SELF-SIMILAR SINGULAR SOLUTION OF DOUBLY SINGULAR PARABOLIC EQUATION WITH GRADIENT ABSORPTION TERM

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 ex-istence, the self-similar singular solution is the self-similar very singular solutions which have compact support. Moreover, the interface relation is obtained. Copyright © 2006 P. Shi and M. Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction an