Exponential Stability for Damped Defocusing Schrödinger Equation in Exterior Domains

What is it about?

The text discusses the existence and exponential stability of solutions for the damped defocusing Schrödinger equation in a two-dimensional exterior domain. The proofs rely on pseudo-differential operators, a Strichartz estimate, and the techniques of Zuazua for stability. The main result is the existence and uniqueness of regular and weak solutions that decay exponentially and uniformly in H1-norm. The paper also covers propagation results for the linear Schrödinger equation in any dimension and boundary conditions using microlocal analysis.

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Why is it important?

This research is important for several reasons: It contributes to the understanding of the damped defocusing Schrödinger equation, which is a fundamental problem in mathematical physics and has applications in various fields, such as quantum mechanics and nonlinear optics. The study provides insights into the existence and uniqueness of solutions, as well as their exponential stability in the H1-level norm. This knowledge helps to improve our understanding of the behavior of solutions for this specific problem. The research also focuses on the propagation results for the linear Schrödinger equation, which can be applied to various dimensional and boundary conditions. This is particularly important in the context of microlocal analysis, as it helps to expand our knowledge in this area. Key Takeaways: 1. The existence and uniqueness of regular and weak solutions to problem (1) are proven, with these solutions decaying exponentially and uniformly to zero in H1-norm. 2. Exponential stability is achieved by combining techniques from the wave equation, global uniqueness theorems, and a Strichartz estimate proved by Anton. 3. The study employs pseudo-differential operators, a Strichartz estimate, and microlocal analysis to prove the well-posedness of the problem and the propagation results for the linear Schrödinger equation.