A Review on Travelling Wave Solutions of One-Dimensional Reaction-Diffusion Equations with Non-Linear Diffusion Term

Faustino Sánchez-Garduño¹, Philip K. Maini² and E. Kappos³

¹Departamento de Matemáticas, Facultad de Ciencias, UNAM, Circuito Exterior, C.U., México 04510, D.F., México
² Centre for Mathematical Biology, Mathematical Institute, University of Oxford, 24-29 St. Giles', Oxford OX1 3LB, U.K.
³ Applied Mathematics Section, School of Mathematics and Statistics, University of Sheffield, Sheffield S10 2UN, U.K.

(Received November 15, 1995; Accepted December 20, 1995)

Keywords: Density Dependent Diffusion, Fisher-KPP, Nagumo, Ill-Posed Problems

Abstract. In this paper we review the existence of different types of travelling wave solutions u(x,t) = f(x - ct) of degenerate non-linear reaction-diffusion equations of the form ut = [D(u)ux]x + g(u) for different density-dependent diffusion coefficients D and kinetic part g. These include the non-linear degenerate generalized Fisher-KPP and the Nagumo equations. Also, we consider an equation whose diffusion coefficient changes sign as the diffusive substance increases. This describes a diffusive-aggregative process. In this case the travelling wave solutions are explored and the ill-posedness of two boundary-value problems associated with the above equation is stated.