Self-similar solutions of the p-Laplace heat equation: the case when p>2
Résumé
We study the self-similar solutions of the equation u_{t}-div(|∇u|^{p-2}∇u)=0, in R^{N}, when p>2. We make a complete study of the existence and possible uniqueness of solutions of the form u(x,t)=(±t)^{-α/β}w((±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)}.
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