Article Contents
Article Contents

Sharper and finer energy decay rate for an elastic string with localized Kelvin-Voigt damping

• *Corresponding author: Zhuangyi Liu

The first author is supported by the Natural Science Foundation of China grant NSFC-62073236

• This paper is on the asymptotic behavior of the elastic string equation with localized Kelvin-Voigt damping

$u_{tt}(x, t)-[u_{x}(x, t)+b(x)u_{x, t}(x, t)]_{x} = 0, \; x\in(-1, 1), \; t>0,$

where $b(x) = 0$ on $x\in (-1, 0]$, and $b(x) = a(x)>0$ on $x\in (0, 1)$. It is known that the Geometric Optics Condition for exponential stability does not apply to Kelvin-Voigt damping. Under the assumption that $a'(x)$ 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 $x^{\alpha}(-\log x)^{-\beta}$ near $x = 0$ for $0\le{\alpha}<1, \;0\le\beta$ or $0<{\alpha}<1, \;\beta<0$, we show that the system can achieve a mixed polynomial-logarithmic decay rate.

As a byproduct, when $\beta = 0$, we obtain the decay rate $t^{-\frac{ 3-\alpha-\varepsilon}{2(1-{\alpha})}}$ of solution for arbitrarily small $\varepsilon>0$, which improves the rate $t^{-\frac{1}{1-{\alpha}}}$ obtained in [14]. The new rate is again consistent with the exponential decay rate in the limit case $\alpha\to 1^-$. This is a step toward the goal of obtaining the optimal decay rate eventually.

Mathematics Subject Classification: Primary: 35B35, 35B40; Secondary: 93D20.

 Citation:

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