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

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.