Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equations - Sánchez-Garduño, F; Maini, P (1994)
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 $u(x, t) = φ(x - c^{*} t)$ of sharp type satisfying $φ(-∞) = 1$, $φ(τ) = 0$ $\forall τ≥τ^{*}$; $φ'(τ^{*-}) = -c^{*}/D'(0),φ'(τ^{*+}) = 0$ and 3. a continuum of travelling wave solutions of monotone decreasing front type for each c > c *. These fronts satisfy the boundary conditions $φ(–∞) = 1$, $φ'(–∞)=φ(+∞) = φ'(+∞) = 0$. We illustrate our analytical results with some numerical solutions.