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

  • * 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

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  • 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.

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


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