Strichartz estimates and Strauss conjecture on non-trapping asymptotically hyperbolic manifolds

Sire, Yannick; Sogge, Christopher; Wang, Chengbo; Zhang, Junyong

doi:10.1090/tran/8210

Strichartz estimates and Strauss conjecture on non-trapping asymptotically hyperbolic manifolds
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by Yannick Sire, Christopher D. Sogge, Chengbo Wang and Junyong Zhang PDF

Trans. Amer. Math. Soc. 373 (2020), 7639-7668 Request permission

Abstract:

We prove global-in-time Strichartz estimates for the shifted wave equations on non-trapping asymptotically hyperbolic manifolds. The key tools are the spectral measure estimates from [Ann. Inst. Fourier, Grenoble 68 (2018), pp. 1011–1075] and arguments borrowed from [Analysis PDE 9 (2016), pp. 151–192], [Adv. Math. 271 (2015), pp. 91–111]. As an application, we prove the small data global existence for any power $p\in (1, 1+\frac {4}{n-1})$ for the shifted wave equation in this setting, involving nonlinearities of the form $\pm |u|^p$ or $\pm |u|^{p-1}u$, which answers partially an open question raised in [Discrete Contin. Dyn. Syst. 39 (2019), pp. 7081–7099].

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