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

This paper is on the asymptotic behavior of the elastic string equation with localized degenerate Kelvin–Voigt damping 0.1 (Formula presented.) where (Formula presented.) on (Formula presented.), and (Formula presented.) on (Formula presented.) for (Formula presented.). It is known that the optimal decay rate of solution is (Formula presented.) in the limit case (Formula presented.) and exponential for (Formula presented.). When (Formula presented.), the damping coefficient (Formula presented.) is continuous, but its derivative has a singularity at the interface (Formula presented.). In this case, the best known decay rate is (Formula presented.), which fails to match the optimal one at (Formula presented.). In this paper, we obtain a sharper polynomial decay rate (Formula presented.). More significantly, it is consistent with the optimal polynomial decay rate at (Formula presented.) and uniform boundedness of the resolvent operator on the imaginary axis at (Formula presented.) (consequently, the exponential decay rate at (Formula presented.) as (Formula presented.)). This is a big step toward the goal of obtaining eventually the optimal decay rate.