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

Andrej Zlatos

Published 2015-03-26Version 1

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 any other hypotheses, but not for all pure bistable reactions. In fact, we construct examples of either spatially or temporally periodic pure bistable reactions (independent of the other space-time variable) for which no transition front exists. These are the first such examples among periodic media which are non-degenerate in a natural sense, and the spatially periodic example also proves a conjecture from \cite{DHZ}.