EXISTENCE AND UNIQUENESS OF A SHARP TRAVELING-WAVE IN DEGENERATE NONLINEAR DIFFUSION FISHER-KPP EQUATIONS

In this paper we use a dynamical systems approach to prove the existence of a unique critical value c* of the speed c for which the degenerate density-dependent diffusion equation u(t)=[D(u)u(x)](x)+g(u) has: 1. no travelling wave solutions for 0<c<c*, 2. a travelling wave solution u(x,t)= phi(x-c*t) of sharp type satisfying phi(-infinity)=1, phi(tau)=0 For All tau greater than or equal to tau*; phi'(tau*(-))=-c*/D'(0), phi'(tau*(+))=0 and 3. a continuum of travelling wave solutions of monotone decreasing front type for each c>c*. These fronts satisfy the boundary conditions phi(-infinity)=1, phi'(-infinity)=phi(+infinity)=phi'(+infinity)=0. We illustrate our analytical results with some numerical solutions.