HOMOGENIZATION OF ATTRACTORS OF THE CHAFEE–INFANTE EQUATIONS IN A PERFORATED DOMAIN

This work addresses the averaging problem for the Chafee–Infante equation in a micro-heterogeneous medium. It analyzes the equation with rapidly oscillating terms and dissipation. The model introduces a small parameter and formulates an averaged problem based on its limits. The study proves that the trajectory attractors of the original equation converge to those of the averaged equation. It establishes the relevant theorem. In cases with unique solutions, it investigates the convergence of global attractors, adding further conditions on the nonlinear terms. The work also analyzes singular problems with nontrivial boundary conditions in perforated domains and on cavity boundaries. Here, the averaged equation differs from the original, reflecting the model’s effective averaged characteristics and possibly an extra potential term. As the small parameter approaches zero, the study shows that the attractors converge in Hausdorff distance. The main result is that the attractors of the original Chafee– Infante equation converge to those of the averaged (limit) equation as the small parameter approaches zero. This work establishes this convergence for the first time in the context of homogenization in a periodically perforated medium, highlighting the scientific novelty and relevance of the findings. This research advances averaging methods for differential operators and may apply to solutions of nonlinear differential equations.

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