We consider the quadratically semilinear wave equation on (R d , g), d ≥ 3. The metric g is non-trapping and approaches the Euclidean metric like x −ρ . Using Mourre estimates and the Kato theory of smoothness, we obtain, for ρ > 0, a Keel-Smith-Sogge type inequality for the linear equation. Thanks to this estimate, we prove long time existence for the nonlinear problem with small initial data for ρ ≥ 1. Long time existence means that, for all n > 0, the life time of the solution is a least δ −n , where δ is the size of the initial data in some appropriate Sobolev space. Moreover, for d ≥ 4 and ρ > 1, we obtain global existence for small data. Contents 1. Introduction 1 2. The general setting 4 3. The wave equation and the Mourre estimate 8 4. Proof of the linear estimates 20 5. Proof of the nonlinear result 29 Appendix A. Regularity 31
i n f o a r t i c l e r é s u m é Historique de l'article : Reçu le 29 septembre 2010 Accepté après révision le 18 octobre 2010 Disponible sur Internet le 10 novembre 2010 Présenté par Jean-Michel BonyDans cette note, on démontre une minoration universelle optimale sur la norme de la résolvante tronquée pour les opérateurs de Schrödinger semiclassiques près d'une énergie captive. En particulier, ce résultat implique que des majorations connues pour des captures hyperboliques sont optimales. La preuve repose sur un argument de X.P. Wang et la propagation en temps d'Ehrenfest des états cohérents.
a b s t r a c tIn this note, we prove an optimal universal lower bound on the truncated resolvent for semiclassical Schrödinger operators near a trapping energy. In particular, this shows that known upper bounds for hyperbolic trapping are optimal. The proof rely on an idea of X.P. Wang, and on propagation of coherent states for Ehrenfest times.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.