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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 < c*, 2. a travelling wave solution u(x, t) = ϕ(x - c*t) of sharp type satisfying ϕ(− ∞) = 1, ϕ(τ) = 0 ∀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. © 1994, Springer-Verlag. All rights reserved.

Original publication




Journal article


Journal of Mathematical Biology

Publication Date





163 - 192