The evolution of travelling waves in a hyperbolic Fisher model. III The intial-value problem when the initial data has exponential decay rates

In this paper, we consider an initial-value problem for a non-linear hyperbolic Fisher equation. The nonlinear 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. The initial data considered is positive, having unbounded support with exponential decay of O(e(-sigma x)) at large x (dimensionless distance), where sigma > 0 is a parameter. 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 of reaction - relaxation type. In particular, the wave speed for the large t (dimensionless time) permanent form travelling wave (PTW), which may be subsonic (reaction-diffusion), sonic (reaction-relaxation) or supersonic (reaction-relaxation), the asymptotic correction to the wave speed and the rate of convergence of the solution onto the PTW are obtained for all values of the parameters epsilon and sigma.