The Glassey conjecture on asymptotically flat manifolds

Abstract

We verify the 3 3 -dimensional Glassey conjecture on asymptotically flat manifolds ( R 1 + 3 , g ) (\mathbb {R}^{1+3},\mathfrak {g}) , where the metric g \mathfrak {g} is a certain small space-time perturbation of the flat metric, as well as the nontrapping asymptotically Euclidean manifolds. Moreover, for radial asymptotically flat manifolds ( R 1 + n , g ) (\mathbb {R}^{1+n},\mathfrak {g}) with n ≥ 3 n\ge 3 , we verify the Glassey conjecture in the radial case. High dimensional wave equations with higher regularity are also discussed. The main idea is to exploit local energy and KSS estimates with variable coefficients, together with the weighted Sobolev estimates including trace estimates.