Homogenization of attractors to reaction–diffusion equations in domains with rapidly oscillating boundary: Critical case
Azhmoldaev G.F. Bekmaganbetov K.A. Chechkin G.A. Chepyzhov V.V.
2024American Institute of Mathematical Sciences
Networks and Heterogeneous Media
2024#19Issue 31381 - 1401 pp.
In the present paper, reaction–diffusion systems (RD-systems) with rapidly oscillating coefficients and righthand sides in equations and in boundary conditions were considered in domains with locally periodic oscillating (wavering) boundary. We proved a weak convergence of the trajectory attractors of the given systems to the trajectory attractors of the limit (homogenized) RD-systems in domain independent of the small parameter, characterizing the oscillation rate. We consider the critical case in which the type of boundary condition was preserved. For this aim, we used the approach of Chepyzhov and Vishik concerning trajectory attractors of evolutionary equations. Also, we applied the homogenization (averaging) method and asymptotic analysis to derive the limit (averaged) system and to prove the convergence. Defining the appropriate axillary functional spaces with weak topology, we proved the existence of trajectory attractors for these systems. Then, we formulated the main theorem and proved it with the help of auxiliary lemmata.
attractors , homogenization , nonlinear PDE, convergence in a weak sense , rapidly oscillating (wavering) boundary , reaction–diffusion equations
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Department of Fundamental Mathematics, L.N. Gumilyov Eurasian National University, Astana, Kazakhstan
Department of Fundamental and Applied Mathematics, Kazakhstan Branch of M.V. Lomonosov Moscow State University, Astana, Kazakhstan
Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
Department of Differential Equations, Faculty of Mechanics and Mathematics, M.V. Lomonosov Moscow State University, Moscow, Russian Federation
Institute of Mathematics with Computing Center Subdivision of the Ufa Federal Research Center of Russian Academy of Science, Ufa, Russian Federation
Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russian Federation
Higher School of Economics (NRU), Nizhny Novgorog, Russian Federation
Department of Fundamental Mathematics
Department of Fundamental and Applied Mathematics
Institute of Mathematics and Mathematical Modeling
Department of Differential Equations
Institute of Mathematics with Computing Center Subdivision of the Ufa Federal Research Center of Russian Academy of Science
Institute for Information Transmission Problems
Higher School of Economics (NRU)
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