Admissible speeds of transition fronts for non-autonomous monostable equations

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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→±∞.