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

Abstract

We consider a linear system of PDEs of the form \begin{equation} \begin{array}{c} \begin{array}{rcl} u_{tt} - c\Delta u_t - \Delta u = 0 & \text{in} & \Omega \times (0,T)\\ u_{tt} + \partial_n (u+cu_t) - \Delta_\Gamma (c \alpha u_t + u) = 0 & \text{on} & \Gamma_1 \times (0,T)\\ u = 0 & \text{on} & \Gamma_0 \times (0,T) \end{array}\\ (u(0),u_t(0),u|_{\Gamma_1}(0),u_t|_{\Gamma_1}(0)) \in \s{H} \end{array} \end{equation} on a bounded domain $\Omega$ with boundary $\Gamma = \Gamma_1 \cup \Gamma_0$. We show that the system generates a strongly continuous semigroup $T(t)$ which is analytic for $\alpha > 0$ and of Gevrey class for $\alpha = 0$. In both cases the flow exhibits a regularizing effect on the data. In particular, we prove quantitative time-smoothing estimates of the form $\|(d/dt)T(t)\| \lesssim |t|^{-1}$ for $\alpha > 0$, $\|(d/dt)T(t)\| \lesssim |t|^{-2}$ for $\alpha = 0$. Moreover, when $\alpha = 0$ we prove a novel result which shows that these estimates hold under relatively bounded perturbations up to $1/2$ power of the generator.