Sharp stability of a string with local degenerate Kelvin–Voigt damping

This paper is on the asymptotic behavior of elastic string equation with localized degenerate Kelvin–Voigt damping u t ( x , ) − [ + b ] = 0 ∈ 1 > \begin{equation} u_{tt}(x,t)-[u_{x}(x,t)+b(x) u_{xt}(x,t) ]_{x}=0,\; x\in (-1,1),\; t>0, \end{equation} (0.1)where $b(x)=0$ $x\in (-1,0]$ and α $b(x)=x^\alpha >0$ (0,1)$ for $\alpha \in . It known that optimal decay rate solution 2 $t^{-2}$ in limit case =0$ exponential ≥ \ge 1$ When coefficient $b(x)$ continuous, but its derivative has a singularity at interface $x=0$ In this case, best 3 $t^{-\frac{3-\alpha }{2(1-\alpha )}}$ which fails to match one paper, we obtain sharper polynomial $t^{-\frac{2-\alpha }{1-\alpha }}$ More significantly, it consistent uniform boundedness resolvent operator imaginary axis (consequently, =1$ as → ∞ $t\rightarrow \infty$ ). big step toward goal obtaining eventually rate.