New patterns of travelling waves in the generalized Fisher-Kolmogorov equation

TL;DR

This paper establishes the existence, uniqueness, and detailed properties of traveling wave solutions in a generalized Fisher-Kolmogorov equation, extending classical models with nonlinear diffusion and nonsmooth reactions.

Contribution

It introduces a rigorous analysis of traveling waves in a degenerate quasilinear PDE, generalizing the classical Fisher-Kolmogorov equation with new existence and shape results.

Findings

Existence and uniqueness of traveling waves proven.

Wave shape and asymptotic behavior characterized.

Dependence on nonlinear diffusion and reaction functions clarified.

Abstract

We prove the existence and uniqueness of a family of travelling waves in a degenerate (or singular) quasilinear parabolic problem that may be regarded as a generalization of the semilinear Fisher-Kolmogorov-Petrovski-Piscounov equation for the advance of advantageous genes in biology. Depending on the relation between the nonlinear diffusion and the nonsmooth reaction function, which we quantify precisely, we investigate the shape and asymptotic properties of travelling waves. Our method is based on comparison results for semilinear ODEs.