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 thep-Laplace equa- tion posed in an exterior domain in space dimensionN < p with boundary conditionsuD 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 D 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 solu- tions. 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. We are concerned with understanding the effect of the presence of one or several holes in the domain on the large-time behaviour of the solutions of nonlinear diffusion equations. In this paper we study the question for the evolutionp-Laplace equation and find interest- ing non-standard asymptotics. To be specific, we consider an exterior domainD R N nG whereG is a bounded domain in R N with smooth boundary, and study the asymptotic be- haviour of the solutions of the exterior Dirichlet problem with zero boundary conditions: 8