Stabilization of the weakly coupled Schrödinger system

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.

Keywords:

• Coupled Schrödinger system
• logarithmic decay
• interpolation inequality
• Carleman estimates

Mathematics Subject Classifications:

• 35Q40
• 93C20
• 93D20

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is partially supported by the NSFC [grant number 11971333], [grant number 11931011].