EVOLUTION EQUATIONS AND doi:10.3934/eect.2013.2.631 CONTROL THEORY Volume 2, Number 4, December 2013 pp. 631–667 REGULARITY AND STABILITY OF A WAVE EQUATION WITH A STRONG DAMPING AND DYNAMIC BOUNDARY CONDITIONS Nicolas Fourrier Department of Mathematics, University of Virginia Charlottesville, VA 22904, USA CGG, Massy, France Irena Lasiecka Department of Mathematics, University of Memphis Memphis, TN 38152-3370 IBS, Polish Academy of Sciences, Warsaw, Poland Abstract. We present an analysis of regularity and stability of solutions cor- responding to wave equation with dynamic boundary conditions. It has been known since the pioneering work by [26, 27, 30] that addition of dynamics to the boundary may change drastically both regularity and stability properties of the underlying system. We shall investigate these properties in the context of wave equation with the damping affecting either the interior dynamics or the boundary dynamics or both. This leads to a consideration of a wave equation acting on a bounded 3-d domain coupled with another second order dynamics acting on the boundary. The wave equation is equipped with a viscoelastic damping, zero Dirichlet boundary conditions on a portion of the boundary and dynamic boundary conditions. These are general Wentzell type of boundary conditions which describe wave equation oscillating on a tangent manifold of a lower dimension. We shall examine regularity and stability properties of the resulting system -as a function of strength and location of the dissipation. Properties such as well- posedness of finite energy solutions, analyticity of the associated semigroup, strong and uniform stability will be discussed. The results obtained analytically are illustrated by numerical analysis. The latter shows the impact of various types of dissipation on the spectrum of the generator as well as the dynamic behavior of the solution on a rectangular domain. 1. Introduction. We consider the following damped wave equation with dynamic boundary conditions: 2010 Mathematics Subject Classification. Primary: 35Lxx, 35Kxx, 35Pxx; Secondary: 65kxx, 76Nxx. Key words and phrases. Wave equation with dynamic boundary conditions, strong damping, Laplace’a Beltrami operator, semigroup generation, analyticity of semigroups, Gevrey’s regularity, strong stability, uniform stability, spectral analysis. The second author is supported by NSF grant DMS-0104305 and AFOSR Grant FA9550-09-1- 0459. 631