Local and global well-posedness for the nonlinear Schrödinger equation with nonhomogeneous boundary conditions

Abstract

In this paper, we study the initial-boundary value problem for the nonlinear Schrödinger equation in ${\mathbb{R}}_{+}^n$ \begin{equation*} i\partial_{t}u+Δu+λ|u|^pu=0, \qquad (x, t) \in \mathbb{R}_{+}^{n} \times \mathbb{R}_{+},\ \ p\in\mathbb{R}_{+} \end{equation*} with nonhomogeneous Dirichlet boundary conditions. For the corresponding linear problem, endpoint Strichartz estimates are derived. For the nonlinear problem, we prove local well-posedness in $H^s({\mathbb{R}}_{+}^n)$ with $s\in [0,\frac{5}{2})$ and $p<\frac{4}{n-2s}$. Moreover, global well-posedness is established in the same regularity range. For $s\in [1,\frac{5}{2})$, the one-dimensional global theory of \cite{figment} in $H^s({\mathbb{R}}_{+})$ is extended to $H^s({\mathbb{R}}_{+}^n)$. Additionally, we obtain global solutions in the lower regularity setting $s\in [0,1)$ for the first time. It is noteworthy that for $s=0$, we overcome the lack of mass conservation resulting from the nonzero boundary data and derive the pivotal $L^2({\mathbb{R}}_{+}^n)$ a priori estimates.

61 pages