Strichartz estimates and Strauss conjecture on non-trapping asymptotically hyperbolic manifolds

We prove global-in-time Strichartz estimates for the shifted wave equations on non-trapping asymptotically hyperbolic manifolds. The key tools are the spectral measure estimates from Ann. Inst. Fourier, Grenoble 68 (2018), pp. 1011–1075 and arguments borrowed from Analysis PDE 9 (2016), pp. 151–192, Adv. Math. 271 (2015), pp. 91–111. As an application, we prove the small data global existence for any power p ∈ (1, 1 + 4 n − 1) p (1, 1+ 4n-1) for the shifted wave equation in this setting, involving nonlinearities of the form ± | u | p |u|ᵖ or ± | u | p − 1 u |u|^p-1u, which answers partially an open question raised in Discrete Contin. Dyn. Syst. 39 (2019), pp. 7081–7099.