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

Marie-Françoise Bidaut-Véron (1)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique

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)}.

Mots clés: Self-similar solutions; quasilinear parabolic equations; dynamical systems

Domaines: Equations aux dérivées partielles [math.AP]

Marie-Françoise Bidaut-Véron. Self-similar solutions of the p-Laplace heat equation: the case when p>2. Proceedings of the Royal Society of Edinburg, 2009, 139A, pp.1-43. ⟨hal-00360982⟩