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For general second order evolution equations, we prove an optimal conditionon the degree of unboundedness of the damping, that rules out finite-timeextinction. We show that control estimates give energy decay rates thatexplicitly depend on the degree of unboundedness, and establish a dilationmethod to turn existing control estimates for one propagator into those foranother in the functional calculus. As corollaries, we prove Schr\"odingerobservability gives decay for unbounded damping, weak monotonicity in damping,and quantitative unique continuation and optimal propagation for fractionalLaplacians. As applications, we establish a variety of novel and explicitenergy decay results to systems with unbounded damping, including singulardamping, linearised gravity water waves and Euler--Bernoulli plates.

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