Attractors for Strongly Damped Wave Equations with Nonlinear Hyperbolic Dynamic Boundary Conditions

Abstract

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.