Long-time dynamics of a hinged-free plate driven by a nonconservative force (Q2693070)

The authors consider the nonlinear nonlocal evolution equation \[ u_{tt}+\delta u_t+\Delta^2 u+\left[ P-S\int_\Omega u_x^2 \right]u_{xx}=f \] where the unknown \(u\) evidently denotes displacement from equilibrium of a two-dimensional plate with hinged opposite edges. The remaining two edges are free. The external force \(f\) communicates aerodynamic forces and damping. The form of the force \(f\) guarantees solutions do not generate a gradient flow. Moreover, the PDE admits a multiplicity of stationary solutions. Several interesting results concerning the qualitative behavior of solutions are given in the article, including compact global attractors and exponential attractors. The existence of attractors follows after applying quasi-stability analysis [\textit{I. Chueshov}, Dynamics of quasi-stable dissipative systems. Cham: Springer (2015; Zbl 1362.37001)]. The existence of finite determining modes is also shown.