Uniform Decay Rates of Energy in the Presence of Overdamping and Underdamping in Second-Order-in-Time Evolutions: Applications to Nonlinear Waves and Plates

Author

• Ling He

Date

2026

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

One of the central questions in the stability theory of differential equations is whether solutions converge to an equilibrium—and if so, how rapidly, as decay rates give a quantitative description of oscillations, which is important in applied sciences. It has been used in a wide range of areas, including acoustic waves, geophysical waves, mechanical oscillations, aeronautics, medical sciences such as ultrasound, electrical circuits, and civil engineering applications like buildings and bridges. For linear systems governed by linear and bounded-in-time semigroups on Banach spaces, precise decay rates can be established using resolvent estimates along the imaginary axis. However, this powerful technique, rooted in linear theory, does not extend to nonlinear settings within the framework of nonlinear semigroups on infinite-dimensional spaces. In the nonlinear context, decay properties of solutions to nonlinear semigroups have become a very active area, particularly in hyperbolic-like dynamics, where the lack of stability is natural and challenging. This is due to the location of the infinite-dimensional essential spectrum on the imaginary axis, making stability and decay closely related to various forms of damping. Many results exist for specific models of wave equations subjected to semilinear damping, using methods involving suitably weighted functionals tailored to particular operators and damping mechanisms. In this work, a general second-order abstract evolution model is studied, encompassing nonlinear waves, plates, and shell dynamics with various forms of nonlinear forcing and damping. Of particular interest are models when the damping mechanism exceeds the usually treated ranges of velocities. This will include the so called overdamped and underdamped models. These are known to compromise the decay rates of the energy function. However, it turns out that the effects of overdamping or underdamping on `small” scales of velocities versus large” scale of velocities are very different. The velocity localization of overdamping and underdamping is a critical factor in the analysis. We will show that even strongly overdamped models, with supercritical ranges of nonlinear velocities, may still exhibit exponential decay of the energy. This contrasts with previously known, in some cases, polynomial decay. In order to approach the problem in this generality, new methodology is needed. In particular, a methodology which allows to localize the effects of dissipation. Our approach is based on reducing the asymptotic behavior of the nonlinear infinite-dimensional system to that of a nonlinear ordinary differential equation through careful energy reconstruction, also related to inverse problems. The obtained results not only extend the theory to much larger classes of dynamical systems, but when applied to some specific models, provide new results. The introduced methods also allow for the treatment of nonlinearly forced models. The latter aspect requires a careful analysis of the time invariance of the stability of higher energies—a rather delicate issue.

Comments

Data is provided by the student.

Library Comment

Dissertation or thesis originally submitted to ProQuest/Clarivate.

Notes

Embargoed until 2031-04-01

Recommended Citation

He, Ling, "Uniform Decay Rates of Energy in the Presence of Overdamping and Underdamping in Second-Order-in-Time Evolutions: Applications to Nonlinear Waves and Plates" (2026). Electronic Theses and Dissertations Archive. 3973.
https://digitalcommons.memphis.edu/etd/3973