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

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 <tex-math id="M1">{document}n{document}</tex-math>-dimensional model is addressed <tex-math id="FE1">{document}uₜₜ-κ Δ u+Δ ²u-γ[∫Ω(|Δ u|²+|uₜ|²)dx ]qΔ uₜ+f(u) = 0 \ in \ Ω × R⁺,{document}</tex-math> where <tex-math id="M2">{document}Ω⊂ Rⁿ{document}</tex-math> is a bounded domain with smooth boundary, the coefficient of extensibility <tex-math id="M3">{document}κ{document}</tex-math> is nonnegative, the damping coefficient <tex-math id="M4">{document}γ{document}</tex-math> is positive and <tex-math id="M5">{document}q≥ 1{document}</tex-math>. The nonlinear source <tex-math id="M6">{document}$ f(u) ${document}</tex-math> 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.