Traveling waves of quasilinear reaction-diffusion equations with discontinuous diffusivity

We are concerned with traveling wave solutions to a class of quasilinear reaction-diffusion equations on the real line. We consider two types of continuous reaction term frequently found in applications - bistable and monostable. The diffusion coefficient is only piecewise continuous in (0,1) and allows for degenerations as well as singularities at 0 and 1. Under these general assumptions, we establish the existence of non-smooth traveling wave profiles. Uniqueness is shown in the bistable case, while a monostable reaction can give rise to a continuum of admissible wave speeds. Our approach is based on the investigation of an equivalent first order problem in the sense of Carathéodory and provides a broad theoretical background for mathematical treatment of various phenomena in population dynamics, chemistry and physics. Assuming power-type behaviour of the reaction and diffusion terms near the equilibria, we also discuss some asymptotic properties of the solutions.

This is a joint work with Pavel Drábek.