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

Abstract

This paper is concerned with the well-posedness and asymptotic behavior of solutions for the following extensible beam equation with non-local energy damping $u_{tt}+\Delta ^2 u-\kappa \phi (\Vert \nabla u\Vert ^2)\Delta u-\varphi (\,\Vert \Delta u\Vert ^2+\Vert u_t\Vert ^2\,)\Delta u_t+f(u)=h.$ 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 $\varphi \in C^1({\mathbb {R}}^+)$ with non-degenerate condition $\varphi (s)>0$, $s\in {\mathbb {R}}^+$. They prove the existence of strong global and exponential attractors and their robustness on the perturbed extensibility parameter $\kappa$. In this paper assuming $\varphi (s)\gtrapprox \gamma s^q$ which contemplates the degenerate condition $\varphi (0)=0$, 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 $(X,S_t)$ 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).