10.1007/978-3-642-84659-5.

[39]

A. A. Ilyin, Averaging principle for dissipative dynamical systems with rapidly oscillatingr ight-hand sides, Sb. Math., 187 (1996), 635-677.  doi: 10.1070/SM1996v187n05ABEH000126.

[40]

A. A. Ilyin, Global averaging of dissipative dynamical systems, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 22 (1998), 165-191. 

[41] M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, GITTL, Moscow, 1956 [in Russia]. Pergamon Press, London, 1964. 
[42]

J. L. Lions and E. Magenes, Problemes oux Limites non Homogénes et Applications, Dunod, Gauthier-Villars, Paris, 1968.

[43]

J. L. Lions, Quelques Méthodes de Résolutions des Problèmes aux Limites non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.

[44]

V. A. Marchenko and E. Ya. Khruslov, Boundary Value Problems in Domains with Fine-Grain Boundary, Naukova Dumka, Kiev, 1974 [in Russian].

[45]

V. A. Marchenko and E. Ya. Khruslov, Homogenization of Partial Differential Equations, Vol. 46, Birkhäuser Boston, Inc., Boston, MA, 2006.

[46]

I. MoiseR. Rosa and X. Wang, Attractors for non-compact semigroups via energy equations, Nonlinearity, 11 (1998), 1369-1393.  doi: 10.1088/0951-7715/11/5/012.

[47]

L. S. Pankratov and I. D. Cheushov, Averaging of attractors of nonlinear hyperbolic equations with asymptotically degenerate coefficients, Sb. Math., 190 (1999), 1325-1352.  doi: 10.1070/SM1999v190n09ABEH000427.

[48]

R. Rosa, The global attractor for the 2D Navier–Stokes flow on some unbounded domains, Nonlinear Anal. Theory Methods Appl., 32 (1998), 71-85.  doi: 10.1016/S0362-546X(97)00453-7.

[49]

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

[50]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, Vol. 68, 2$^nd$ edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[51]

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