ISSN:
1432-1416
Keywords:
Travelling waves
;
Non-linear diffusion equations
;
Sharp solutions
;
Wavespeed
;
Degenerate diffusion
Source:
Springer Online Journal Archives 1860-2000
Topics:
Biology
,
Mathematics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00160178
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