Traveling Wave Phenomena in Some Degenerate Reaction-Diffusion Equations

In this paper we study the existence of travelling wave solutions (t.w.s.), u(x, t)=φ(x−ct) for the equation [formula]+g(u), (*) where the reactive part g(u) is as in the Fisher-KPP equation and different assumptions are made on the non-linear diffusion termD(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 u (0, 1], D and g C2[0,1], D′(0)≠0 and D′′(0)≠0. 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 0c*, 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(0)=D′(0)=0, D and g C2[0,1], D′′(0)≠0. If, in addition, we impose D′′(0)>0 with D(u)>0 u (0, 1], We give sufficient conditions on c for the existence of t.w.s. of front type. Meanwhile if D′′(0)<0 with D(u)<0 u (0, 1] we analyse just one example (D(u)=−u2, and g(u)=u(1−u)) which has oscillatory t.w.s. for 02. 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.