Similarity solutions for a quasilinear parabolic equation

Abstract

AbstractIn this paper, we use an ordinary differential equation approach to study the existence of similarity solutions for the equationu1= Δ(uα) + θu–βin Rn× (0, ∞) where β > 0, θ ∈ [0, 1}, andn≥ 1. This includes the slow diffusion equation when α > = 1, and the standard heat equation when α = 1, and the fast diffusion equation when 0 < α < 1. We prove that there are forward self-similar solutions for this equation with initial data of the formc|x|p, wherep= 2/(α + β) if θ = 1;p≥ 0 and 2 + (1 – α)p> 0 if θ = 0, for some positive constantc.