Characterization of the limit of some higher dimensional thin domain problems

Abstract

A reaction-diffusion equation on a family of three dimensional thin domains, collapsing onto a two dimensional subspace, is considered. In [The effect of domain squeezing upon the dynamics of reaction-diffusion equations, J. Differential Equations 173 (2001), 271–320] it was proved that, as the thickness of the domains tends to zero, the solutions of the equations converge in a strong sense to the solutions of an abstract semilinear parabolic equation living in a closed subspace of $H^1$. Also, existence and upper semicontinuity of the attractors was proved. In this work, for a specific class of domains, the limit problem is completely characterized as a system of two-dimensional reaction-diffusion equations, coupled by mean of compatibility and balance boundary conditions.