Complete classification of shape functions of self-similar solutions

In this paper we study some asymptotic profiles of shape functions of self-similar solutions to the initial-boundary value problem with Neumann boundary condition for the generalized KPZ equation: ut=uxx−q|ux|, where q is positive number. The shapes of solutions of the corresponding nonlinear ordinary differential equation are of very different nature. The properties depend on the critical value and initial data as usual.