  This work is concerned with new results on long-time dynamics of a class of
hyperbolic evolution equations related to extensible beams with three
distinguished nonlocal nonlinear damping terms. In the first possibly
degenerate case, the results feature the existence of a family of compact
global attractors and a thickness estimate for their Kolmogorov's
$\varepsilon$-entropy. Then, in the non-degenerate context, the structure of
the helpful nonlocal damping leads to the existence of finite-dimensional
compact global and exponential attractors. Lastly, in a degenerate and critical
framework, it is proved the existence of a bounded closed global attractor but
not compact. To the proofs, we provide several new technical results by means
of refined estimates that open up perspectives for a new branch of nonlinearly
damped problems.
