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 utt+Au−Aαw=0,wt+Aαut+Aβw=0,u(0)=u0,ut(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.