本報告為我們對作者Soonsik Kwon和Tristan Roy於二零一一年發表的論文〈Bilinear Local Smoothing Estimate for Airy Equation〉以及作者Hans Lindblad 和 Avy Soffer於二零零五年時發表的論文〈A Remark on Asymptotic Completeness for the Critical Nonlinear Klein-Gordon Equation〉所作之補充證解。我們詳細論述闡明他們的想法和概念，使之更加清晰明白。
　　在第一章，我們介紹兩篇論文的研究動機、主要工作及我們有興趣的原因，也提供了我們對這兩篇論文的修改及其補充。在第二章，我們給了一些預備知識，包含了證明中所需要的定理和引理，以及對常用到的符號加以定義。在第三章，我們將定理1.2的證明流程做了詳細說明，並補充了作者省略的證明細節，我們也修改了作者提供的定理1.2的例子。在第四章，我們先介紹Paley-Littlewood decomposition,再詳細補充作者證明Corollary 1.3的每一個步驟。在第五章，我們分成三個小節。首先，我們給予linear Klein-Gordon equation的漸進行為的證明，這是作者略過的部分。其次，我們詳細解說[L5]中第二節的內容。最後，我們對[L5]中第三節的編排做了些修正，並補充了作者略過的引理證明。在第六章，我們重新編排[L5]中第四節的敘述，並試著將[L5]的完整流程解釋清楚。
　　誠希望本報告能幫助其他讀者充分且簡易地了解兩篇論文的內涵。

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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