The Evolution of Travelling Wave-Fronts in a Hyperbolic Fisher Model. IV. Generalised Fisher Kinetics

In this article, an initial-value problem for a nonlinear hyperbolic generalised Fisher equation is considered in detail. The nonlinear hyperbolic generalised Fisher equation of order p(> 0) is given by$\epsilon u_{tt}+u_t+F(u)+\epsilon F(u)_t$where ∈> 0 is a parameter and F(u)= u^p(1 – u) is the generalised Fisher kinetics. When p= 1 the equation is the classical hyperbolic Fisher equation which has been examined in detail elsewhere, and we restrict attention to p ∈ (0,1) ∪ (1,∞). It is established 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 wavefront is of reaction–diffusion type when (1< p< 2, ∈ < ∈2(p)) or (p ≥ 2, ∈ < 2^p), where ${\epsilon _2}(p)=\frac{p^p}{(p-1)^{(p-1)}}$. While when (0< p< 1, ∈ > 0), (1< p< 2, ∈ ≥ ∈2(p)) or (p = 2, ∈ = 2^p) the wavefront is of reaction–relaxation type.