The Mathematical Institute, University of Oxford, Eprints Archive

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

Sánchez-Garduño, F. and Maini, P. K. (1994) Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equations. Journal of Mathematical Biology, 33 (10.1007/BF00160178). pp. 163-192.

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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 $u_t = [D(u)u_x]_x + g(u)$ has: 1. no travelling wave solutions for 0 < c < c *, 2. a travelling wave solution $u(x, t) = \phi(x - c^{*} t)$ of sharp type satisfying $\phi(-\infty) = 1$, $\phi(\tau) = 0$ $\forall \tau\geq\tau^{*}$; $\phi'(\tau^{*-}) = -c^{*}/D'(0),\phi'(\tau^{*+}) = 0$ and 3. a continuum of travelling wave solutions of monotone decreasing front type for each c > c *. These fronts satisfy the boundary conditions $\phi(–\infty) = 1$, $\phi'(–\infty)=\phi(+\infty) = \phi'(+\infty) = 0$. We illustrate our analytical results with some numerical solutions.

Item Type:Article
Uncontrolled Keywords:Travelling waves - Non-linear diffusion equations - Sharp solutions - Wavespeed - Degenerate diffusion
Subjects:A - C > Biology and other natural sciences
Research Groups:Centre for Mathematical Biology
ID Code:491
Deposited By:Philip Maini
Deposited On:10 Dec 2006
Last Modified:20 Jul 2009 14:21

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