## 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 a^{0}(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 x^{α}(− 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 t^{−}^{2(1}−α) of solution 1 for arbitrarily small ε > 0, which improves the rate t^{−}^{1−}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 language | English (US) |
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Pages (from-to) | 1455-1467 |

Number of pages | 13 |

Journal | Discrete and Continuous Dynamical Systems - Series S |

Volume | 15 |

Issue number | 5 |

DOIs | |

State | Published - 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