On morawetz estimates with time-dependent weights for the Klein-Gordon equation

Jungkwon Kim; Hyeongjin Lee; Ihyeok Seo; Jihyeon Seok

doi:10.3934/dcds.2020279

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2020, Volume 40, Issue 11: 6295-6308. Doi: 10.3934/dcds.2020279

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On morawetz estimates with time-dependent weights for the Klein-Gordon equation

Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea

^* Corresponding author: Ihyeok Seo
^* Corresponding author: Ihyeok Seo

Received: November 2019

Revised: April 2020

Published: July 2020

This research was supported by NRF-2019R1F1A1061316

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Abstract

We obtain some new Morawetz estimates for the Klein-Gordon flow of the form

$ \begin{equation*} \big\| |\nabla|^{\sigma} e^{it \sqrt{1-\Delta}}f \big\|_{L^{2}_{x, t}(|(x, t)|^{-\alpha})} \lesssim \| f \|_{H^s} \end{equation*} $

where $ \sigma, s\geq0 $ and $ \alpha>0 $. The conventional approaches to Morawetz estimates with $ |x|^{-\alpha} $ are no longer available in the case of time-dependent weights $ |(x, t)|^{-\alpha} $. Here we instead apply the Littlewood-Paley theory with Muckenhoupt $ A_2 $ weights to frequency localized estimates thereof that are obtained by making use of the bilinear interpolation between their bilinear form estimates which need to carefully analyze some relevant oscillatory integrals according to the different scaling of $ \sqrt{1-\Delta} $ for low and high frequencies.

Keywords:

Morawetz estimates,
Klein-Gordon equation.

Mathematics Subject Classification: Primary: 35B45; Secondary: 35Q40.

Citation:

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Figure 1. The range of $ (\alpha, \sigma) $ for (1) and (4) with $ s = 1/2 $

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