The evolution of reaction–diffusion waves in a class of scalar reaction–diffusion equations: algebraic decay rates

In this paper, we consider an initial-boundary value problem for the generalized Fisher–Kolmogorov–Petrovskii–Piscounov (FKPP) equation of order m>1, with the non-negative initial data being of O(x−α) at large x (dimensionless distance), where α≥1/(m−1) (the corresponding problem when α<1/(m−1) having been considered in detail by Needham and Barnes [Nonlinearity 12 (1999) 41–58]. Using matched asymptotic expansions we are able to obtain the complete structure of the solution to the initial-boundary value problem for large t (time), which exhibits the formation of a permanent form reaction–diffusion travelling wave structure. This being in contrast to the case when α<1/(m−1), where the large t (time) structure exhibits the formation of an accelerating phase wave.