J Austral Math Soc Ser B 37 pp253--266, 1995.

Similarity solutions for a quasilinear parabolic equation

Jong-Sheng Guo

Abstract

In this paper, we use an ordinary differential approach to study the existence of similarity solutions for the equation u_t = D(u^a) + qu^-b in Rⁿ × (0, ¥), where a > 0, b > 0, q Î {0, 1}, and n ³ 1. This includes the slow diffusion equation when a > 1, the standard heat equation when a = 1, and the fast diffusion equation when 0 < a < 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 / (a + b) if q = 1; p ³ 0 and 2 + (1 - a) p > 0 if q = 0, for some positive constant c.

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Author

Jong-Sheng Guo

Institute of Applied Mathematics, National Tsing Hua University, Hsinchu, Taiwan 30043, R.O.C.