在本篇論文中，我們補述 Kenji Nakanishi 在[18]中所探討的相關細節。這份論文的主要論述是透過Bourgain在[3]的方法來避免在於一維和二維的空間下所遭遇到的時間遞減為t_{-n/2}階的非收歛性。首先，利用在特定挑選的時空區域下所作的近似能量估計的結果，並且藉由乘上挑選過的權重函數的調整，我們可獲得在整個時空下的Morawetz估計。

接著，我們將解釋Bourgain可在一個相當長的時間區間上控制局部性能量，而讓我們可以透過一個線性的解來分離局部的能量的想法。由於在這些的時空測度下的波產生衰變，讓我們可以去估計剩下的部份。因此，藉由歸納法的證明，我們可以獲得對於證明非線性解的漸進完備性所需要的全域估計。

In this thesis, we endeavor to replenish some details for the result established on [18] by Kenji Nakanishi. The main purpose of this paper is inspired by Bourgain’s
idea [3], which can avoid the unboundedness decay order t−n/2 of the free equations. First, applying some variants of the energy estimate in each chosen cones and multiplied with a special weight function, we obtain the Morawetz-type estimates.Then, there exists a very long subinterval with the mean density is very low.

Next, we explain the idea which can control some localized energy conservation on some long interval. For the long interval with small space-time norm, we can separate the wave component corresponding to the localized energy by a free solution. Since the separated wave component has decayed in the space-time norms, we can estimate the remaining component. Therefore, by induction on the energy size, we obtain the global estimate for the nonlinear solutions in the proof of asymptotic completeness.

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