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.
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Juan Luis Vázquez, Razvan Gabriel Iagar, Anomalous large-time behaviour of the -Laplacian flow in an exterior domain in low dimension. J. Eur. Math. Soc. 12 (2010), no. 1, pp. 249–277DOI 10.4171/JEMS/197