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A semi-linear shifted wave equation on the hyperbolic spaces with application on a quintic wave equation on $ \mathbb{R}^2$


Authors: Ruipeng Shen and Gigliola Staffilani
Journal: Trans. Amer. Math. Soc. 368 (2016), 2809-2864
MSC (2010): Primary 35L71; Secondary 35L05
DOI: https://doi.org/10.1090/S0002-9947-2015-06513-1
Published electronically: March 24, 2015
MathSciNet review: 3449259
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Abstract: In this paper we consider a semi-linear, defocusing, shifted wave equation on the hyperbolic space

$\displaystyle \partial _t^2 u - (\Delta _{{\mathbb{H}}^n} + \rho ^2) u = - \vert u\vert^{p-1} u, \quad (x,t)\in {\mathbb{H}}^n \times {\mathbb{R}}, $

and we introduce a Morawetz-type inequality

$\displaystyle \int _{-T_-}^{T_+} \int _{{\mathbb{H}}^n} \vert u\vert^{p+1} d\mu dt < C \mathcal E, $

where $ \mathcal E$ is the energy. Combining this inequality with a well-posedness theory, we can establish a scattering result for solutions with initial data in $ H^{1/2,1/2} \times H^{1/2,-1/2}({\mathbb{H}}^n)$ if $ 2 \leq n \leq 6$ and $ 1<p<p_c = 1+ 4/(n-2)$. As another application we show that a solution to the quintic wave equation $ \partial _t^2 u - \Delta u = - \vert u\vert^4 u$ on $ {\mathbb{R}}^2$ scatters if its initial data are radial and satisfy the conditions

$\displaystyle \vert\nabla u_0 (x)\vert, \vert u_1 (x)\vert \leq A(\vert x\vert+... ...rt u_0 (x)\vert \leq A(\vert x\vert)^{-1/2-\varepsilon },\quad \varepsilon >0. $


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Additional Information

Ruipeng Shen
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
Address at time of publication: Department of Mathematics and Statistics, McMaster University, 1280 Main Street, Hamilton, Ontario L8S 4K1, Canada
Email: srpgow@gmail.com

Gigliola Staffilani
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139

DOI: https://doi.org/10.1090/S0002-9947-2015-06513-1
Received by editor(s): February 25, 2014
Received by editor(s) in revised form: June 10, 2014
Published electronically: March 24, 2015
Additional Notes: The second author was funded in part by NSF DMS 1068815.
Article copyright: © Copyright 2015 American Mathematical Society