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

<p style='text-indent:20px;'>This paper is on the asymptotic behavior of elastic string equation with localized Kelvin-Voigt damping</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \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, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ b(x) 0 $\end{document}</tex-math></inline-formula> id="M2">\begin{document}$ x\in (-1, 0] $\end{document}</tex-math></inline-formula>, and id="M3">\begin{document}$ a(x)>0 id="M4">\begin{document}$ (0, 1) $\end{document}</tex-math></inline-formula>. It known that Geometric Optics Condition for exponential stability does not apply to damping. Under assumption id="M5">\begin{document}$ a'(x) has a singularity at id="M6">\begin{document}$ x we investigate decay rate solution which depends order singularity.</p><p style='text-indent:20px;'>When id="M7">\begin{document}$ a(x) behaves like id="M8">\begin{document}$ x^{\alpha}(-\log x)^{-\beta} near id="M9">\begin{document}$ id="M10">\begin{document}$ 0\le{\alpha}<1, \;0\le\beta or id="M11">\begin{document}$ 0<{\alpha}<1, \;\beta<0 show system can achieve mixed polynomial-logarithmic rate.</p><p style='text-indent:20px;'>As byproduct, when id="M12">\begin{document}$ \beta obtain id="M13">\begin{document}$ t^{-\frac{ 3-\alpha-\varepsilon}{2(1-{\alpha})}} arbitrarily small id="M14">\begin{document}$ \varepsilon>0 improves id="M15">\begin{document}$ t^{-\frac{1}{1-{\alpha}}} obtained in [<xref ref-type="bibr" rid="b14">14</xref>]. The new again consistent limit case id="M16">\begin{document}$ \alpha\to 1^- This step toward goal obtaining optimal eventually.</p>