Self-similar solutions of the p-Laplace heat equation: the case when p>2
Abstract
We study the self-similar solutions of the equation \[ ut-div(| ∇ u| p-2∇ u)=0, \] in RN, when p>2. We make a complete study of the existence and possible uniqueness of solutions of the form \[ u(x,t)=( t)-α/βw(( t)-1/β| x|) \] 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)=CN,p(| x| p/(-t))1/(p-2).
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