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.

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