Asymptotic behavior of the nonlinear Schrödinger equation on exterior domain

We consider the following nonlinear Schrödinger equation on exterior domain:\[\begin{cases}iu_t+ \Delta_g u + ia(x)u-|u|^{p-1}u=0 & (x,t) \in \Omega \times (0,+ \infty), \\u {\big \vert}_{\Gamma}=0 & t \in (0,+ \infty), \\u (x,0) = u_0(x) & x \in \Omega,\end{cases}\]

where $1\lt p \lt \frac{n+2}{n-2}$, $\Omega \subset \mathbb{R}^n (n \geq 3)$ is an exterior domain and $(\mathbb{R}^n , g)$ is a complete Riemannian manifold. We establish Morawetz estimates for the system without dissipation ($a(x) \equiv 0$ and meanwhile prove exponential stability of the system 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 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 with non-uniform energy decay rate, which is dependent on the initial data, but also prove exponential stability of the system with uniform energy decay rate. The main methods are the development of Morawetz multipliers in non (asymptotically) Euclidean spaces and compactness-uniqueness arguments.

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This work is supported by the National Science Foundation of China, grants no.61473126 and no.61573342, and Key Research Program of Frontier Sciences, CAS, no. QYZDJ-SSW-SYS011.

Received 20 December 2018

Accepted 30 March 2020

Published 17 February 2021