本論文研究半直線上具有一般非線性項 $f(u)$ 之非線性薛丁格方程的初始邊界值問題。對於$\displaystyle s\in \bigg(\frac{1}{2}, \frac{3}{2}\bigg]$，我們證明當 $f\in C^{\lceil s\rceil +1}(\mathbb{C}; \mathbb{C})$, $f(0) = 0$，且初始資料與邊界資料 $\displaystyle (u(x, 0), u(0, t))\in H^s_x(\mathbb{R}_+)\times H^\frac{2s+1}{4}_t(0, T)$時，該初始邊界值問題具有局部適定性。

在此基礎上，我們進一步探討解的整體適定性與爆破解。考慮非線性項具有形式 $\displaystyle f(u) = \frac{u}{|u|}\Tilde{f}(|u|)$ 且 $\Tilde{f}$ 為實數值函數。並令 $F'(x) = \Tilde{f}(x)$。我們證明，若對所有 $v\in H^1(\mathbb{R}_+)$，$\displaystyle \int^\infty_0F(|v|)dx\leq\|v\|^p_{L^2(\mathbb{R_+})}\|v_x\|^q_{L^2(\mathbb{R_+})}$皆成立，且指數滿足 $\displaystyle \frac{p}{3}+q\leq 2$，則對任意 $u_0(x)\in H^1(\mathbb{R}_+)$，$g_0(t)\in H^1(0, T)$，其對應之非線性薛丁格方程解皆可延拓至所有時間，因此具有整體適定性。另一方面，我們證明若 $u(0, t)\in H^\frac{2s+1}{4}_{loc}(0, T)$ 且滿足 $\displaystyle \lim_{t\to T}|g_0(t)| =\infty$，則解 $u$ 必於時間 $T$ 發生爆破。

此外，根據文獻\cite{himonas2019}，對於 $\displaystyle 0\leq s<\frac{1}{2}$，若$\displaystyle (u(x, 0), u(0, t))\in H^s_x(\mathbb{R}_+)\times H^\frac{2s+1}{4}_t(0, T)$，則非線性薛丁格方程具有局部適定性。我們進一步證明，當 $s=0$ 與 $s=\dfrac{1}{4}$ 時，指數 $\dfrac{2s+1}{4}$ 是 sharp 的。具體而言，我們構造一組資料 $\displaystyle (u(x, 0), u(0, t))\in H^s_x(\mathbb{R}_+)\times H^m_t(0, T)$，其中 $m<\dfrac{2s+1}{4}$，使得非線性薛丁格方程在$H^s$空間中是不適定的。

In this thesis, we consider the initial boundary value problem for nonlinear Schr\"odinger equation (NLS) on the half line with general forced term $f(u)$. For $\displaystyle s\in\bigg(\frac{1}{2}, \frac{3}{2}\bigg]$, we will show the local well-posedness of the NLS with $f\in C^{\lceil s\rceil+1}(\mathbb{C}; \mathbb{C})$, $f(0) = 0$ and data $(u(x, 0), u(0, t))$ in $H_x^s(\mathbb{R}_+)\times H^\frac{2s+1}{4}_t(0, T)$.

Following this result, we will discuss the global well-posedness of NLS and blow-up solution. Consider $f(u)$ with the form $\displaystyle \frac{u}{|u|}\Tilde{f}(|u|)$ and let $F'(x) = \Tilde{f}(x)$. For global well-posedness, we will show that if $\displaystyle \int^\infty_0F(|v|)dx\leq \|v\|^p_{L^2(\mathbb{R_+})}\|v_x\|^q_{L^2(\mathbb{R_+})}$ for $v\in H^1(\mathbb{R_+})$ and $\displaystyle \frac{p}{3} + q\leq 2$, then for $u_0\in H^1_x(\mathbb{R}_+)$ and $g_0\in H_t^1(0, T)$, the solution $u$ of NLS is globally defined. For blow-up solution, we will show that if $u(0, t)\in H^\frac{2s+1}{4}_{loc}(0, T)$ and $\lim\limits_{t\to T}|u(0, t)|=\infty$, the solution $u$ blows up at $T$.

As for $0\leq s<\dfrac{1}{2}$, based on \cite{himonas2019}, we have local well-posedness of NLS with data $(u(x, 0), u(0, t))$ in $H_x^s(\mathbb{R}_+)\times H^\frac{2s+1}{4}_t(0, T)$. We will show that the number $\dfrac{2s+1}{4}$ is sharp for $s=0$ and $s=\dfrac{1}{4}$ by showing that there exists a pair of $(u_0(x), g_0(t))\in H^s_x(\mathbb{R}_+)\times H_t^m(0, T)$, where $m<\dfrac{2s+1}{4}$, such that NLS is ill-posedness in $H^s$.

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