Self-similar solutions to the generalized deterministic KPZ equation

Abstract.

Geometric properties of shape functions of self-similar solution to the equation $u_t = u_{xx} + \lambda\vert u_x\vert^q$ are studied, $\lambda$ and q are positive numbers. These shapes-the solutions of the corresponding nonlinear ODE-are of very different nature. The properties usually depend on three critical values of q (1, 3/2 and 2). For the range 1<q<2 the dependence of $\lambda$ is more remarkable, for example there is no global existence in general.