Traveling waves for unbalanced bistable equations with density dependent diffusion

Abstract

We study the existence and qualitative properties of traveling wave solutions for the unbalanced bistable reaction-diffusion equation with a rather general density dependent diffusion coefficient. In particular, it allows for singularities and/or degenerations as well as discontinuities of the first kind at a finite number of points. The reaction term vanishes at equilibria and it is a continuous, possibly non-Lipschitz function. We prove the existence of a unique speed of propagation and a unique traveling wave profile (up to translation) which is a non-smooth function in general. In the case of the power-type behavior of the diffusion and reaction near equilibria we provide detailed asymptotic analysis of the profile. For more information see https://ejde.math.txstate.edu/Volumes/2021/76/abstr.html