The asymptotic critical wave speed in a family of scalar reaction-diffusion equations

Abstract

We study traveling wave solutions for the class of scalar reaction-diffusion equations au/at = a(2)u/ax(2) + f(m)(u), where the family of potential functions {f(m)} is given by f(m)(u) = 2u(m) (1 - u). For each m >= 1 real, there is a critical wave speed C-crit(m) that separates waves of exponential structure from those which decay only algebraically. We derive a rigorous asymptotic expansion for c(crit)(m) in the limit as m -> infinity. This expansion also seems to provide a useful approximation to ccrit(m) over a wide range of m-values. Moreover, we prove that c(crit)(m) is C-infinity-smooth as a function of m(-1). Our analysis relies on geometric singular perturbation theory, as well as on the blow-up technique, and confirms the results obtained by means of asymptotic methods in [D.J. Needham, A.N. Barries, Reaction-diffusion and phase waves occurring in a class of scalar reaction-diffusion equations, Nonlinearity 12 (1) (1999) 41-58; T.P. Witelski, K. Ono, T.J. Kaper, Critical wave speeds for a family of scalar reaction-diffusion equations, Appl. Math. Lett. 14 (1) (2001) 65-73]. (c) 2006 Elsevier Inc. All rights reserved.