The Glassey conjecture on asymptotically flat manifolds

Abstract

We verify the 3-dimensional Glassey conjecture on asymptotically flat manifolds $(R^{1+3},g)$, where the metric $g$ is certain small space-time perturbation of the flat metric, as well as the nontrapping asymptotically Euclidean manifolds. Moreover, for radial asymptotically flat manifolds $(R^{1+n},g)$ with $n\ge 3$, 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.

Final version, to appear in Transactions of the American Mathematical Society. 22 pages. For Theorem 1.4, we have revised the proof and enhanced the result to include nonlinearities involving the full space-time gradient. Typos fixed and references updated

References in corpus (6)

• The Strauss conjecture on Kerr black hole backgrounds
• The Glassey conjecture with radially symmetric data
• Angular Regularity and Strichartz Estimates for the Wave Equation
• Global existence of null-form wave equations on small asymptotically Euclidean manifolds
• Concerning the Strauss conjecture on asymptotically Euclidean manifolds
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Cited by in corpus (8)

• Combined effects of two nonlinearities in lifespan of small solutions to semi-linear wave equations
• Blow-up and lifespan estimate for generalized Tricomi equations related to Glassey conjecture
• Long time existence for semilinear wave equations on asymptotically flat space-times
• The Strauss conjecture on asymptotically flat space-times
• Lifespan of Solutions to Wave Equations on de Sitter Spacetime
• The Glassey conjecture for nontrapping obstacles
• Blow up for small-amplitude semilinear wave equations with mixed nonlinearities on asymptotically Euclidean manifolds
• Long time existence for semilinear wave equations with the inverse-square potential