Attractors for strongly damped wave equations with nonlinear hyperbolic dynamic boundary conditions

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 $\alpha \in \left[0,1\right],$ and we show that the family of global attractors is upper-semicontinuous as $\alpha \to 0.$ Furthermore, for each $\alpha \in \left[0,1\right]$, 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 ensures the corresponding global attractor also possesses finite fractal dimension in the weak topology; moreover, the dimension is independent of the perturbation parameter α. In both settings, attractors are found under minimal assumptions on the nonlinear terms.