  We establish the well-posedness of a strongly damped semilinear wave equation
equipped with nonlinear hyperbolic dynamic boundary conditions. Results are
carried out with the presence of a parameter distinguishing whether the
underlying operator is analytic, $\alpha>0$, or only of Gevrey class,
$\alpha=0$. We establish the existence of a global attractor for each
$\alpha\in[0,1],$ and we show that the family of global attractors is
upper-semicontinuous as $\alpha\rightarrow0.$ Furthermore, for each
$\alpha\in[0,1]$, we show the existence of a weak exponential attractor. A weak
exponential attractor is a finite dimensional compact set in the weak topology
of the phase space. This result insures the corresponding global attractor also
possess finite fractal dimension in the weak topology; moreover, the dimension
is independent of the perturbation parameter $\alpha$. In both settings,
attractors are found under minimal assumptions on the nonlinear terms.
