Travelling waves in the Fisher-KPP equation with nonlinear diffusion and a non-Lipschitzian reaction term

TL;DR

This paper studies how nonlinear and possibly degenerate diffusion, combined with non-Lipschitz reaction terms, affects the existence and properties of travelling wave solutions in a one-dimensional Fisher-KPP type equation.

Contribution

It analyzes the impact of degenerate or singular diffusion coefficients and non-differentiable reaction terms on travelling wave existence and characteristics.

Findings

Degenerate or singular diffusion influences wave existence.

Non-Lipschitz reaction terms alter wave properties.

Conditions for existence and nonexistence of travelling waves.

Abstract

We consider a one-dimensional reaction-diffusion equation of Fisher-Kolmogoroff-Petrovsky-Piscounoff type. We investigate the effect of the interaction between the nonlinear diffusion coefficient and the reaction term on the existence and nonexistence of travelling waves. Our diffusion coefficient is allowed to be degenerate or singular at both equilibrium points, 0 and 1, while the reaction term need not be differentiable. These facts influence the existence and qualitative properties of travelling waves in a substantial way.