Travelling wave phenomena in some degenerate reaction-diffusion equations

Journal article

In this paper we study the existence of travelling wave solutions (t.w.s.), $u(x, t)=\phi(x−ct)$ for the equation $u_t=[D(u)u_x]_x+g(u) (*)$ 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 $\forall u \in (0, 1]$, D and $g\in C^{2}_{[0,1]}$, $D'(0)\neq0$ and $D''(0)\neq 0$. 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 $g \in C^{2}_{[0,1]}$, $D''(0)\neq 0$. If, in addition, we impose $D''(0)>0$ with $D(u)>0$ $\forall u(0, 1]$, We give sufficient conditions on c for the existence of t.w.s. of front type. Meanwhile if $D''(0)<0$ with $D(u)<0$ $\forall u\in (0, 1]$ we analyse just one example ($D(u)=-u^2$, 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.

Authors: Sánchez-Garduño, F; Maini, P

• Publication date: 1994-01-01