Wave Asymptotic Solutions of Degenerate Semilinear Parabolic and Hyperbolic Equations

Abstract

This chapter presents a method for constructing nonsmooth asymptotic solutions uniformly bounded as a small parameter ε tends to zero. These solutions have semi-bounded (or compact) supports and week singularities on the boundary of supports. This is the difference from well-known asymptotic solutions describing the propagation of shock waves in media with low viscosity, or soliton-like waves in weakly dispersive media [55, 58, 59, 94]. The solutions constructed in this chapter generalize simple diffusion waves, known since the publication of the classical work by Kolmogorov, Petrovskii and Piskunov [41] (see also [107]).