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The Glassey conjecture on asymptotically flat manifolds


Author: Chengbo Wang
Journal: Trans. Amer. Math. Soc. 367 (2015), 7429-7451
MSC (2010): Primary 35L70, 35L15
DOI: https://doi.org/10.1090/S0002-9947-2014-06423-4
Published electronically: September 23, 2014
MathSciNet review: 3378835
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Abstract: We verify the $ 3$-dimensional Glassey conjecture on asymptotically flat manifolds $ (\mathbb{R}^{1+3},\mathfrak{g})$, where the metric $ \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 $ (\mathbb{R}^{1+n},\mathfrak{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.


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Additional Information

Chengbo Wang
Affiliation: Department of Mathematics, Zhejiang University, Hangzhou 310027, People’s Republic of China
Email: wangcbo@gmail.com

DOI: https://doi.org/10.1090/S0002-9947-2014-06423-4
Keywords: Glassey conjecture, semilinear wave equations, local energy estimates, KSS estimate, asymptotically Euclidean manifold, asymptotically flat manifold
Received by editor(s): June 28, 2013
Received by editor(s) in revised form: March 10, 2014
Published electronically: September 23, 2014
Additional Notes: The author was supported by the Zhejiang Provincial Natural Science Foundation of China LR12A01002, the Fundamental Research Funds for the Central Universities, NSFC 11301478, 11271322 and J1210038.
Article copyright: © Copyright 2014 American Mathematical Society

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