*Forma,* Vol. 11 (No. 1), pp. 45-59, 1996

*Review*

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

Faustino Sánchez-Garduño^{1}, Philip K. Maini^{2} and E. Kappos^{3}

^{1}Departamento 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.