Non-homogeneous Problems for Nonlinear Schr\"odinger Equations in a Strip Domain

Abstract

This paper studies the initial-boundary-value problem (IBVP) of a nonlinear Schr\"odinger equation posed on a strip domain $\mathbb{R}\times[0,1]$ with non-homogeneous Dirichlet boundary conditions. For any $s\ge0$, if the initial data $\varphi(x,y)$ is in Sobolev space $H^s(\mathbb{R}\times[0,1])$ and the boundary data $h(x,t)$ is in $$ {\cal H}^s (\mathbb{R} ) = \left \{ h (x, t) \in L^2 ( \mathbb{R}^2 ) \ \big | \ ( 1 + |\lambda | + |\xi|)^{\frac12} ( 1+ |\lambda | + |\xi |^2 )^{\frac{s}{2}}\hat h ( \lambda, \xi ) \in L^2 (\mathbb{R}^2 ) \right \} $$ where $\hat h $ is the Fourier transform of $h$ with respect to $t$ and $ x$, the local well-posedness of the IBVP in $C([0,T]; H^s(\mathbb{R} \times [0,1]))$ is proved. The global well-posedness is also obtained for $s = 1$. The basic idea used here relies on the derivation of an integral operator for the non-homogeneous boundary data and the proof of the series version of Strichartz's estimates for this operator. After the problem is transformed to finding a fixed point of an integral operator, the contraction mapping argument then yields a fixed point using the Strichartz's estimates for initial and boundary operators. The global well-posedness is proved using {\it a-priori} estimates of the solutions.