Abstract
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 ct = [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 ∀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.
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Sánchez-Garduño, F., Maini, P.K. Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equations. J. Math. Biol. 33, 163–192 (1994). https://doi.org/10.1007/BF00160178
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DOI: https://doi.org/10.1007/BF00160178