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

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 $0 < \epsilon < 1$ , the wavefront is of reaction–diffusion type.