Rigorous Asymptotic Expansions for Critical Wave Speeds in a Family of Scalar Reaction-Diffusion Equations

We investigate traveling wave solutions in a family of reaction-diffusion equations which includes the Fisher–Kolmogorov–Petrowskii–Piscounov (FKPP) equation with quadratic nonlinearity and a bistable equation with degenerate cubic nonlinearity. It is known that, for each equation in this family, there is a critical wave speed which separates waves of exponential decay from those of algebraic decay at one of the end states. We derive rigorous asymptotic expansions for these critical speeds by perturbing off the classical FKPP and bistable cases. Our approach uses geometric singular perturbation theory and the blow-up technique, as well as a variant of the Melnikov method, and confirms the results previously obtained through asymptotic analysis in [J.H. Merkin and D.J. Needham, (1993). J. Appl. Math. Phys. (ZAMP) A, vol. 44, No. 4, 707–721] and [T.P. Witelski, K. Ono, and T.J. Kaper, (2001). Appl. Math. Lett., vol. 14, No. 1, 65–73].