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Journal of Mathematical Biology

, Volume 33, Issue 2, pp 163–192 | Cite as

Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equations

  • Faustino Sánchez-Garduño
  • Philip K. Maini
Article

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.

Key words

Travelling waves Non-linear diffusion equations Sharp solutions Wavespeed Degenerate diffusion 

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Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Faustino Sánchez-Garduño
    • 1
    • 2
  • Philip K. Maini
    • 1
  1. 1.Centre for Mathematical Biology, Mathematical InstituteUniversity of OxfordOxfordUK
  2. 2.Departamento de Matemáticas. Facultad de CienciasUNAMMexicoD.F. Mexico

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