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(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),where A is a self-adjoint, positive definite operator on a complex Hilbert space H, and (alpha, beta) is an element of [0,1] x [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 S-3 := {(alpha, beta) is an element of [0, 1] x [0, 1]; beta < 2 alpha - 1} taking into account the presence of a singularity at zero.(c) 2023 Elsevier Ltd. All rights reserved.