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

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

2009-02-13

arxiv.org/abs/0902.2311v1

Proceedings of the Royal Society of Edinburg 139A (2009) 1-43

Mathematics

Analysis of PDEs

Scientific paper

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 \[ u(x,t)=(\pm t)^{-\alpha/\beta}w((\pm t)^{-1/\beta}| x|) \] 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 $U(x,t)=C_{N,p}(| x| ^{p}/(-t))^{1/(p-2)}.$

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