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.2010 Mathematics Subject Classification. 35L70, 35L15.

Global existence for semilinear damped wave equations in relation with the Strauss conjecture

“…For example, Bony-Häfner [1], Sogge-Wang [14] and Yang [24] studied the analogs of the John-Klainerman theorem [6] and global existence under null conditions for semilinear wave equations (see also Wang-Yu [23] and Yang [25] for quasilinear wave equations). Sogge-Wang [14], Wang-Yu [21], Metcalfe-Wang [12] and Wang [19] proved the analogs of the global existence part of the Strauss conjecture when n = 3, 4 (see also [20] for a review of recent results and Wakasa-Yordanov [16] for the recent blow-up results with critical power). For the analogs of the Glassey conjecture, see Wang [18] and references therein.…”

“…The first main theorem of this paper states as follows: For more precise statement, see Theorem 4.1. Concerning the proof, the idea is to adapt the recent approach of using local energy and weighted Strichartz estimates, which has been very successful in the recent resolution of the Strauss conjecture on various space-times, including Schwarzschild/Kerr black-hole space-times ( [2], [4], [9], [12]). In particular, we revisit the proof of [12, Theorem 4.1] to extract the key weighted Strichartz estimates, Lemma 3.2, which, combined with the local energy estimates ( [1], [14], [18], [10]), is good enough to treat the lower order terms in (1.3) in a perturbative way.…”

Global existence for a coupled wave system related to the Strauss conjecture

“…Here we seek to establish the same using techniques that are sufficiently robust so as to allow background geometries. Specifically, we shall use a variant of the weighted Strichartz estimates of [22], [19], which were further developed in [27], [35], and the localized energy estimate to prove such global existence.…”

“…We shall also assume that the perturbations admit a (weak) localized energy decay. More specifically, we assume that there is R 1 (with R 1 > R 0 in the case 1 Here, as in [35], for a norm A, we set…”

“…We note that the techniques to prove Theorem 1.1 also work in four spatial dimensions, but as they require p, q ≥ 2 and p c = 2 in this case, nothing new is gained over [35].…”