Asymptotic behavior of the nonlinear Schr\"{o}dinger equation on exterior domain

TL;DR

This paper studies the asymptotic behavior of the nonlinear Schrödinger equation on exterior domains, deriving Morawetz estimates independent of Euclidean assumptions and proving exponential stability with both uniform and non-uniform decay rates.

Contribution

It introduces Morawetz estimates based on the metric without Euclidean assumptions and establishes exponential stability with both uniform and non-uniform decay rates.

Findings

Morawetz estimates derived directly from the metric g.

Exponential stability with non-uniform energy decay.

Exponential stability with uniform energy decay.

Abstract

{\bf Abstract} \,\, We consider the following nonlinear Schr\"{o}dinger equation on exterior domain. \begin{equation} \begin{cases} iu_t+\Delta_g u + ia(x)u - |u|^{p-1}u = 0 \qquad (x,t) \in \Omega\times (0,+\infty), \qquad (1)\cr u\big|_\Gamma = 0\qquad t \in (0,+\infty), \cr u(x,0) = u_0(x)\qquad x \in \Omega, \end{cases} \end{equation} where $1<p<\frac{n+2}{n-2}$, $\Omega\subset\mathbb{R}^n$ ($n\ge3$) is an exterior domain and $(\mathbb{R}^n,g)$ is a complete Riemannian manifold. We establish Morawetz estimates for the system (1) without dissipation ($a(x)\equiv 0$ in (1)) and meanwhile prove exponential stability of the system (1) with a dissipation effective on a neighborhood of the infinity. It is worth mentioning that our results are different from the existing studies. First, Morawetz estimates for the system (1) are directly derived from the metric $g$ and are independent on the assumption of an (asymptotically) Euclidean metric. In addition, we not only prove exponential stability of the system (1) with non-uniform energy decay rate, which is dependent on the initial data, but also prove exponential stability of the system (1) with uniform energy decay rate. The main methods are the development of Morawetz multipliers in non (asymptotically) Euclidean spaces and compactness-uniqueness arguments.