ABSTRACT

In this paper, we investigate the energy decay for solutions of the weakly coupled dissipative Schrödinger system. Among the m-coupled equations, only one equation is directly damped. Under some assumptions about the damping and the coupling terms, it is shown that sufficiently smooth solutions of the system decay logarithmically with mixed boundary conditions, including the coupling of the Schrödinger system subject to Dirichlet and Robin type boundary conditions, respectively. The proof is based on some frequency estimates with an exponential loss on the resolvent operators, which will be solved by establishing an interpolation inequality for a suitable weakly coupled elliptic system.

摘要

在本文中，我们研究了弱耦合耗散薛定谔系统解的能量衰减。在m个耦合方程中，只有一个方程是直接阻尼的。在关于阻尼和耦合项的一些假设下，表明系统的足够光滑的解在混合边界条件下呈对数衰减，包括分别服从 Dirichlet 和 Robin 型边界条件的薛定谔系统的耦合。证明基于一些频率估计，在分解算子上具有指数损失，这将通过为合适的弱耦合椭圆系统建立插值不等式来解决。