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July  2019, 24(7): 3281-3298. doi: 10.3934/dcdsb.2018320

## 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

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

Received  November 2017 Revised  August 2018 Published  January 2019

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
 $n$
 $u_{tt}-\kappa \Delta u+\Delta ^2u-\gamma\left[\int_{\Omega}\left(|\Delta u|^2+|u_t|^2\right)dx \right]^q\Delta u_t+f(u) = 0 \ in \ \Omega \times \mathbb{R}^+,$
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.
Citation: Marcio A. Jorge Silva, Vando Narciso, André Vicente. On a beam model related to flight structures with nonlocal energy damping. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3281-3298. doi: 10.3934/dcdsb.2018320
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##### References:
 [1] Marcio Antonio Jorge da Silva, Vando Narciso. Attractors and their properties for a class of nonlocal extensible beams. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 985-1008. doi: 10.3934/dcds.2015.35.985 [2] Filippo Dell'Oro, Vittorino Pata. Memory relaxation of type III thermoelastic extensible beams and Berger plates. Evolution Equations & Control Theory, 2012, 1 (2) : 251-270. doi: 10.3934/eect.2012.1.251 [3] Marcio Antonio Jorge da Silva, Vando Narciso. Long-time dynamics for a class of extensible beams with nonlocal nonlinear damping*. Evolution Equations & Control Theory, 2017, 6 (3) : 437-470. doi: 10.3934/eect.2017023 [4] Chunxiang Zhao, Chunyan Zhao, Chengkui Zhong. The global attractor for a class of extensible beams with nonlocal weak damping. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019197 [5] Franco Maceri, Michele Marino, Giuseppe Vairo. Equilibrium and stability of tensegrity structures: A convex analysis approach. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 461-478. doi: 10.3934/dcdss.2013.6.461 [6] E. N. Dancer. Some examples on solution structures for weakly nonlinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 817-826. doi: 10.3934/dcds.2005.12.817 [7] Gen Qi Xu, Siu Pang Yung. Stability and Riesz basis property of a star-shaped network of Euler-Bernoulli beams with joint damping. Networks & Heterogeneous Media, 2008, 3 (4) : 723-747. doi: 10.3934/nhm.2008.3.723 [8] Dominika Pilarczyk. Asymptotic stability of singular solution to nonlinear heat equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 991-1001. doi: 10.3934/dcds.2009.25.991 [9] Michele Coti Zelati. Global and exponential attractors for the singularly perturbed extensible beam. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 1041-1060. doi: 10.3934/dcds.2009.25.1041 [10] Alessia Berti, Maria Grazia Naso. Vibrations of a damped extensible beam between two stops. Evolution Equations & Control Theory, 2013, 2 (1) : 35-54. doi: 10.3934/eect.2013.2.35 [11] Philippe Jaming, Vilmos Komornik. Moving and oblique observations of beams and plates. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020013 [12] Xiao-Bing Li, Xian-Jun Long, Zhi Lin. Stability of solution mapping for parametric symmetric vector equilibrium problems. Journal of Industrial & Management Optimization, 2015, 11 (2) : 661-671. doi: 10.3934/jimo.2015.11.661 [13] Mi-Young Kim. Uniqueness and stability of positive periodic numerical solution of an epidemic model. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 365-375. doi: 10.3934/dcdsb.2007.7.365 [14] Jose Carlos Camacho, Maria de los Santos Bruzon. Similarity reductions of a nonlinear model for vibrations of beams. Conference Publications, 2015, 2015 (special) : 176-184. doi: 10.3934/proc.2015.0176 [15] Patrick Ballard, Bernadette Miara. Formal asymptotic analysis of elastic beams and thin-walled beams: A derivation of the Vlassov equations and their generalization to the anisotropic heterogeneous case. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1547-1588. doi: 10.3934/dcdss.2019107 [16] William D. Kalies, Konstantin Mischaikow, Robert C.A.M. Vandervorst. Lattice structures for attractors I. Journal of Computational Dynamics, 2014, 1 (2) : 307-338. doi: 10.3934/jcd.2014.1.307 [17] Paulo Antunes, Joana M. Nunes da Costa. Hypersymplectic structures on Courant algebroids. Journal of Geometric Mechanics, 2015, 7 (3) : 255-280. doi: 10.3934/jgm.2015.7.255 [18] Javier de la Cruz, Michael Kiermaier, Alfred Wassermann, Wolfgang Willems. Algebraic structures of MRD codes. Advances in Mathematics of Communications, 2016, 10 (3) : 499-510. doi: 10.3934/amc.2016021 [19] Francesco Maddalena, Danilo Percivale, Franco Tomarelli. Adhesive flexible material structures. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 553-574. doi: 10.3934/dcdsb.2012.17.553 [20] Haifeng Hu, Kaijun Zhang. Stability of the stationary solution of the cauchy problem to a semiconductor full hydrodynamic model with recombination-generation rate. Kinetic & Related Models, 2015, 8 (1) : 117-151. doi: 10.3934/krm.2015.8.117

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