Asymptotic behavior for the filtration equation in domains with noncompact boundary

We consider the initial value boundary problem with zero Neumann data for an equation modelled after the porous media equation, with variable coefficients. The spatial domain is unbounded and shaped like a (general) paraboloid, and the solution u is integrable in space and non-negative. We show that the asymptotic profile for large times of u is one-dimensional and given by an explicit function, which can be regarded as the fundamental solution of a one-dimensional differential equation with weights. In the case when the domain is a cone or the whole space (Cauchy problem) we obtain a genuine multi-dimensional profile given by the well known Barenblatt solution.