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

TL;DR

This paper thoroughly investigates self-similar solutions of the p-Laplace heat equation for p>2, revealing diverse behaviors including expanding, shrinking, and oscillating solutions, and establishing existence and uniqueness results.

Contribution

It provides a complete classification of self-similar solutions for p>2, including regular, singular, and oscillatory cases, with detailed existence and uniqueness analysis.

Findings

Solutions with expanding and shrinking support identified

Existence of oscillating solutions around a particular solution proven

Classification of solutions based on sign and support behavior

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