Self-similar solutions of quasilinear parabolic equations with nonlinear gradient terms

This paper is devoted to study the classification of self-similar solutions to the m ≥ 1,p,q ＞ 0 and p + q ＞ m. For m = 1, it is shown that the very singular self-similar solution exists if and only if nq + (n + 1)p ＜ n + 2, and in case of existence, such solution is unique. For m ＞ 1, it is shown that very singular self-similar solutions exist if and only if 1 ＜ m ＜ 2 and nq + (n + 1)p ＜ 2 + mn, and such solutions have compact support if they exist. Moreover, the interface relation is obtained.