The Strauss conjecture on asymptotically flat space-times

Abstract

By assuming a certain localized energy estimate, we prove the existence portion of the Strauss conjecture on asymptotically flat manifolds, possibly exterior to a compact domain, when the spatial dimension is 3 or 4. In particular, this result applies to the 3 and 4-dimensional Schwarzschild and Kerr (with small angular momentum) black hole backgrounds, long range asymptotically Euclidean spaces, and small time-dependent asymptotically flat perturbations of Minkowski space-time. We also permit lower order perturbations of the wave operator. The key estimates are a class of weighted Strichartz estimates, which are used near infinity where the metrics can be viewed as small perturbations of the Minkowski metric, and the assumed localized energy estimate, which is used in the remaining compact set.

Final version, to appear in SIAM Journal on Mathematical Analysis. 17 pages

References in corpus (7)

• Hidden symmetries and decay for the Vlasov equation on the Kerr spacetime
• Semilinear wave equations on the Schwarzschild manifold I: Local decay estimates
• A note on energy currents and decay for the wave equation on a Schwarzschild background
• Errata for ``Global existence and scattering for the nonlinear Schrodinger equation on Schwarzschild manifolds'', ``Semilinear wave equations on the Schwarzschild manifold I: Local Decay Estimates'', and ``The wave equation on the Schwarzschild metric II: Local Decay for the spin 2 Regge Wheeler equation''
• Concerning the Strauss conjecture on asymptotically Euclidean manifolds
• Local energy decay for scalar fields on time dependent non-trapping backgrounds
• Decay for solutions of the wave equation on Kerr exterior spacetimes I-II: The cases |a| << M or axisymmetry

Cited by in corpus (6)

• Global existence and lifespan for semilinear wave equations with mixed nonlinear terms
• Lifespan of solutions to the Strauss type wave system on asymptotically flat space-times
• Waves in cosmological background with static Schwarzschild radius in the expanding universe
• Global existence for a coupled wave system related to the Strauss conjecture
• Long time existence for semilinear wave equations with the inverse-square potential
• Lifespan estimates for $2$-dimensional semilinear wave equations in asymptotically Euclidean exterior domains