SHARPER AND FINER ENERGY DECAY RATE FOR AN ELASTIC STRING WITH LOCALIZED KELVIN-VOIGT DAMPING

Zhong Jie Han, Zhuangyi Liu, Jing Wang

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

This paper is on the asymptotic behavior of the elastic string equation with localized Kelvin-Voigt damping utt(x, t) − [ux(x, t) + b(x)ux,t(x, t)]x = 0, x ∈ (−1, 1), t > 0, where b(x) = 0 on x ∈ (−1, 0], and b(x) = a(x) > 0 on x ∈ (0, 1). It is known that the Geometric Optics Condition for exponential stability does not apply to Kelvin-Voigt damping. Under the assumption that a0(x) has a singularity at x = 0, we investigate the decay rate of the solution which depends on the order of the singularity. When a(x) behaves like xα(− log x)−β near x = 0 for 0 ≤ α < 1, 0 ≤ β or 0 < α < 1, β < 0, we show that the system can achieve a mixed polynomial-logarithmic decay rate. 3−α−ε As a byproduct, when β = 0, we obtain the decay rate t2(1−α) of solution 1 for arbitrarily small ε > 0, which improves the rate t1−α obtained in [14]. The new rate is again consistent with the exponential decay rate in the limit case α → 1. This is a step toward the goal of obtaining the optimal decay rate eventually.

Original languageEnglish (US)
Pages (from-to)1455-1467
Number of pages13
JournalDiscrete and Continuous Dynamical Systems - Series S
Volume15
Issue number5
DOIs
StatePublished - May 2022

Bibliographical note

Funding Information:
2020 Mathematics Subject Classification. Primary: 35B35, 35B40; Secondary: 93D20. Key words and phrases. Polynomial-logarithmic decay, frequency domain analysis, localized Kelvin-Voigt damping, wave equation, C0 semigroup. The first author is supported by the Natural Science Foundation of China grant NSFC-62073236. ∗Corresponding author: Zhuangyi Liu.

Publisher Copyright:
© 2022 American Institute of Mathematical Sciences. All rights reserved.

Keywords

  • Polynomial-logarithmic decay
  • frequency domain analysis
  • localized Kelvin-Voigt damping
  • semigroup semigroup.
  • wave equation

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