Analysis Seminar: Abstract unbounded damping and finite-time extinction

Abstract: Energy decay rates for the damped wave equation can be obtained from resolvent estimates for the associated stationary operator. In this talk, I will discuss how, for unbounded damping, abstract control estimates can be used to produce the desired resolvent estimates. I will also describe how these resolvent estimates can be rescaled to apply to powers of the Laplacian. As applications, new decay rates are obtained for the damped wave equation with singular damping, damped linearized water waves and Euler-Bernoulli plates. Also obtained is a Global Carleman estimate for fractional Laplacians without potential. Finally, I will discuss the surprising phenomenon of finite-time extinction of solutions and a sharp condition on the degree of unboundedness that rules this out. This is joint work with Ruoyu P.T. Wang.

Host: Walton Green