Generalized traveling waves on complete Riemannian manifolds

Henri Berestycki (1) , Mauricio Labadie (1) , François Hamel (2)

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

In the article of H. Berestycki and F. Hamel, On a general definition of transition waves, there is a generalization of the classical definition of a transition wave in Euclidean spaces (e.g. a traveling wave or an invasive front) to the case where the level sets of the wave are no longer planes but surfaces. We will prove that the same results and properties on general transition waves that appear in the cited article hold in the case of a non-compact complete Riemannian manifold, namely: (1) the wave is associated to a generalized front, which moves “close” to the level sets of the wave; (2) there is a mean propagation speed of the wave, which is independent of the choice of the associated front; (3) in the case of an invasion the wave is an increasing function in time.

Mots clés: complete Riemannian manifolds; traveling waves; Reaction-diffusion equations; parabolic equations on manifolds; complete Riemannian manifolds.

Domaines: Equations aux dérivées partielles [math.AP]

Henri Berestycki, Mauricio Labadie, François Hamel. Generalized traveling waves on complete Riemannian manifolds. 2010. ⟨hal-00465620⟩