Large time and space size behaviour of the heat equation in non-cylindrical domains

In this note we consider a linear parabolic problem defined on a non-cylindrical unbounded domain Q. If Ωt denotes the section of Q above t, the Ωt size goes to +∞, when t → +∞, i.e. the sections Ωt become unbounded in some directions when the time t becomes large. Here a model problem is studied, but the technique used can be applied for a wide class of problems, as nonlinear ones, defined on more general domains Q as those introduced by Lions [11]. An asymptotically exponential convergence of the solutions of such problems towards the solution of an elliptic problem defined on a lower dimensional domain is established.