Optimal backward uniqueness and polynomial stability of second order equations with unbounded damping

TL;DR

This paper establishes optimal conditions for unbounded damping in second order equations, linking control estimates to energy decay, and applies these results to various physical systems with unbounded damping.

Contribution

It introduces a dilation method and optimal conditions on damping unboundedness, advancing the understanding of energy decay and control in second order evolution equations.

Findings

Derived explicit energy decay rates depending on damping unboundedness

Proved control estimates translate between different propagators

Applied results to systems like water waves and plates

Abstract

For general second order evolution equations, we prove an optimal condition on the degree of unboundedness of the damping, that rules out finite-time extinction. We show that control estimates give energy decay rates that explicitly depend on the degree of unboundedness, and establish a dilation method to turn existing control estimates for one propagator into those for another in the functional calculus. As corollaries, we prove Schr\"odinger observability gives decay for unbounded damping, weak monotonicity in damping, and quantitative unique continuation and optimal propagation for fractional Laplacians. As applications, we establish a variety of novel and explicit energy decay results to systems with unbounded damping, including singular damping, linearised gravity water waves and Euler--Bernoulli plates.