Travelling wave phenomena in some degenerate reaction-diffusion equations

Abstract

In this paper we study the existence of travelling wave solutions (t.w.s.), for the equation



where the reactive part g(u) is as in the Fisher-KPP equation and different assumptions are made on the non-linear diffusion term D(u). Both functions D and g are defined on the interval [0, 1]. The existence problem is analysed in the following two cases.

Case 1. D(0)=0, D(u)>0 , D and , and . We prove that if there exists a value of c, c*, for which the equation (*) possesses a travelling wave solution of sharp type, it must be unique. By using some continuity arguments we show that: for 0<c<c*, there are no t.w.s., while for c>c*, the equation (*) has a continuum of t.w.s. of front type. The proof of uniqueness uses a monotonicity property of the solutions of a system of ordinary differential equations, which is also proved.

Case 2. , D and , . If, in addition, we impose with , We give sufficient conditions on c for the existence of t.w.s. of front type. Meanwhile if with we analyse just one example (, and ) which has oscillatory t.w.s. for and t.w.s. of front type for c>2. In both the above cases we use higher order terms in the Taylor series and the Centre Manifold Theorem in order to get the local behaviour around a non-hyperbolic point of codimension one in the phase plane.