Admissible speeds of transition fronts for nonautonomous monostable equations

We consider a reaction-diffusion equation with a nonlinear term of the Fisher-KPP type, depending on time t and admitting two limits as t → ±∞. We derive the set of admissible asymptotic past and future speeds of transition fronts for such equation. We further show that any transition front which is non-critical as t → −∞ always admits two asymptotic past and future speeds. We finally describe the asymptotic profiles of the non-critical fronts as t → ±∞.