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

This paper is on the asymptotic behavior of the elastic string equation with localized Kelvin-Voigt damping\begin{document}$ u_{tt}(x, t)-[u_{x}(x, t)+b(x)u_{x, t}(x, t)]_{x} = 0, \; x\in(-1, 1), \; t>0, $\end{document}where \begin{document}$ b(x) = 0 $\end{document} on \begin{document}$ x\in (-1, 0] $\end{document} , and \begin{document}$ b(x) = a(x)>0 $\end{document} on \begin{document}$ x\in (0, 1) $\end{document} . It is known that the Geometric Optics Condition for exponential stability does not apply to Kelvin-Voigt damping. Under the assumption that \begin{document}$ a'(x) $\end{document} has a singularity at \begin{document}$ x = 0 $\end{document} , we investigate the decay rate of the solution which depends on the order of the singularity. When \begin{document}$ a(x) $\end{document} behaves like \begin{document}$ x^{\alpha}(-\log x)^{-\beta} $\end{document} near \begin{document}$ x = 0 $\end{document} for \begin{document}$ 0\le{\alpha}<1, \;0\le\beta $\end{document} or \begin{document}$ 0<{\alpha}<1, \;\beta<0 $\end{document} , we show that the system can achieve a mixed polynomial-logarithmic decay rate. As a byproduct, when \begin{document}$ \beta = 0 $\end{document} , we obtain the decay rate \begin{document}$ t^{-\frac{ 3-\alpha-\varepsilon}{2(1-{\alpha})}} $\end{document} of solution for arbitrarily small \begin{document}$ \varepsilon>0 $\end{document} , which improves the rate \begin{document}$ t^{-\frac{1}{1-{\alpha}}} $\end{document} obtained in [ 14 ]. The new rate is again consistent with the exponential decay rate in the limit case \begin{document}$ \alpha\to 1^- $\end{document} . This is a step toward the goal of obtaining the optimal decay rate eventually.