Analysis of Existence, Regularity and Stability of Solutions to Wave Equations with Dynamic Boundary Conditions

Fourrier, Nicolas, Mathematics - Graduate School of Arts and Sciences, University of Virginia
Lasiecka, Irena, Department of Mathematics, University of Virginia

In this dissertation, we present an analysis of existence, smoothing properties and long-time behavior of solutions corresponding to wave equation with dynamic boundary conditions. Different damping mechanisms acting on either the interior dynamics or the boundary dynamics or both will be considered.
This leads to a consideration of a wave equation acting on a bounded 3-d domain, equipped with zero-Dirichlet boundary condtions on a portion of the boundary, coupled with another second order dynamics acting on a portion of the boundary. These are general Wentzell type of boundary conditions which describe wave equation oscillating on a tangent manifold of a lower dimension. Both interior and boundary dynamics are subject to viscoleastic and/or frictional dampings.
Chapter 1 provides the physical motivation for the model as well as mathematical background. Then, we shall examine the regularity and stability properties of the resulting system as a function of strength and location of the dissipation. Properties such as wellposedness 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.

PHD (Doctor of Philosophy)
Differential equations, wave equations, dynamic boundary condtions, Wentzell boundary conditions,, strong damping, Laplace-Beltrami operator, semigroup generation, analyticity of semigroups, Gevrey's regularity, strong stability, exponential stability, spectral analysis, wave equations with dynamic boundary conditions
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