Transition fronts for inhomogeneous monostable reaction-diffusion equations via linearization at zero

We prove the existence of transition fronts for a large class of reaction-diffusion equations in one dimension, with inhomogeneous monostable reactions. We construct these as perturbations of corresponding front-like solutions to the linearization of the PDE at u = 0. While a close relationship between the solutions to the two PDEs is well known and has been exploited for KPP reactions (and our method is an extension of such ideas from Zlatoš A 2012 (J. Math. Pure Appl. 98 89-102)), to the best of our knowledge this is the first time such an approach has been used in the construction and study of fronts for non-KPP monostable reactions.