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

We study traveling wave solutions for the class of scalar reaction–diffusion equations∂u∂t=∂2u∂x2+fm(u), where the family of potential functions {fm}{fm} is given by fm(u)=2um(1−u)fm(u)=2um(1−u). For each m⩾1m⩾1 real, there is a critical wave speed ccrit(m)ccrit(m) that separates waves of exponential structure from those which decay only algebraically. We derive a rigorous asymptotic expansion for ccrit(m)ccrit(m) in the limit as m→∞m→∞. This expansion also seems to provide a useful approximation to ccrit(m)ccrit(m) over a wide range of m  -values. Moreover, we prove that ccrit(m)ccrit(m) is C∞C∞-smooth as a function of m−1m−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. Barnes, 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].