HOMOGENIZATION OF ATTRACTORS TO THE REACTION-DIFFUSION SYSTEM IN A DOMAIN WITH ROUGH BOUNDARY

Abstract

In this paper, we consider the homogenization problem in a micro inhomogeneous domain with a rapidly oscillating
boundary. It is assumed that a system of nonlinear reaction--diffusion equations with rapidly oscillating terms
and dissipation is considered in the domain. On the locally periodic oscillating part of the boundary, the third
boundary condition with rapidly oscillating coefficients and a small parameter characterizing the oscillation of
the boundary to some degree is imposed. Depending on the degree of the small parameter in the boundary condition,
various homogenized (limit) problems are obtained and the convergence of the trajectory attractors of the given
system to the attractors of the homogenized system is proved. Critical, subcritical and supercritical cases of
attractor behavior as the small parameter tends to zero are carefully studied. The paper also considers problems
in a domain with a random rapidly oscillating boundary. In this case, a homogenized system of reaction--diffusion
equations with deterministic coefficients is obtained in the case of a statistically homogeneous random structure
of the boundary. A theorem on the convergence of random trajectory attractors of the initial given system of
reaction-diffusion equations to a deterministic attractor of the homogenized (limit) system of reaction--diffusion
equations is also proved. The paper also proves the convergence of global attractors in the case of uniqueness of
solutions, which in turn is proved for nonlinearity in a system of equations of a special type.