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

César Augusto Bortot; Wellington José Corrêa; Ryuichi Fukuoka; Thales Maier Souza

doi:10.3934/cpaa.2020067

Article Contents

2020, Volume 19, Issue 3: 1367-1386. Doi: 10.3934/cpaa.2020067

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

1.
Department of Mobility Engineering, Federal University of Santa Catarina, Joinville-SC, 89219-600, Brazil
2.
Academic Department of Mathematics, Federal Technological University of Paraná, Campo Mourão-PR, 87301-899, Brazil
3.
Department of Mathematics, State University of Maringá, Maringá-PR, 87020-900, Brazil

^* Corresponding author
^* Corresponding author

Received: March 2019

Revised: September 2019

Published: March 2020

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

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Abstract

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.

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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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