Generalized transition waves and their properties

Henri Berestycki (1) , Francois Hamel (2)

1 CAMS - Centre d'Analyse et de Mathématique sociales
2 LATP - Laboratoire d'Analyse, Topologie, Probabilités

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 classical situations, involve uniform limits, with respect to the geodesic distance, to a family of hypersurfaces which 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.

Mots clés: Waves; Generalized notions; Transition waves; Reaction-diffusion equations; Propagation speeds; Qualitative properties

Domaines: Equations aux dérivées partielles [math.AP]

Henri Berestycki, Francois Hamel. Generalized transition waves and their properties. Communications on Pure and Applied Mathematics, 2012, 65, pp.592-648. ⟨hal-00505135⟩