The paper constructs generalized cosine and sine families on Banach spaces for the abstract damped wave equation with unbounded damping, establishing existence, uniqueness, regularity, and trigonometric identities for solutions.
Long time energy averages and a lowe r resolvent estimate for damped waves
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abstract
We consider the damped wave equation on a compact manifold. We propose different ways of measuring decay of the energy (time averages of lower energy levels, decay for frequency localized data...) and exhibit links with resolvent estimates on the imaginary axis. As an application we prove a universal logarithmic lower resolvent bound on the imaginary axis for the damped wave operator when the Geometric Control Condition (GCC) is not satisfied. This is to be compared to the uniform boundedness of the resolvent on that set when GCC holds. The proofs rely on (i) various (re-)formulations of the damped wave equation as a conservative hyperbolic part perturbed by a lower order damping term;(ii) a "Plancherel-in-time" argument as in classical proofs of the Gearhart-Huang-Pr{\"u}ss theorem; and (iii) an idea of Bony-Burq-Ramond of propagating a coherent state along an undamped trajectory up to Ehrenfest time.
fields
math.AP 2years
2026 2representative citing papers
Links non-uniform energy decay in free damped systems to existence of periodic solutions under periodic forcing, while relating resolvent growth to regularity loss, with illustrations from damped wave equations and a resonance counterexample.
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The Damped Waves Equation and generalized Cosine and Sine families on Banach spaces
The paper constructs generalized cosine and sine families on Banach spaces for the abstract damped wave equation with unbounded damping, establishing existence, uniqueness, regularity, and trigonometric identities for solutions.
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Periodic solutions for weakly damped systems
Links non-uniform energy decay in free damped systems to existence of periodic solutions under periodic forcing, while relating resolvent growth to regularity loss, with illustrations from damped wave equations and a resonance counterexample.