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Semigroup Forum

, Volume 88, Issue 2, pp 333–365 | Cite as

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

  • Philip Jameson GraberEmail author
  • Irena Lasiecka
RESEARCH ARTICLE

Abstract

We consider a linear system of PDEs of the form
$$\begin{aligned} & \begin{aligned} u_{tt} - c\Delta u_t - \Delta u &= 0 \quad\text{in } \varOmega\times (0,T)\\ u_{tt} + \partial_n (u+cu_t) - \Delta_\varGamma(c \alpha u_t + u)& = 0 \quad\text{on } \varGamma_1 \times(0,T)\\ u &= 0 \quad\text{on } \varGamma_0 \times(0,T) \end{aligned} \\ &\quad (u(0),u_t(0),u|_{\varGamma_1}(0),u_t|_{\varGamma_1}(0)) \in {\mathcal{H}} \end{aligned}$$
(1)
on a bounded domain Ω with boundary Γ=Γ 1Γ 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.

Keywords

Gevrey’s semigroups Analytic semigroups Wave equation with dynamic boundary conditions Pseudodifferential operators 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Unité de Mathématiques AppliquéesCommands (ENSTA ParisTech, INRIA Saclay)Palaiseau CedexFrance
  2. 2.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA

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