Weak convergence of attractors of reaction–diffusion systems with randomly oscillating coefficients

References

• Marchenko VA , Khruslov EYa . Boundary value problems in domains with Fine-Grain boundary. Kiev: Naukova Dumka; 1974. Russian.
   Google Scholar
• Bensoussan A , Lions J-L , Papanicolau G . Asymptotic analysis for periodic structures. Amsterdam: North-Holland; 1978.
   Google Scholar
• Bakhvalov NS , Panasenko GP . Averaging processes in periodic media. Dordrecht: Kluwer Academic Publishers; 1989.
   Google Scholar
• Sanchez-Palencia É . Homogenization techniques for composite media. Berlin: Springer-Verlag; 1987.
   Google Scholar
• Jikov VV , Kozlov SM , Oleinik OA . Homogenization of differential operators and integral functionals. Berlin: Springer-Verlag; 1994.
   Google Scholar
• Marchenko VA , Khruslov EYa . Homogenization of partial differential equations. Boston (MA): Birkhäuser; 2006.
   Google Scholar
• Chechkin GA , Piatnitski AL , Shamaev AS . Homogenization: methods and applications. Providence (RI): American Mathematical Society; 2007.
   Google Scholar
• Chechkin GA , Chechkina TP , D’Apice C , et al . Homogenization in domains randomly perforated along the boundary. Disc Contin Dyn Syst Ser B. 2009;12(4):713–730.
   Web of Science ®Google Scholar
• Chechkin GA , Chechkina TP , D’Apice C , et al . Asymptotic analysis of a boundary value problem in a cascade thick junction with a random transmission zone. Appl Anal. 2009;88(10–11):1543–1562.
   Web of Science ®Google Scholar
• Chechkin GA , Chechkina TP , D’Apice C , et al . Homogenization of 3D thick cascade junction with the random transmission zone periodic in one direction. Russ J Math Phys. 2010;17(1):35–55.
   Web of Science ®Google Scholar
• Amirat Y , Bodart O , Chechkin GA , et al . Boundary homogenization in domains with randomly oscillating boundary. Stochastic Processes Appl. 2011;121(1):1–23.
   Web of Science ®Google Scholar
• Chechkin GA , D’Apice C , De Maio U , et al . On the rate of convergence of solutions in domain with random multilevel oscillating boundary. Asymptotic Anal. 2014;87(1–2):1–28.
   Web of Science ®Google Scholar
• Chechkin GA , Chechkina TP , Ratiu TS , et al . Nematodynamics and random homogenization. Appl Anal. 2016;95(10):2243–2253.
   Web of Science ®Google Scholar
• Babin AV , Vishik MI . Attractors of evolution equations. Amsterdam: North-Holland; 1992. Moscow: Nauka; 1989.
   Google Scholar
• Chepyzhov VV , Vishik MI . Attractors for equations of mathematical physics. Providence (RI): American Mathematical Society; 2002.
   Google Scholar
• Temam R . Infinite-dimensional dynamical systems in mechanics and physics. Applied mathematics series V. Vol. 68. New York (NY): Springer-Verlag; 1988.
   Google Scholar
• Bogolyubov NN , Mitropolski YaA . Asymptotic methods in the theory of non-linear oscillations. New York (NY): Gordon & Breach; 1961.
   Google Scholar
• Hale JK , Verduyn Lunel SM . Averaging in infinite dimensions. J Int Equ Appl. 1990;2:463–494.
   Google Scholar
• Ilyin AA . Averaging principle for dissipative dynamical systems with rapidly oscillating right-hand sides. Sb Math. 1996;187:635–677.
   Web of Science ®Google Scholar
• Ilyin AA . Global averaging of dissipative dynamical systems. Rend Accad Naz Sci XL Mem Mat Appl. 1998;22:165–191.
   Google Scholar
• Efendiev M , Zelik S . Attractors of the reaction-diffusion systems with rapidly oscillating coefficients and their homogenization. Ann Inst H Poincaré Anal Non Linéaire. 2002;19:961–989.
   Web of Science ®Google Scholar
• Efendiev M , Zelik S . The regular attractor for the reaction-diffusion system with a nonlinearity rapidly oscillating in time and its averaging. Adv Differ Equ. 2003;8:673–732.
   Google Scholar
• Fiedler B , Vishik MI . Quantitative homogenization of analytic semigroups and reaction-diffusion equations with Diophantine spatial sequences. Adv Differ Equ. 2001;6:1377–1408.
   Google Scholar
• Fiedler B , Vishik MI . Quantitative homogenization of global attractors for reaction-diffusion systems with rapidly oscillating terms. Asymptotic Anal. 2003;34:159–185.
