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(-alpha)) at large x (dimensionless distance), where alpha greater than or equal to 1/(m - 1) (the corresponding problem when alpha < 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 travellina wave structure. This being in contrast to the case when alpha < 1(m - 1), where the large t (time) structure exhibits the formation of an accelerating phase wave. (C) 2002 Elsevier Science B.V. All rights reserved.