Classification of weak and unbounded traveling wave solutions for a Porous–Fisher-KPP equation

This paper reports results on the classification of non-negative traveling wave solutions, including the weak sense, in the Fisher-KPP equation with nonlinear diffusion in one-dimensional space. This is done using dynamical systems theory and geometric approaches (in particular, Poincaré compactification). The classification of traveling wave solutions refers to the enumeration of those that exist and the presentation of information about each solution, such as its shape and asymptotic behavior. The key idea is to give all dynamical systems, including the 2-dimensional ordinary differential equation systems to infinity that characterize traveling waves, and to classify all connecting orbits. The classification then gives a classification of traveling waves that combines the concept of weak solutions and the flux condition proposed in previous studies. This idea not only gives a previously unexplained classification of unbounded traveling waves, but also points out that the information about the shape of the traveling wave near the singularity of a sharp-type traveling wave and its asymptotic behavior varies depending on the parameters included in the equation.