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) = phi(x-ct) for the equation () partial derivative u/partial derivative t=partial derivative/partial derivative x[D(u)partial derivative u/partial derivative x]+g(u), where the reactive part g(u) is as in the Fisher-KPP equation and different assumptions are made on the Mon-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 For All u is an element of(0, 1], D and g is an element of C-[0 ,C-1] (2), D'(0)not equal 0 and D ''(0)not equal 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 is an element of C-[0,C- 1] (2), D ''(0)not equal 0. If, in addition, we impose D ''(0)>0 with D(u)>0 For All u is an element of(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 For All u is an element of(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 less than or equal to 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. (C) 1995 Academic Press, Inc.