The self-similar profiles of generalized KPZ equation

Abstract

In this paper we consider, for 1 m < p < 2, the generalized KPZ equation u t = (u m ) -|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 p indicates that the self-similar solutions are crucially important in study intermediate asymptotic behavior of general solutions.