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

Abstract

We study the self-similar solutions of the equation \[ u_{t}-div(| \nabla u| ^{p-2}\nabla u)=0, \] in ${\mathbb{R}}^N,$ when $p>2.$ We make a complete study of the existence and possible uniqueness of solutions of the form \[ u(x,t)=(\pm t)^{-α/β}w((\pm t)^{-1/β}| x|) \] of 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 focussing hole (for $t<0).$ When $t<0,$ we show the existence of positive solutions oscillating around the particular solution $U(x,t)=C_{N,p}(|x{|}^p/(-t){)}^{1/(p-2)}.$