Generalized transition waves and their properties

In this paper, we generalize the usual notions of waves, fronts, and propagation speeds in a very general setting. These new notions, which cover all usual situations, involve uniform limits, with respect to the geodesic distance, to a family of hypersurfaces that are parametrized by time. We prove the existence of new such waves for some time-dependent reaction-diffusion equations, as well as general intrinsic properties, some monotonicity properties, and some uniqueness results for almost-planar fronts. The classification results, which are obtained under some appropriate assumptions, show the robustness of our general definitions. (c) 2012 Wiley Periodicals, Inc.