Homogenization of attractors to reaction - diffusion equations in domains with rapidly oscillating boundary: supercritical case

This paper is devoted to studying the reaction - diffusion systems with rapidly oscillating coefficients in the equations and in boundary conditions in domains with locally periodic oscillating boundary; on this boundary a Robin boundary condition is imposed. We consider the supercritical case, when the homogenization changes the Robin boundary condition on the oscillating boundary is to the homogeneous Dirichlet boundary condition in the limit as the small parameter, which characterizes oscillations of the boundary, tends to zero. In this case, we prove that the trajectory attractors of these systems converge in a weak sense to the trajectory attractors of the limit (homogenized) reaction - diffusion systems in the domain independent of the small parameter. For this aim we use the homogenization theory, asymptotic analysis and the approach of V.V. Chepyzhov and M.I. Vishik concerning trajectory attractors of dissipative evolution equations. The homogenization method and asymptotic analysis are used to derive the homogenized reaction - diffusion system and to prove the convergence of solutions. First we define the appropriate auxiliary functional spaces with weak topology, then, we prove the existence of trajectory attractors for these systems and formulate the main Theorem. Finally, we prove the main convergence result with the help of auxiliary lemmas.