Traveling Wave Phenomena in Some Degenerate Reaction-Diffusion Equations

Abstract 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 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 ∀ u ∈(0, 1], D and g ∈ C 2 [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 ∈ C 2 [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) D ( u ) u ∈(0, 1] we analyse just one example ( D ( u )=− u 2 , 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.