Attractors and asymptotic behavior for an energy-damped extensible beam model

This paper is concerned with the well-posedness and asymptotic behavior of solutions for the following extensible beam equation with non-local energy damping \(u_{tt}+\Delta ^2 u-\kappa \phi (\Vert \nabla u\Vert ^2)\Delta u-\varphi (\,\Vert \Delta u\Vert ^2+\Vert u_t\Vert ^2\,)\Delta u_t+f(u)=h.\) $u_{\mathit{tt}}+{\mathrm{\Delta }}^{2}u-{\kappa \varphi (\Vert \mathrm{\nabla }u\Vert }^2{)\mathrm{\Delta }u-\phi (\Vert \mathrm{\Delta }u\Vert }^2+\Vert u_t{\Vert }^2)\mathrm{\Delta }{u}_t+f(u)=h.$ More specifically, this is a complementary work to the paper by Sun and Yang (Discrete Contin Dyn Syst Ser B 27(6):3101–3129, 2022), where the authors consider this model assuming the hypothesis that \(\varphi \in C^1({\mathbb {R}}^+)\) $\phi \in C^1({\mathbb{R}}^{+})$ with non-degenerate condition \(\varphi (s)>0\) $\phi (s)>0$ , \(s\in {\mathbb {R}}^+\) $s\in {\mathbb{R}}^{+}$ . They prove the existence of strong global and exponential attractors and their robustness on the perturbed extensibility parameter \(\kappa \) $\kappa$ . In this paper assuming \(\varphi (s)\gtrapprox \gamma s^q\) $\phi (s)⪆\gamma s^q$ which contemplates the degenerate condition \(\varphi (0)=0\) $\phi (0)=0$ , we prove the existence of weak and regular solutions to the problem proposed and employing the method given in Temam (Springer-Verlag, New York, 1998) we show that the dynamic system \((X,S_t)\) $(X,S_t)$ given by the weak solutions of the problem has a compact global attractor in the weak topology of the phase space X. This class of nonlinear beams arising in connection with models for flight structures with non-local energy damping is proposed by Balakrishnan and Taylor (Proceedings Damping 89, Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989).