Self-similar solutions of the p-Laplace heat equation: the fast diffusion case

Marie Françoise Bidaut-Véron

Vol. 227 (2006), No. 2, 201–269

DOI: 10.2140/pjm.2006.227.201

Abstract

We study the self-similar solutions of the equation u_t − div(|∇u|^p−2∇u) = 0 in ℝ^N, where N ≥ 1, p ∈ (1,2). We provide a complete description of the signed solutions of the form u(x,t) = (±t)^−α∕βw((±t)^−1∕β|x|), regular or singular at x = 0, with α,β real, β≠0, and possibly not defined on all of ℝ^N × (0,±∞).

Keywords

degenerate parabolic equations, self-similar solutions

Mathematical Subject Classification 2000

Primary: 35K65

Milestones

Received: 4 February 2005

Revised: 2 January 2006

Published: 1 October 2006