Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equations |
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Authors: | Faustino Sánchez-Garduño Philip K. Maini |
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Affiliation: | (1) Centre for Mathematical Biology, Mathematical Institute, University of Oxford, 24-29 St Giles', OX1 3LB Oxford, UK;(2) Departamento de Matemáticas. Facultad de Ciencias, UNAM, Circuito Exterior, C.U., 04510 Mexico, D.F. Mexico |
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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 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) = (x - c*t) of sharp type satisfying (– ) = 1, () = 0 *; '(*–) = – c*/D'(0), '(*+) = 0 and 3. a continuum of travelling wave solutions of monotone decreasing front type for each c > c*. These fronts satisfy the boundary conditions (– ) = 1, '(– ) = (+ ) = '(+ ) = 0. We illustrate our analytical results with some numerical solutions. |
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Keywords: | Travelling waves Non-linear diffusion equations Sharp solutions Wavespeed Degenerate diffusion |
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