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

Abstract

<p style='text-indent:20px;'>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 <i>energy damping</i> first proposed by Balakrishnan-Taylor [<xref ref-type="bibr" rid="b2">2</xref>]. More precisely, the following <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional model is addressed <p style='text-indent:20px;'><disp-formula><label/><tex-math id="FE1">\begin{document}$ 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}^+, $\end{document}</tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ \Omega\subset \mathbb{R}^n $\end{document}</tex-math></inline-formula> is a bounded domain with smooth boundary, the coefficient of extensibility <inline-formula><tex-math id="M3">\begin{document}$ \kappa $\end{document}</tex-math></inline-formula> is nonnegative, the damping coefficient <inline-formula><tex-math id="M4">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula> is positive and <inline-formula><tex-math id="M5">\begin{document}$ q\ge 1 $\end{document}</tex-math></inline-formula>. The nonlinear source <inline-formula><tex-math id="M6">\begin{document}$ f(u) $\end{document}</tex-math></inline-formula> 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.