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 uct = [D(u)ux]x + g(u) has: 1. no travelling wave solutions for 0 c*. These fronts satisfy the boundary conditions ϕ(− ∞) = 1, ϕ'(− ∞) = ϕ(+ ∞) = ϕ'(+ ∞) = 0. We illustrate our analytical results with some numerical solutions.