Electronic Theses and Dissertations

Date

2025

Document Type

Dissertation

Degree Name

Doctor of Philosophy

Department

Mathematical Sciences

Committee Chair

Irena Lasiecka

Committee Member

Hongqiu Chen

Committee Member

Roberto Triggiani

Committee Member

Stephen Guffey

Abstract

This dissertation investigates the long-time behavior of solutions to nonlinear evolution equations, emphasizing stability and energy decay within the framework of functional analysis and nonlinear partial differential equations (PDEs). The primary focus is on hyperbolic-like dynamical systems with dissipation, where unchecked energy propagation can result in persistent oscillations or blow-up phenomena. Unlike parabolic models, which naturally diffuse energy, hyperbolic systems require external dissipation—applied either internally or on the boundary—to induce effective energy decay. A central challenge in stabilization theory arises from the infinite-dimensional nature of hyperbolic instabilities, which necessitates carefully designed damping mechanisms with particular geometric and topological properties. While compact perturbations can successfully induce strong energy decay, they frequently fall short of ensuring uniform decay rates across various system configurations. Consequently, this dissertation evaluates the effectiveness of diverse dissipative mechanisms in achieving uniform energy decay relative to the topology of the underlying phase space. Particular emphasis is placed on examining "overdamping" and "underdamping," both of which can significantly affect the rate at which energy diminishes. A qualitative analysis of how these phenomena impact energy decay is presented through rigorous stability criteria and measured dissipation rates. Moreover, more damping does not always equate to faster or more stable energy decay. In fact, excessive damping can sometimes provoke instability rather than enhance stability—a phenomenon that reveals a counterintuitive stability paradox within nonlinear dynamical systems. Understanding the nuanced long-term behavior of dissipative dynamical systems is thus crucial for effectively controlling energy decay and maintaining stability in nonlinear PDE frameworks. Hyperbolic systems, such as wave equations and dynamic elasticity models—including plates and shells—require precisely tailored damping strategies to mitigate instability and achieve controlled energy dissipation. This research addresses critical questions regarding how different damping mechanisms influence energy decay rates and the fundamental question of how rapidly energy in a dissipative system diminishes over time. By developing sophisticated analytical tools capable of establishing explicit decay rates and robust stability criteria, the dissertation systematically investigates the conditions under which optimal energy dissipation is attained. The findings notably resolve the observed stability paradox, providing valuable insights into why greater damping does not necessarily produce better stability outcomes. By bridging rigorous mathematical theory with practical applications, this work advances stabilization theory for infinite-dimensional systems. It highlights the necessity of balanced damping design, demonstrating that carefully moderated dissipation often yields superior stability and decay properties compared to excessive damping measures. The broader implications of these findings extend significantly into mathematical physics, engineering, and materials science, specifically impacting areas such as vibration control, structural dynamics, and energy optimization in dynamic systems. Thus, this dissertation contributes to a deeper understanding of energy dissipation mechanisms, establishing a foundation for future research aimed at improving stability and energy management across various nonlinear systems.

Comments

Data is provided by the student

Library Comment

Dissertation or thesis originally submitted to ProQuest.

Notes

Open Access

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