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We study transition fronts for one-dimensional reaction-diffusion equationswith compactly perturbed ignition-monostable reactions. We establish an almostsharp condition on reactions which characterizes the existence andnon-existence of fronts. In particular, we prove that a strong inhomogeneity inthe reaction prevents formation of transition fronts, while a weakinhomogeneity gives rise to a front. Our work extends results and methodsintroduced by J. Nolen, J.M. Roquejoffre, L. Ryzhik, and A. Zlato\v{s}.

Existence and non-existence of transition fronts in mixed ignition-monostable media