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

We study reaction-diffusion equations in one spatial dimension and with general (space- or time-) inhomogeneous mixed bistable-ignition reactions. For those satisfying a simple quantitative hypothesis, we prove existence and uniqueness of transition fronts, as well as convergence of “typical” solutions to the unique transition front (the existence part even extends to mixed bistable-ignition-monostable reactions). These results also hold for all pure ignition reactions without the extra hypothesis, but not for all pure bistable reactions. In fact, we find examples of either spatially or temporally periodic pure bistable reactions (independent of the other space-time variable) for which we can prove non-existence of transition fronts. The latter are the first such examples in periodic media which are non-degenerate in a natural sense, and they also prove a conjecture from [7].