On a beam model with degenerate nonlocal nonlinear damping

This paper contains results about the existence, uniqueness and stability of solutions for the damped nonlinear extensible beam equation

\begin{document}$ u_{tt}+\Delta ^2u-M(\|\nabla u(t)\|^2)\Delta u+\|\Delta u(t)\|^{2\alpha}\,|u_t|^{\gamma}u_t = 0\ \mbox{ in } \ \Omega \times \mathbb{R}^+, $\end{document}

where \begin{document}$ \alpha>0 $\end{document}, \begin{document}$ \gamma\ge 0 $\end{document}, \begin{document}$ \Omega\subset \mathbb{R}^n $\end{document} is a bounded domain with smooth boundary \begin{document}$ \Gamma = \partial \Omega $\end{document}, and \begin{document}$ M $\end{document} is a nonlocal function that represents beam's extensibility term. The novelty of the work is to consider the damping as a product of a degenerate and nonlocal term with a nonlinear function. This work complements the recent article by Cavalcanti et al. [8] who treated this model with degenerate nonlocal weak (and strong) damping. The main result of the work is to show that for regular initial data the energy associated with the problem proposed goes to zero when \begin{document}$ t $\end{document} goes to infinity.