| Summary: | 碩士 === 國立成功大學 === 數學系應用數學碩博士班 === 101 === The report is mainly our supplementary proofs and explanation of central purposes of Soonsik Kwon and Tristan Roy's paper, emph{Bilinear Local Smoothing Estimate for Airy Equation} which was issued on 2011 and Hans Lindblad and Avy Soffer’s paper, emph{A Remark on Asymptotic Completeness for the Critical Nonlinear Klein-Gordon Equation} which was issued on 2005. We also added some details in order to elaborate the authors' ideas clearer.
In Section 1, we will introduce the papers' motives of studying, main purposes, the reason of why we are interested, and also provided our correction and supplement of the two papers. In Section 2, we will give some preliminaries including propositions and lemmas which are necessary for proofing and also definitions in some common signs. In Section 3, we will explain the proofing processes of Theorem 1.2 thoroughly and supplemented the proofing details omitted by the authors, we also edited the examples given in Theorem 1.2. In Section 4, we will first introduce Littlewood-Paley decomposition, and then is supplement to the proof of Corollary 1.3 by giving every steps. In Section 5, we will separate the section into three parts. Firstly, we will give the proof of linear Klein-Gordon equation's asymptotic behavior. Secondly, it will be the detailed explanation of Section 2 in [L5]. Thirdly, we will do some correction on the arrangement of section 3 in [L5], and will also complete the skipped proof of lemmas. In Section 6, we will rearrange Section 4 in [L5], and try to make the complete process of [L5] be clearly explained.
Furthermore, we hope that it can help others who will study these two papers to understand and get their connotation easier.
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