The Self-similar Profiles of Generalized Kpz Equation

In this paper we consider, for 1 ≤ m < p < 2, the generalized KPZ equation ut = (u) − |∇u|p. For m = 1, we show existence and uniqueness of the so called very singular solution which is self-similar. A complete classification of self-similar solutions is also given. For m > 1, we establish the existence of very singular self-similar solution and prove that such a solution must have compact support. Moreover, we derive the interface relation. Recent experience with parallel equations where the gradient term |∇u|p is replaced by u indicates that the self-similar solutions are crucially important in study intermediate asymptotic behavior of general solutions.