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July/August 2016 Energy-critical semi-linear shifted wave equation on the hyperbolic spaces
Ruipeng Shen
Differential Integral Equations 29(7/8): 731-756 (July/August 2016). DOI: 10.57262/die/1462298683

Abstract

In this paper, we consider a semi-linear, energy-critical, shifted wave equation on the hyperbolic space ${\mathbb H}^n$ with $3 \leq n \leq 5$: \[ \partial_t^2 u - (\Delta_{{\mathbb H}^n} + \rho^2) u = \zeta |u|^{4/(n-2)} u, \quad (x,t)\in {\mathbb H}^n \times {\mathbb R}. \] Here, $\zeta = \pm 1$ and $\rho = (n-1)/2$ are constants. We introduce a family of Strichartz estimates compatible with initial data in the energy space $H^{0,1} \times L^2 ({\mathbb H}^n)$ and then establish a local theory with these initial data. In addition, we prove a Morawetz-type inequality \[ \int_{-T_-}^{T_+} \int_{{\mathbb H}^n} \frac{\rho (\cosh |x|) |u(x,t)|^{2n/(n-2)}} {\sinh |x|} d\mu(x) dt \leq n {\mathcal E}, \] in the defocusing case $\zeta = -1$, where ${\mathcal E}$ is the energy. Moreover, if the initial data are also radial, we can prove the scattering of the corresponding solutions by combining the Morawetz-type inequality, the local theory and a pointwise estimate on radial $H^{0,1}({\mathbb H}^n)$ functions.

Citation

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Ruipeng Shen. "Energy-critical semi-linear shifted wave equation on the hyperbolic spaces." Differential Integral Equations 29 (7/8) 731 - 756, July/August 2016. https://doi.org/10.57262/die/1462298683

Information

Published: July/August 2016
First available in Project Euclid: 3 May 2016

zbMATH: 1363.35252
MathSciNet: MR3498875
Digital Object Identifier: 10.57262/die/1462298683

Subjects:
Primary: 35L05 , 35L71

Rights: Copyright © 2016 Khayyam Publishing, Inc.

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Vol.29 • No. 7/8 • July/August 2016
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