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 ≥ p, a range in which the large-time influence of the holes is less dramatic. © European Mathematical Society 2010.