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Exponential stability for the locally damped defocusing Schrödinger equation on compact manifold

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Research of Wellington J. Corrêa partially supported by the CNPq Grant 438807/2018-9

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  • In this paper we study the asymptotic dynamics for semilinear defocusing Schrödinger equation subject to a damping locally distributed on a n-dimentional compact Riemannian manifold $ M^n $ without boundary. The proofs are based on a result of unique continuation property, in the construction of a function $ f $ whose Hessian is positive definite and $ \Delta f = C_0 $ in some region contained in $ M $ and about the smoothing effect due to Aloui adapted to the present context.

    Mathematics Subject Classification: Primary: 35L60; Secondary: 35B40.

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  • Figure 1.  $ \mathbb{V} $ is a non-dissipative area (in white) arbitrarily large while the demarcated area (in black) contains dissipative effects and can be considered arbitrarily small, both totally distributed on $ M $

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