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

Abstract

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