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

TL;DR

This paper investigates the detailed large-time behavior of solutions to the porous medium equation in a two-dimensional exterior domain, focusing on the near field scale to provide sharp decay rates and asymptotic profiles, extending previous work.

Contribution

It provides a precise characterization of the near field asymptotics for the porous medium equation in exterior domains, completing earlier results by Gilding and Goncerzewicz.

Findings

Sharp decay rates in the near field scale.

Asymptotic profiles for large time behavior.

Extension of previous asymptotic results.

Abstract

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, $x=\xi t^{\frac1{2m}}/(\log t)^{\frac{m-1}{2m}}$, $\xi\ne 0$, 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 $|x|=o\big(t^{\frac1{2m}}/(\log t)^{\frac{m-1}{2m}}\big)$, in the so called near field scale, except that the solution is $o\big((t\log t)^{-\frac1m}\big)$ 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].