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We study reaction-diffusion equations in one spatial dimension and withgeneral (space- or time-) inhomogeneous mixed bistable-ignition reactions. Forthose satisfying a simple quantitative hypothesis, we prove existence anduniqueness of transition fronts, as well as convergence of "typical" solutionsto the unique transition front (the existence part even extends to mixedbistable-ignition-monostable reactions). These results also hold for all pureignition reactions without any other hypotheses, but not for all pure bistablereactions. In fact, we find examples of either spatially or temporally periodicpure bistable reactions (independent of the other space-time variable) forwhich we can prove non-existence of transition fronts. These are the first suchresults for periodic media which are non-degenerate in a natural sense, and thespatially periodic examples also prove a conjecture from \cite{DHZ}.

Existence and non-existence of transition fronts for bistable and ignition reactions