Anomalous large-time behaviour of the $p$-Laplacian flow in an exterior domain in low dimension

Abstract

We study the large time behaviour of weak nonnegative solutions of the $p$-Laplace equation posed in an exterior domain in space dimension $N<p$ with boundary conditions $u=0$. The description is done in terms of matched asymptotics: the outer asymptotic profile is a dipole-like self-similar solution with a singularity at $x=0$ and anomalous similarity exponents. The inner asymptotic behaviour is given by a separate-variable profile. We gather both estimates in a global approximant and we also study the behaviour of the free boundary for compactly supported solutions. We complete in this way the analysis made in a previous work for high space dimensions $N\ge p$, a range in which the large-time influence of the holes is less dramatic.