Homogenization of Attractors to Reaction–Diffusion Equations in Domains with Rapidly Oscillating Boundary: Subcritical Case

Abstract

We consider the reaction–diffusion system of equations with rapidly oscillating terms in the equation and in boundary conditions in a domain with locally periodic oscillating boundary. In the subcritical case (the Fourier boundary condition is changed to the Neumann boundary condition in the limit) we proved that the trajectory attractors of this system converge in a weak sense to the trajectory attractors of the limit (homogenized) reaction–diffusion systems in domain independent of the small parameter, characterizing the oscillation rate. To obtain the results we use the approach of homogenization theory, asymptotic analysis and methods of the theory concerning trajectory attractors of evolution equations. Defining the appropriate functional and topological spaces with weak topology, we prove the existence of trajectory attractors and global attractors for these systems. Then we formulate the main Theorem and prove it with the help of auxiliary Lemmata. Applying the homogenization method and asymptotic analysis we derive the homogenized (limit) system of equations, prove the existence of trajectory attractors and global attractors and show the convergence of trajectory and global attractors.