On the asymptotic behaviour of the pure Neumann problem in cylinder-like domains and its applications

We consider in this paper the pure Neumann problem in n-dimensional cylinder-like domains. We are interested in the asymptotic behaviour of the solution of this kind of problems when the domain becomes infinite in p-directions, 1 ≤ p < n. We show that this solution converges exponentially to the solution of a Neumann problem in the corresponding unbounded domain. We distinguish between the case p = 1 and 1 < p < n the latter requiring a more involved analysis. For p = 1 we consider also the special situation when the domain and the initial data are periodic.