The semilinear wave equation on asymptotically euclidean manifolds

Jean Francois Bony (1) , Dietrich Häfner (1)

1 IMB - Institut de Mathématiques de Bordeaux

We consider the quadratically semilinear wave equation on R^d, d>=3, equipped with a Riemannian metric. This metric is non-trapping and approaches the Euclidean metric polynomially at infinity. Using Mourre estimates and the Kato theory of smoothness, we obtain a Keel-Smith-Sogge type inequality for the linear equation. Thanks to this estimate, we prove long time existence (d=3) and global existence (d>3) for the nonlinear problem with small initial data.

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

Jean Francois Bony, Dietrich Häfner. The semilinear wave equation on asymptotically euclidean manifolds. Comm. Partial Differential Equations, 2010, 35 (1), pp.23-67. ⟨hal-00384722⟩