Existence and uniqueness of a sharp travelling wave in degenerate non-linear 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 $\phi(-\infty) = 1$, $\phi(\tau) = 0$ $\forall \tau\geq\tau^{*}$; $\phi'(\tau^{*-}) = -c^{*}/D'(0),\phi'(\tau^{*+}) = 0$ and 3. a continuum of travelling wave solutions of monotone decreasing front type for each c > c *. These fronts satisfy the boundary conditions $\phi(–\infty) = 1$, $\phi'(–\infty)=\phi(+\infty) = \phi'(+\infty) = 0$. We illustrate our analytical results with some numerical solutions