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 [9] (or Tataru [29]) 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 and Pierfelice [1] and Metcalfe and Taylor [22]. 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 [29] for $ {\mathbb H}^3 $ and settle a dispute, in his favor, raised in [21] about his proof. Our proof is slightly more self-contained than the one in [29] 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.