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 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(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 0<c≤2 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. © 1995 by Academic Press, Inc.