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Qualitative aspects for the cubic nonlinear Schrödinger equations with localized damping: exponential and polynomial stabilization. (English) Zbl 1190.35206

Summary: Results of exponential/polynomial decay rates of the energy in \(L^{2}\)-level, related to the cubic nonlinear Schrödinger equation with localized damping posed on the whole real line, are established. We use Kato’s theory and a priori estimates to obtain the result of global well-posedness and we determine exponential/polynomial stabilization combining the ideas of unique continuation due to Zhang, semigroup property and Komornik’s approach.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
93D15 Stabilization of systems by feedback
93B05 Controllability
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