This paper investigates the well-posedness and stability of the beam model with degenerate nonlocal damping: utt+Δ2u-M(‖∇u‖2)Δu+(‖Δu‖θ+q‖ut‖ρ)(-Δ)δut+f(u)=0inΩ×R+,\documentclass[12pt]{minimal}
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\begin{document}$$ u_{tt}+\Delta ^2u-M(\Vert \nabla u\Vert ^2)\Delta u+(\Vert \Delta u\Vert ^\theta +q\Vert u_t\Vert ^\rho )(-\Delta )^\delta u_t+f(u)=0\ \ \hbox {in} \ \ \Omega \times {\mathbb {R}}^+,$$\end{document} where Ω⊂Rn\documentclass[12pt]{minimal}
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\begin{document}$$\Omega \subset {\mathbb {R}}^n$$\end{document} is a bounded domain with smooth boundary, θ≥1,q≥0,ρ>0\documentclass[12pt]{minimal}
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\begin{document}$$\theta \ge 1,~q\ge 0,~\rho >0$$\end{document} and 0≤δ≤1\documentclass[12pt]{minimal}
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\begin{document}$$0\le \delta \le 1$$\end{document}. The main purpose in the present paper is to show that the transition from the case q=0\documentclass[12pt]{minimal}
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\begin{document}$$q=0$$\end{document} to the case q>0\documentclass[12pt]{minimal}
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\begin{document}$$q>0$$\end{document} produces an explicit influence on the stability of energy solutions. More precisely, when q=0\documentclass[12pt]{minimal}
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\begin{document}$$q=0$$\end{document}, we conclude that the energy goes to zero as t goes to infinity without an explicit decay rate; while when q>0\documentclass[12pt]{minimal}
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\begin{document}$$q>0$$\end{document}, we present a polynomial decay rate of type (1+t)-2ρ\documentclass[12pt]{minimal}
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\begin{document}$$(1+t)^{-\frac{2}{\rho }}$$\end{document} that depends only on the exponent ρ\documentclass[12pt]{minimal}
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\begin{document}$$\rho $$\end{document} of the velocity term, not on θ\documentclass[12pt]{minimal}
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\begin{document}$$\theta $$\end{document} and δ\documentclass[12pt]{minimal}
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\begin{document}$$\delta $$\end{document}. Furthermore, we prove that the energy cannot be exponentially stable and derive more accurate decay rates of the energy.
