Sharp estimates and finite speed of propagation for a Neumann problem in domains narrowing at infinity

Abstract

We investigate the connection between the geometry of an unbounded domain $\Omega$ and the existence and qualitative behaviour of solutions to a degenerate doubly non linear parabolic equation posed in $\Omega$. The domain $\Omega$ is assumed to be ``narrowing'' at infinity in a suitable sense, so that it has infinite volume. On the boundary of $\Omega$ we prescribe a homogeneous Neumann condition. Among other results, we prove sharp estimates for the finite speed of propagation of the support of positive solutions originating from initial data with bounded support. This is done by means of a new approach, which is flexible enough to be applied to the geometry at hand, and to cover the case of initial data measures. We also show that even if the initial datum has finite mass, the solution need not be globally bounded over $\Omega$ for a fixed positive time. We provide sharp estimates for such solutions. Our main tool is a new embedding inequality connected with the geometry of $\Omega$.