Near-field asymptotics for the porous medium equation in exterior domains: The critical two-dimensional case

Abstract

We consider the porous medium equation in an exterior two-dimensional domain that excludes a hole, with zero Dirichlet data on its boundary. Gilding and Goncerzewicz proved in 2007 that in the far-field scale, which is the adequate one to describe the movement of the free boundary, solutions to this problem with integrable and compactly supported initial data behave as an instantaneous point-source solution for the equation with a variable mass that decays to 0 in a precise way, determined by the initial data and the hole. In this paper, starting from their result in the far field, we study the large time behavior in the near field, in scales that evolve more slowly than the free boundary. In this way we get, in particular, the final profile and decay rate on compact sets. Spatial dimension two is critical for this problem, and involves logarithmic corrections.