In this thesis, we discuss classical and recent results around the damped wave equation on compact and noncompact manifolds. We firstly show that on asymptotically cylindrical and conic manifolds, the geometric control condition and the network control condition give exponential and logarithmic decay rates respectively. We then show that a transversely geometrically controlling boundary damping strip is sufficient but not necessary for a inverse square root of time decay of waves on product manifolds, and give a general scheme to turn resolvent estimates for impedance problems on cross-sections to wave decay on product manifolds. We also show that on non-product cylinders, there is a inverse cubic root of time decay when the damping is uniformly bounded from below on the cylindrical wall. We then prove sharp polynomial energy decay for polynomially controlled singular damping on the torus. We also prove that for normally p-integrable damping on compact manifolds, the Schrödinger observability gives p-dependent polynomial decay, and finite time extinction cannot occur. We eventually show that polynomially controlled singular damping on the circle gives exponential decay. We also provide two appendices on semiclassical analysis on manifolds of bounded geometry, and semigroup theory respectively.

Creator

• Wang, Ruoyu P. T.

DOI

• https://doi.org/10.21985/n2-7eqs-cv18

Subject

• Mathematics

Language

• en

Alternate Identifier

• etdadmin_upload_986144
• http://dissertations.umi.com/northwestern:16577

Keyword

• noncompact manifolds
• Carleman estimates
• sharp energy decay
• damped waves

Date created

• 2023-01-01

Resource type

• Dissertation

Rights statement

• In Copyright