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Asymptotics for Wave Equations with Damping Only on the Dynamical Boundary

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Abstract

In this work, we use a semigroup approach to study the asymptotics of the linear wave equation with frictional damping only on the dynamic boundary. We reformulate the model into an abstract Cauchy problem and show that the spectrum of the differential operator corresponding to the Cauchy problem has no purely imaginary values. Moreover, by controlling the trace of the first derivative, we establish the estimate for the order of unboundedness of the resolvent on the imaginary axis and obtain the asymptotics for the system.

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Acknowledgements

The author thank the reviewers very much for their truly professional and valuable comments and suggestions, especially for offering a more elementary and insightful method to prove Proposition 3.1.

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  1. School of Sciences, Hangzhou Dianzi University, Hangzhou, 310018, China

    Chan Li

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Correspondence to Chan Li.

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The research is supported by the Natural Science Foundation of Zhejiang Province, China (Grant No. LQ19A010009) and National Natural Science Foundation of China (Grant Nos. 12101167, 11947004)

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Li, C. Asymptotics for Wave Equations with Damping Only on the Dynamical Boundary. Appl Math Optim 84 (Suppl 2), 2011–2026 (2021). https://doi.org/10.1007/s00245-021-09818-z

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