Stabilization of the linear Kawahara equation with localized damping

Vasconcellos, Carlos Frederico; Silva, Patrı́cia Nunes da

doi:10.3233/asy-2010-0987

Cited by 9 publications

(6 citation statements)

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“…To prove the above theorem we use multiplier techniques and the Holmgren's Uniqueness Theorem. This proof follows the same method developed in Menzala et al [5] for the KdV linear system and Vasconcellos-Silva [11] for the Kawahara linear system.…”

Section: Theorem 22 (Stabilization For the Linear Case)mentioning

confidence: 90%

The stabilization of the nonlinear model arising in pulse propagation in optical fiber

Silva¹,

Vasconcellos²

2015

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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We analyze the stabilization of the third order nonlinear Schrödinger equation in a bounded interval under the effect of a localized damping mechanism. That is, we consider the following equation: iu t + αu xx + iβu xxx + |u| 2 u + ia(x)u = 0. Where u = u(x, t) is a complex valued function defined in (0, L) × (0, +∞) and α, β are real constants.This equation models a pulse propagation in a long-distance and high-speed optical fiber transmission system. Using multiplier techniques and a special uniform continuation theorem we prove the exponential decay of the total energy associated with the above system.

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Section: Theorem 22 (Stabilization For the Linear Case)mentioning

confidence: 90%

The stabilization of the nonlinear model arising in pulse propagation in optical fiber

Silva¹,

Vasconcellos²

2015

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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“…Inspired by the ideas developed in [16], [19] and [20], we have the following result: Theorem 3.3. Let X be the set of the values L of the length of interval which satisfies the following conditions:…”

Section: Exponential Decay Of the Energymentioning

confidence: 99%

“…Vasconcellos-Silva [21] studied the existence, regularity of the solutions and stabilization for the Kawahara system. The stabilization and controllability for linear Kawahara system is proved in Vasconcellos-Silva [19] and [20].…”

Section: Introductionmentioning

confidence: 99%

On the stabilization and controllability for a third order linear equation

Silva¹,

Vasconcellos²

2011

Port. Math.

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We analyze the stabilization and the exact controllability of a third order linear equation in a bounded interval. That is, we consider the following equation:where u ¼ uðx; tÞ is a complex valued function defined in ð0; LÞ Â ð0; þlÞ and a, b and g are real constants. Using multiplier techniques, HUM method and a special uniform continuation theorem, we prove the exponential decay of the total energy and the boundary exact controllability associated with the above equation. Moreover, we characterize a set of lengths L, named X, in which it is possible to find non null solutions for the above equation with constant (in time) energy and we show it depends strongly on the parameters a, b and g.

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“…Recently global stabilization of the generalized KdV system have been obtained by Rosier-Zhang [14] and Linares-Pazoto [10] studied the stabilization of the generalized KdV system with critical exponents. For the stabilization of global solutions of the Kawahara under the effect of a localized damping mechanism, see Vasconcellos and Silva [18,19,20].…”

Section: Palavras-chave: Kawahara Equation Periodic Boundary Conditimentioning

confidence: 99%