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 existen ce of a unique critical value c of the speed c for which the degenera te 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 soluti on u(x,t)= phi(x-ct) of sharp type satisfying phi(-infinity)=1, phi(t au)=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 th e boundary conditions phi(-infinity)=1, phi'(-infinity)=phi(+infinity) =phi'(+infinity)=0. We illustrate our analytical results with some num erical solutions.