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 = ∂u ∂x + fm(u), where the family of potential functions {fm} is given by fm(u) = 2u (1 − u). For each m ≥ 1 real, there is a critical wave speed ccrit(m) that separates waves of exponential structure from those which decay only algebraically. We derive a rigorous asymptotic expansion for ccrit(m) in the limit as m → ∞. This expansion also seems to provide a useful approximation to ccrit(m) over a wide range of m-values. Moreover, we prove that ccrit(m) is C ∞-smooth as a function of m. 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 and A.N. Barnes, Nonlinearity, 12(1):41-58, 1999] and in [T.P. Witelski, K. Ono, and T.J. Kaper, Appl. Math. Lett., 14(1):65-73, 2001].