Stability of the wave equation with localized Kelvin–Voigt damping and dynamic Wentzell boundary conditions with delay

In a bounded domain, we consider the wave equation with localized Kelvin–Voigt damping and dynamic boundary conditions of Wentzell type with delay. First of all, using semigroup theory, we prove the existence and uniqueness of a solution in a suitable energy space. Secondly, via Arendt–Batty's criteria, we prove the strong stability under some suitable conditions between the interior damping and the boundary dynamic. Finally, assuming some regularity on the damping coefficient, we show that the exponential stability holds by virtue of the frequency domain approach due to Huang and Prüss, combined with a perturbation argument.