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

We consider the porous medium equation in an exterior two-dimensional domain which excludes a hole, with zero Dirichlet data on its boundary. Gilding and Goncerzewicz proved in [Gilding-Goncerzewicz-2007] that in the far field scale, $x=\xi t^{\frac{1}{2m}}\mathrm{/}(\log t{)}^{\frac{m-1}{2m}}$, $\xi \mathrm{\ne }0$, solutions to this problem with an 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. However, their result does not say much about the behavior when $\mathrm{\mid }x\mathrm{\mid }=o(t^{\frac{1}{2m}}\mathrm{/}(\log t{)}^{\frac{m-1}{2m}})$, in the so called near field scale, except that the solution is $o((t\log t{)}^{-\frac{1}{m}})$ there. In particular, it does not give a sharp decay rate, neither a nontrivial asymptotic profile, on compact sets. In this paper we characterize the large time behavior in such scale, thus completing the results of [Gilding-Goncerzewicz-2007].