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 u_tt+Au−A^αw=0,w_t+A^αu_t+A^βw=0,u(0)=u₀,u_t(0)=u₁,w(0)=w₀,where A is a self-adjoint, positive definite operator on a complex Hilbert space H, and (α,β)∈[0,1]×[0,1], which is considered in Ammar-Khodja et al. (1999), and after, in Hao et al. (2013). Our contribution is to identify a fine scale of polynomial stability of the solution in the region S₃:=(α,β)∈[0,1]×[0,1];β<2α−1 taking into account the presence of a singularity at zero.