Parabolic problems in highly oscillating thin domains

Abstract

In this work, we consider the asymptotic behavior of the nonlinear semigroup defined by a semilinear parabolic problem with homogeneous Neumann boundary conditions posed in a region of \({\mathbb {R}}^2\) that degenerates into a line segment when a positive parameter \(\epsilon \) goes to zero (a thin domain). Here we also allow that its boundary presents highly oscillatory behavior with different orders and variable profile. We take thin domains possessing the same order \(\epsilon \) to the thickness and amplitude of the oscillations, but assuming different order to the period of oscillations on the top and the bottom of the boundary. Combining methods from linear homogenization theory and the theory on nonlinear dynamics of dissipative systems, we obtain the limit problem establishing convergence properties for the nonlinear semigroup, as well as the upper semicontinuity of the attractors and stationary states.