The evolution of travelling wave-fronts in a hyperbolic Fisher model. II. The initial-value problem

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Abstract

In this paper an initial-value problem for a non-linear hyperbolic Fisher equation is considered in detail. The non-linear hyperbolic Fisher equation is given by epsilon u(tt) + u(t) = u(xx) + F(u) + epsilon F(u)(t), where epsilon > 0 is a parameter and F(u) = u (1-u) is the classical Fisher kinetics. It is established, via the method of matched asymptotic expansions, that the large-time structure of the solution to the initial-value problem involves the evolution of a propagating wavefront which is either of reaction-diffusion or reaction-relaxation type. It is demonstrated that the case epsilon = 1 is a bifurcation point in the sense that for epsilon > 1 the wavefront is of reaction-relaxation type, whereas for 0 <c <1, the wavefront is of reaction-diffusion type.

Details

Original languageEnglish
Pages (from-to)171-193
Number of pages23
JournalJournal of Engineering Mathematics
Volume59
Issue number2
Early online date6 Jun 2007
Publication statusPublished - 1 Oct 2007

Keywords

  • asymptotics, travelling waves, hyperbolic Fisher equation