On a beam model related to flight structures with nonlocal energy damping

Marcio A. Jorge Silva; Vando Narciso; André Vicente

doi:10.3934/dcdsb.2018320

Article Contents

2019, Volume 24, Issue 7: 3281-3298. Doi: 10.3934/dcdsb.2018320

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On a beam model related to flight structures with nonlocal energy damping

1.
Department of Mathematics, State University of Londrina, 86057-970, Londrina, PR, Brazil
2.
Nucleus of Exact and Technological Sciences, State University of Mato Grosso do Sul, 79804-970, Dourados, MS, Brazil
3.
Center of Exact and Technological Sciences, State University of Paraná West, 85819-110, Cascavel, PR, Brazil

^* Corresponding author. M. A. Jorge Silva has been supported by CNPq, grant 441414/2014-1
^* Corresponding author. M. A. Jorge Silva has been supported by CNPq, grant 441414/2014-1

^† V. Narciso has been supported by FUNDECT, grant 219/2016

Received: November 2017

Revised: August 2018

Early access: January 2019

Published: May 2019

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Abstract

This paper deals with new results on existence, uniqueness and stability for a class of nonlinear beams arising in connection with nonlocal dissipative models for flight structures with energy damping first proposed by Balakrishnan-Taylor [2]. More precisely, the following $ n $-dimensional model is addressed

$ u_{tt}-\kappa \Delta u+\Delta ^2u-\gamma\left[\int_{\Omega}\left(|\Delta u|^2+|u_t|^2\right)dx \right]^q\Delta u_t+f(u) = 0 \ in \ \Omega \times \mathbb{R}^+, $

where $ \Omega\subset \mathbb{R}^n $ is a bounded domain with smooth boundary, the coefficient of extensibility $ \kappa $ is nonnegative, the damping coefficient $ \gamma $ is positive and $ q\ge 1 $. The nonlinear source $ f(u) $ can be seen as an external forcing term of lower order. Our main results feature global existence and uniqueness, polynomial stability and a non-exponential decay prospect.

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Mathematics Subject Classification: Primary: 35B35, 35B40; Secondary: 35L76.

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References

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