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

TL;DR

This paper proves the existence and properties of global and weak exponential attractors for strongly damped wave equations with nonlinear hyperbolic boundary conditions, considering different operator regularities and showing robustness as a parameter varies.

Contribution

It introduces the analysis of attractors for such wave equations under minimal nonlinear assumptions, including the case of non-analytic operators, and demonstrates upper-semicontinuity and finite dimensionality of attractors.

Findings

Existence of global attractors for all considered cases.

Existence of weak exponential attractors with finite fractal dimension.

Upper-semicontinuity of attractors as the operator regularity parameter approaches zero.

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.