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

Abstract

In this paper we try to complete the stability analysis for an abstract system of coupled hyperbolic and parabolic equations $$ \left\{ \begin{array}{lll} \ds u_{tt} + Au - A^αw = 0, \\ w_t + A^αu_t + A^βw = 0,\\ u(0) = u_0, u_t(0) = u_1, w(0) = w_0, \end{array} \right. $$ 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 $S_3:=\left\{(\alpha ,\beta )\in [0,1]\times [0,1];\beta <2\alpha -1\right\}$ taking into account the presence of a singularity at zero.