On the Form of Smooth-Front Travelling Waves in a Reaction-Diffusion Equation with Degenerate Nonlinear Diffusion

Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK

^* Corresponding author: E-mail: jas@ma.hw.ac.uk

Abstract

Reaction-diffusion equations with degenerate nonlinear diffusion are in widespread use as models of biological phenomena. This paper begins with a survey of applications to ecology, cell biology and bacterial colony patterns. The author then reviews mathematical results on the existence of travelling wave front solutions of these equations, and their generation from given initial data. A detailed study is then presented of the form of smooth-front waves with speeds close to that of the (unique) sharp-front solution, for the particular equation u_t = (uu_x)_x + u(1 − u). Using singular perturbation theory, the author derives an asymptotic approximation to the wave, which gives valuable information about the structure of smooth-front solutions. The approximation compares well with numerical results.