TRAVELING-WAVE PHENOMENA IN NONLINEAR DIFFUSION DEGENERATE NAGUMO EQUATIONS

In this paper we study the existence of one-dimensional travelling wav e solutions u(x, t) = phi(x - ct) for the non-linear degenerate (at u = 0) reaction-diffusion equation u(t) = [D(u)u(x)](x) + g(u) where g i s a generalisation of the Nagumo equation arising in nerve conduction theory, as well as describing the Allee effect. We use a dynamical sys tems approach to prove: 1. the global bifurcation of a heteroclinic cy cle (two monotone stationary front solutions), for c = 0, 2. The exist ence of a unique value c > 0 of c for which phi(x - c*t) is a travell ing wave solution of sharp type and 3. A continuum of monotone and osc illatory fronts for c not equal c. We present some numerical simulati ons of the phase portrait in travelling wave coordinates and on the fu ll partial differential equation.