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

$ 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.