The Strauss conjecture on negatively curved backgrounds

Abstract

This paper is devoted to several small data existence results for semi-linear wave equations on negatively curved Riemannian manifolds. We provide a simple and geometric proof of small data global existence for any power $p\in (1, 1+\frac{4}{n-1}]$ for the shifted wave equation on hyperbolic space ${\mathbb H}^n$ involving nonlinearities of the form $\pm |u|^p$ or $\pm|u|^{p-1}u$. It is based on the weighted Strichartz estimates of Georgiev-Lindblad-Sogge (or Tataru) on Euclidean space. We also prove a small data existence theorem for variably curved backgrounds which extends earlier ones for the constant curvature case of Anker-Pierfelice and Metcalfe-Taylor. We also discuss the role of curvature and state a couple of open problems. Finally, in an appendix, we give an alternate proof of dispersive estimates of Tataru for ${\mathbb H}^3$ and settle a dispute, in his favor, raised in Metcalfe-Taylor about his proof. Our proof is slightly more self-contained than the one in Tataru since it does not make use of heavy spherical analysis on hyperbolic space such as the Harish-Chandra $c$-function; instead it relies only on simple facts about Bessel potentials.