Patterns in Parabolic Problems with Nonlinear Boundary Conditions

In this paper we show the existence of stable nonconstant equilibrium (patterns) for reaction-diffusion equations with nonlinear boundary conditions on small domains connected by thin channels. We prove the convergence of eigenvalues and eigenfunctions of the Laplace operator is such domains. This information is used to show that the asymptotic dynamic of the heat equations in this domain is equivalent to the asymptotic dynamics of a two ordinary differential equations diffusively (weakly) coupled. The main tools employed are the invariant manifold theory and uniform trace theorem.