   Web of Science ®Google Scholar
• Chepyzhov VV , Goritsky AYu , Vishik MI . Integral manifolds and attractors with exponential rate for nonautonomous hyperbolic equations with dissipation. Russ J Math Phys. 2005;12:17–39.
   Web of Science ®Google Scholar
• Chepyzhov VV , Vishik MI , Wendland WL . On non-autonomous sine-Gordon type equations with a simple global attractor and some averaging. Disc Contin Dyn Syst. 2005;12:27–38.
   Web of Science ®Google Scholar
• Vishik MI , Chepyzhov VV . Approximation of trajectories lying on a global attractor of a hyperbolic equation with an exterior force that oscillates rapidly over time. Sb Math. 2003;194:1273–1300.
   Web of Science ®Google Scholar
• Zelik S . Global averaging and parametric resonances in damped semilinear wave equations. Proc R Soc Edinburgh Sect A. 2006;136:1053–1097.
   Web of Science ®Google Scholar
• Pankratov LS , Chueshov ID . Averaging of attractors of nonlinear hyperbolic equations with asymptotically degenerate coefficients. Sb Math. 1999;190:1325–1352.
   Web of Science ®Google Scholar
• Vishik MI , Chepyzhov VV . Averaging of trajectory attractors of evolution equations with rapidly oscillating terms. Sb Math. 2001;192:11–47.
   Web of Science ®Google Scholar
• Vishik MI , Fiedler B . Quantitative averaging of global attractors of hyperbolic wave equations with rapidly oscillating coefficients. Russ Math Surv. 2002;57:709–728.
   Web of Science ®Google Scholar
• Chepyzhov VV , Vishik MI . Non-autonomous 2D Navier-Stokes system with a simple global attractor and some averaging problems. ESAIM Control Optim Calc Var. 2002;8:467–487.
   Web of Science ®Google Scholar
• Chepyzhov VV , Vishik MI . Non-autonomous 2D Navier-Stokes system with singularly oscillating external force and its global attractor. J Dyn Differ Equ. 2007;19:655–684.
   Web of Science ®Google Scholar
• Chepyzhov VV , Pata V , Vishik MI . Averaging of nonautonomous damped wave equations with singularly oscillating external forces. J Math Pures Appl. 2008;90:469–491.
   Web of Science ®Google Scholar
• Chepyzhov VV , Pata V , Vishik MI . Averaging of 2D Navier-Stokes equations with singularly oscillating forces. Nonlinearity. 2009;22:351–370.
   Web of Science ®Google Scholar
• Chepyzhov VV , Vishik MI . Global attractors for non-autonomous Ginzburg-Landau equation with singularly oscillating terms. Rend Accad Naz Sci XL Mem Mat Appl. 2005;29:123–148.
   Google Scholar
• Vishik MI , Chepyzhov VV . Attractors of dissipative hyperbolic equations with singularly oscillating external forces. Math Notes. 2006;79:483–504.
   Web of Science ®Google Scholar
• Chueshov ID . SchmalfußB. Averaging of attractors and inertial manifolds for parabolic PDE with random coefficients. Adv. Nonlinear Stud. 2005;5:461–492.
   Web of Science ®Google Scholar
• Chepyzhov VV , Vishik MI . Evolution equations and their trajectory attractors. J Math Pures Appl. 1997;76(10):913–964.
   Web of Science ®Google Scholar
• Vishik MI , Chepyzhov VV . Trajectory attractors of equations of mathematical physics. Russ Math Surv. 2011;66:3–102.
   Web of Science ®Google Scholar
• Birkhoff GD . Proof of the ergodic theorem. Proc Natl Acad Sci USA. 1931;17(12):656–660.
   PubMedGoogle Scholar
• Arnol’d VI , Avez A . Ergodic problems of classical mechanics. New York (NY): W.A. Benjamin; 1968.
   Google Scholar
• Dunford N , Schwartz JT . Linear operators. Part I: general theory. New York (NY): Wiley; 1988.
   Google Scholar
• Zhikov VV . On Two-scale convergence. J Math Sci. 2003;120(3):1328–1352.
   Google Scholar
• Boyer F , Fabrie P . Mathematical tools for the study of the incompressible Navier-Stokes equations and related models. Applied mathematical sciences. vol. 183. New York (NY): Springer; 2013.
   Google Scholar
• Chepyzhov VV , Vishik MI . Trajectory attractors for reaction-diffusion systems. Top Meth Nonlinear Anal J Julius Schauder Center. 1996;7(1):49–76.
   Google Scholar
• Lions J-L . Quelques méthodes de résolutions des problèmes aux limites non linéaires. Paris: Dunod, Gauthier-Villars; 1969.
   Google Scholar