On a beam model with degenerate nonlocal nonlinear damping

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