Semi-uniform stability of operator semigroups and energy decay of damped waves

Abstract

Only in the last fifteen years or so has the notion of semi-uniform stability, which lies between exponential stability and strong stability, become part of the asymptotic theory of $C_0$-semigroups. It now lies at the very heart of modern semigroup theory. After briefly reviewing the notions of exponential and strong stability, we present an overview of some of the best known (and often optimal) abstract results on semi-uniform stability. We go on to indicate briefly how these results can be applied to obtain (sometimes optimal) rates of energy decay for certain damped second-order Cauchy problems.

arXiv commentTo appear in Philosophical Transactions of the Royal Society A

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Cited by in corpus (12)

• Non-uniform Stability of Damped Contraction Semigroups
• Some developments around the Katznelson-Tzafriri theorem
• A short elementary proof of the Gearhart-Prüss theorem for bounded semigroups
• Stability of the multidimensional wave equation in port-Hamiltonian modelling
• Stability of abstract coupled systems
• Resolvent estimates for the one-dimensional damped wave equation with unbounded damping
• Improved polynomial decay for unbounded semigroups
• Optimal decay of semi-uniformly stable operator semigroups with empty spectrum
• Robustness of Polynomial Stability with Respect to Sampling
• Equivalence between the energy decay of fractional damped Klein-Gordon equations and geometric conditions for damping coefficients
• A universal example for quantitative semi-uniform stability
• Well-posedness and stability of the Lagrange representation of the n-D wave equation via boundary triples