Self-similar solutions of the p-Laplace heat equation: the case when p > 2

Abstract

We study the self-similar solutions of the equationin ℝN, when p > 2. We make a complete study of the existence and possible uniqueness of solutions of the formof any sign, regular or singular at x = 0. Among them we find solutions with an expanding compact support or a shrinking hole (for t > 0), or a spreading compact support or a focusing hole (for t < 0). When t < 0, we show the existence of positive solutions oscillating around the particular solution $\smash{U(x,t)=C_{N,p}(|x|^{p}/(-t))^{1/(p-2)}}$.