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Self-similar solutions to the generalized deterministic KPZ equation. (English) Zbl 1024.35039
Summary: Geometric properties of shape functions of self-similar solution to the equation \(u_t = u_{xx} + \lambda|u_x|^q\) 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.

MSC:
35K55 Nonlinear parabolic equations
34A05 Explicit solutions, first integrals of ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
35K65 Degenerate parabolic equations
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