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Similarity solutions for a quasilinear parabolic equation

Published online by Cambridge University Press:  17 February 2009

Jong-Sheng Guo
Jong-Sheng Guo
Affiliation:
Institute of Applied Mathematics, National Tsing Hua University, Hsinchu, Taiwan30043, R.O.C
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Abstract

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In this paper, we use an ordinary differential equation approach to study the existence of similarity solutions for the equation u1 = Δ(uα) + θu–β in Rn × (0, ∞) where β > 0, θ ∈ [0, 1}, and n ≥ 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 form c|x|p, where p = 2/(α + β) if θ = 1; p ≥ 0 and 2 + (1 – α)p > 0 if θ = 0, for some positive constant c.

Type
Research Article
Information
The ANZIAM Journal , Volume 37 , Issue 2 , October 1995 , pp. 253 - 266
Copyright
Copyright © Australian Mathematical Society 1995

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