Inhomogeneous Neumann-boundary value problem for nonlinear Schrödinger equations in the upper half-space

Abstract

We consider the inhomogeneous Neumann boundary value problem for nonlinear Schrödinger equations $$ \begin{cases} i\partial _{t} u+\frac{1}{2}\Delta u=\lambda | u | ^{p-1}u, & t > 0,\ x= ( x^{\prime },x_{n} ) \in \mathbb{R}_{+}^{n}, \\ u(0,x)=u_{0}(x), & x\in \mathbb{R}_{+}^{n}, \\ \partial _{x_{n}}u ( t,x^{\prime },0 ) =h ( t,x^{\prime } ) , & t > 0,\text{ }x^{\prime }\in \mathbb{R}^{n-1}, \end{cases} $$ where $1+\frac{4}{n+2} < p < 1+\frac{4}{n-2}$. We present some results on asymptotic behavior in time of small solutions to the integral equation associated with the original problem.