Pointwise wave behavior of the initial-boundary value problem for the nonlinear damped wave equation in $\mathbb{R}_{+}^{n} $
Stochastic invariance for neutral functional differential equation with non-lipschitz coefficients
On a beam model related to flight structures with nonlocal energy damping
Marcio A. Jorge Silva 1,, , Vando Narciso 2,†, and André Vicente 3,‡,
Department of Mathematics, State University of Londrina, 86057-970, Londrina, PR, Brazil
Nucleus of Exact and Technological Sciences, State University of Mato Grosso do Sul, 79804-970, Dourados, MS, Brazil
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 $
-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}^+, $
$ \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.
Keywords: Extensible beams, flight structures, solution, stability.
Mathematics Subject Classification: Primary: 35B35, 35B40; Secondary: 35L76.
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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Marcio A. Jorge Silva Vando Narciso André Vicente