Journal of Inequalities and Applications 
Volume 2006 (2006), Article ID 20438, 21 pages
doi:10.1155/JIA/2006/20438

Self-similar singular solution of doubly singular parabolic equation with gradient absorption term

Peihu Shi1 and Mingxin Wang1,2

 1Department of Mathematics, Southeast University, Nanjing 210096, China
2Department of Mathematics, Xuzhou Normal University, Xuzhou 221116, China
Received 6 February 2004; Revised 8 January 2005; Accepted 8 January 2005

Abstract

We deal with the self-similar singular solution of doubly singular parabolic equation with a gradient absorption term ut=div(|∇um|p−2∇um)−|∇u|q for p>1, m(p−1)>1 and q>1 in ℝn×(0,∞). By shooting and phase plane methods, we prove that when p>1+n/(1+mn)q+mn/(mn+1) there exists self-similar singular solution, while p≤n+1/(1+mn)q+mn/(mn+1) there is no any self-similar singular solution. In case of existence, the self-similar singular solution is the self-similar very singular solutions which have compact support. Moreover, the interface relation is obtained.