Note on stability of an abstract coupled hyperbolic-parabolic system: singular case

In this paper we try to complete the stability analysis for an abstract system of coupled hyperbolic and parabolic equations $$\{\begin{array}{l}\text{ds}{u}_{tt}+Au-A^{\alpha }w=0,\\ w_t+A^{\alpha }{u}_t+A^{\beta }w=0,\\ u(0)=u_0,u_t(0)=u_1,w(0)=w_0,\end{array}$$ where $A$ is a self-adjoint, positive definite operator on a complex Hilbert space $H$, and $(\alpha ,\beta )\in [0,1]\times [0,1]$, which is considered in \cite{Amk}, and after, in \cite{liu1}. Our contribution is to identify a fine scale of polynomial stability of the solution in the region $S_3:=\{(\alpha ,\beta )\in [0,1]\times [0,1];\beta <2\alpha -1\}$ taking into account the presence of a singularity at zero.