Analyticity and Gevrey class regularity for a strongly damped wave equation with hyperbolic dynamic boundary conditions

We consider a linear system of PDEs of the form 1 $$begin{aligned} & begin{aligned} u_{tt} - cDelta u_t - Delta u &= 0 quadtext{in } varOmegatimes (0,T) u_{tt} + partial_n (u+cu_t) - Delta_varGamma(c alpha u_t + u)& = 0 quadtext{on } varGamma_1 times(0,T) u &= 0 quadtext{on } varGamma_0 times(0,T) end{aligned} &quad (u(0),u_t(0),u|_{varGamma_1}(0),u_t|_{varGamma_1}(0)) in {mathcal{H}} end{aligned}$$ on a bounded domain Ω with boundary Γ=Γ ₁∪Γ ₀. We show that the system generates a strongly continuous semigroup T(t) which is analytic for α>0 and of Gevrey class for α=0. In both cases the flow exhibits a regularizing effect on the data. In particular, we prove quantitative time-smoothing estimates of the form ∥(d/dt)T(t)∥?|t|^?1 for α>0, ∥(d/dt)T(t)∥?|t|^?2 for α=0. Moreover, when α=0 we prove a novel result which shows that these estimates hold under relatively bounded perturbations up to 1/2 power of the generator.