Admissible speeds of transition fronts for non-autonomous 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 $t\to \pm \mathrm{\infty }$. 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 $t\to -\mathrm{\infty }$ always admits two asymptotic past and future speeds. We finally describe the asymptotic profiles of the non-critical fronts as $t\to \pm \mathrm{\infty }$.